positive forms and stability of linear time delay systems we consider the problem of constructing lyapunov functions for linear differential equations with delays for such systems it is known that exponential stability implies the existence of a positive lyapunov function which is quadratic on the space of continuous functions we give an explicit parameterization of a sequence of finite dimensional subsets of the cone of positive lyapunov functions using positive semidefinite matrices this allows stability analysis of linear time delay systems to be formulated as a semidefinite program

a geometric optimization approach to detecting and intercepting dynamic targets using a mobile sensor network a methodology is developed to deploy a mobile sensor network for the purpose of detecting and capturing mobile targets in the plane the sensing pursuit problem considered in this paper is analogous to the marco polo game in which a pursuer marco must capture multiple mobile targets that are sensed intermittently and with very limited information the competing objectives exhibited by this problem arise in a number of surveillance and monitoring applications in this paper the mobile sensor network consists of a set of robotic sensors that must track and capture mobile targets based on the information obtained through cooperative detections when these detections form a satisfactory target track a mobile sensor is switched to pursuit mode and deployed to capture the target in minimum time since the sensors are installed on robotic platforms and have limited range the geometry of the platforms and of the sensors fields of view play a key role in obstacle avoidance and target detection a new cell decomposition approach is presented to determine the probability of detection and the cost of operating the sensors from the geometric properties of the network and its workspace the correctness and complexity of the algorithm are analyzed proving that the termination time is a function of the network parameters and of the number of required detections

searching and pebbling abstract we relate the search number of an undirected graph g with the minimum and maximum of the progressive pebble demands of the directed acyclic graphs obtained by orienting g towards this end we introduce node searching a slight variant of searching in which an edge is cleared by placing searchers on both of its endpoints we also show that the minimum number of searchers necessary to node search a graph equals its vertex separator plus one

connected searching of weighted trees in this paper we consider the problem of connected edge searching of weighted trees barriere et al claim in l barriere p flocchini p fraigniaud n santoro capture of an intruder by mobile agents in spaa 02 proceedings of the fourteenth annual acm symposium on parallel algorithms and architectures acm new york ny usa 2002 pp 200 209 that there exists a polynomial time algorithm for finding an optimal search strategy that is a strategy that minimizes the number of used searchers however due to some flaws in their algorithm the problem turns out to be open it is proven in this paper that the considered problem is strongly np complete even for node weighted trees the weight of each edge is 1 with one vertex of degree greater than 2 it is also shown that there exists a polynomial time algorithm for finding an optimal connected search strategy for a given bounded degree tree with arbitrary weights on the edges and on the vertices this is an fpt algorithm with respect to the maximum degree of a tree

on the relationship between nlc width and linear nlc width in this paper we consider nlc width nlct width and linear nlc width bounded graphs we show that the set of all complete binary trees has unbounded linear nlc width and that the set of all co graphs has unbounded nlct width since trees have nlct width 3 and co graphs have nlc width 1 it follows that the family of linear nlc width bounded graph classes is a proper subfamily of the family of nlct width bounded graph classes and that the family of nlct width bounded graph classes is a proper subfamily of the family of nlc width bounded graph classes

tradeoffs in process strategy games with application in the wdm reconfiguration problem we consider a variant of the graph searching games that models the routing reconfiguration problem in wdm networks in the digraph processing game a team of agents aims at processing or clearing the vertices of a digraph d we are interested in two different measures 1 the total number of agents used and 2 the total number of vertices occupied by an agent during the processing of d these measures respectively correspond to the maximum number of simultaneous connections interrupted and to the total number of interruptions during a routing reconfiguration in a wdm network previous works have studied the problem of independently minimizing each of these parameters in particular the corresponding minimization problems are apx hard and the first one is known not to be in apx in this paper we give several complexity results and study tradeoffs between these conflicting objectives in particular we show that minimizing one of these parameters while the other is constrained is np complete then we prove that there exist some digraphs for which minimizing one of these objectives arbitrarily impairs the quality of the solution for the other one we show that such bad tradeoffs may happen even for a basic class of digraphs on the other hand we exhibit classes of graphs for which good tradeoffs can be achieved we finally detail the relationship between this game and the routing reconfiguration problem in particular we prove that any instance of the processing game i e any digraph corresponds to an instance of the routing reconfiguration problem

multicommodity max flow min cut theorems and their use in designing approximation algorithms in this paper we establish max flow min cut theorems for several important classes of multicommodity flow problems in particular we show that for any n node multicommodity flow problem with uniform demands the max flow for the problem is within an o log n factor of the upper bound implied by the min cut the result which is existentially optimal establishes an important analogue of the famous 1 commodity max flow min cut theorem for problems with multiple commodities the result also has substantial applications to the field of approximation algorithms for example we use the flow result to design the first polynomial time polylog n times optimal approximation algorithms for well known np hard optimization problems such as graph partitioning min cut linear arrangement crossing number vlsi layout and minimum feedback arc set applications of the flow results to path routing problems network reconfiguration communication in distributed networks scientific computing and rapidly mixing markov chains are also described in the paper categories and subject descriptors f 2 2 analysis of algorithms and problem complexity

offline variants of the lion and man problem consider the following safe path planning problem given a set of trajectories paths of k point robots with maximum unit speed in a bounded region over a long time interval 0 t find another trajectory if it exists subject to the same maximum unit speed limit that avoids that is stays at a safe distance of each of the other k trajectories over the entire time interval we call this variant the continuous model of the safe path planning problem the discrete model of this problem is given a set of trajectories paths of k point robots in a graph over a long time interval 0 1 2 t find a trajectory path for another robot that avoids each of the other k at any time instant in the given time interval we introduce the notions of the avoidance number of a region and that of a graph respectively as the maximum number of trajectories which can be avoided in the region respectively graph we give the first estimates on the avoidance number of the nxn grid g n and also devise an efficient algorithm for the corresponding safe path planning problem in arbitrary graphs we then show that our estimates on the avoidance number of g n can be extended for the avoidance number of a bounded fat region in the final part of our paper we consider other related offline questions such as the maximum number of men problem and the spy problem

characterizations for restricted graphs of nlc width 2 in this paper we give a finite forbidden subgraph characterization of graphs defined by nlc width 2 expressions by nlct width 2 expressions or by linear nlc width 2 expressions that have tree width 1

the complexity of some polynomial network consistency algorithms for constraint satisfaction problems abstract constraint satisfaction problems play a central role in artificial intelligence a class of network consistency algorithms for eliminating local inconsistencies in such problems has previously been described we analyze the time complexity of several node arc and path consistency algorithms and prove that arc consistency is achievable in time linear in the number of binary constraints the waltz filtering algorithm is a special case of the arc consistency algorithm in the edge labelling computational vision application the constraint graph is planar and so the time complexity is linear in the number of variables

finite graph automata for linear and boundary graph languages graph grammars can be regarded as a generalization of context free grammars from strings to graphs over the past 30 years a rich theory of graph grammars and their languages has been developed however there are no graph automata there is no duality between generative and recognizing devices as it is known for the chomsky hierarchy of formal languages here we introduce graph automata as devices for the recognition of sets of undirected node labelled graphs a graph automaton consists of a finite state control a finite set of instructions and a collection of heads or guards it reads an input graph in a systematic way and performs a graph search directed by the instructions as our main results we show that finite graph automata recognize exactly the set of graph languages generated by linear nce graph grammars and that alternating finite graph automata recognize exactly the languages of boundary graph grammars finally we generalize some automata theoretic properties from string to graph automata integrate the connectivity of graphs into graph automata and explain why graph automata cannot be generalized to deal with dynamic edge relabellings and ence graph languages

the complexity of searching a graph t parsons originally proposed and studied the following pursuit evasion problem on graphs members of a team of searchers traverse the edges of a graph g in pursuit of a fugitive who moves along the edges of the graph with complete knowledge of the locations of the pursuers what is the smallest number s g of searchers that will suffice for guaranteeing capture of the fugitive it is shown that determining whether s g k for a given integer k is np complete for general graphs but can be solved in linear time for trees we also provide a structural characterization of those graphs g with s g k for k 1 2 3

linear rank width and linear clique width of trees we show that for every forest t the linear rank width of t is equal to the path width of t and the linear clique width of t equals the path width of t plus two provided that t contains a path of length three it follows that both linear rank width and linear clique width of forests can be computed in linear time using our characterization of linear rank width of forests we determine the set of minimal excluded acyclic vertex minors for the class of graphs of linear rank width at most k

non deterministic graph searching in trees non deterministic graph searching was introduced by fomin et al to provide a unified approach for pathwidth treewidth and their interpretations in terms of graph searching games given q 0 the q limited search number s q g of a graph g is the smallest number of searchers required to capture an invisible fugitive in g when the searchers are allowed to know the position of the fugitive at most q times the search parameter s 0 g corresponds to the pathwidth of a graph g and s g to its treewidth determining s q g is np complete for any fixed q 0 in general graphs and s 0 t can be computed in linear time in trees however the complexity of the problem on trees has been unknown for any q 0 we introduce a new variant of graph searching called restricted non deterministic the corresponding parameter is denoted by rs q and is shown to be equal to the non deterministic graph searching parameter s q for q 0 1 and at most twice s q for any q 2 for any graph g our main result is a polynomial time algorithm that computes rs q t for any tree t and any q 0 this provides a 2 approximation of s q t for any tree t and shows that the decision problem associated to s 1 is polynomial in the class of trees our proofs are based on a new decomposition technique for trees which might be of independent interest

approximate search strategies for weighted trees the problems of classical searching and connected searching of weighted trees are known to be computationally hard in this work we give a polynomial time 3 approximation algorithm that finds a connected search strategy of a given weighted tree this in particular yields constant factor approximation algorithms for the non connected classical searching problems and for the weighted pathwidth problem for this class of graphs

the treewidth of proofs abstract so called ordered variants of the classical notions of pathwidth and treewidth are introduced and proposed as proof theoretically meaningful complexity measures for the directed acyclic graphs underlying proofs ordered pathwidth is roughly the same as proof space and the ordered treewidth of a proof is meant to serve as a measure of how far it is from being treelike length space lower bounds for k dnf refutations are generalized to arbitrary infinity axioms and strengthened in that the space measure is relaxed to ordered treewidth

edge and node searching problems on trees in this paper we show that there is a natural correspondence between a tree s edge search strategy and its node search strategy by doing so we simplify the previous linear time algorithm for determining the edge search number of a tree and improve the running time of a previous algorithm for constructing an edge search strategy or plan for a tree containing n vertices from o n log n to o n time

the vertex separation and search number of a graph abstract we relate two concepts in graph theory and algorithmic complexity namely the search number and the vertex separation of a graph let s g denote the search number and vs g denote the vertex separation of a connected undirected graph g we show that vs g s g vs g 2 and we give a simple transformation from g to g such that vs g s g we characterize those trees having a given vertex separation and describe the smallest such trees we also note that there exist trees for which the difference between search number and vertex separation is indeed 2 we give algorithms that for any tree t compute vs t in linear time and compute an optimal layout with respect to vertex separation in time o n log n vertex separation has previously been related to progressive black white pebble demand and has been shown to be identical to a variant of search number node search number and to path width which has been related to gate matrix layout cost all these properties are known to be computationally intractable for fixed k an o n log 2 n algorithm is known which decides whether a graph has path width at most k

digraph decompositions and monotonicity in digraph searching graph decompositions such as tree decompositions and associated width measures have been the focus of much attention in structural and algorithmic graph theory in particular it has been found that many otherwise intractable problems become tractable on graph classes of bounded tree width more recently proposals have been made to define a similar notion to tree width for directed graphs several proposals have appeared so far supported by algorithmic applications in this paper we explore the limits of algorithmic applicability of digraph decompositions and show that various natural candidates for problems which potentially could benefit from digraphs having small directed width remain np complete even on almost acyclic graphs closely related to graph and digraph decompositions are graph searching games an important property of graph searching games is monotonicity and a large number of papers addresses the question whether particular variants of these games are monotone however so far for two natural types of graph searching games underlying dag and kelly decompositions the question whether they are monotone was still open we settle this issue by showing that both variants the visible and the inert invisible graph searching games on directed graphs are non monotone

complexity of node coverage games modern software systems may exhibit a nondeterministic behavior due to many unpredictable factors in this work we propose the node coverage game a two player turn based game played on a finite game graph as a formalization of the problem to test such systems each node in the graph represents a functional equivalence class of the software under test sut one player the tester wants to maximize the node coverage measured by the number of nodes visited when exploring the game graphs while his opponent the sut wants to minimize it an optimal test would maximize the cover and it is an interesting problem to find the maximal number of nodes that the tester can guarantee to visit irrespective of the responses of the sut we show that the decision problem of whether the guarantee is less than a given number is np complete we also discuss two extensions of our result and present a testing procedure based on our result when the sut is not so hostile we establish that the decision problem of maximal node coverage guarantee of non deterministic game graphs is np complete we present two variations of node coverage game with np complete decision problem complexity we also provide an adaptive procedure that employs an np procedure to generate test cases with maximal coverage guarantee and reevaluate the mcg and regenerate test cases when the sut is not so hostile

a linear recognition algorithm for cographs cographs are the graphs formed from a single vertex under the closure of the operations of union and complement another characterization of cographs is that they are the undirected graphs with no induced paths on four vertices cographs arise naturally in such application areas as examination scheduling and automatic clustering of index terms furthermore it is known that cographs have a unique tree representation called a cotree using the cotree it is possible to design very fast polynomial time algorithms for problems which are intractable for graphs in general such problems include chromatic number clique determination clustering minimum weight domination isomorphism minimum fill in and hamiltonicity in this paper we present a linear time algorithm for recognizing cographs and constructing their cotree representation

on partitioning a graph into two connected subgraphs suppose a graph g is given with two vertex disjoint sets of vertices z 1 and z 2 can we partition the remaining vertices of g such that we obtain two connected vertex disjoint subgraphs of g that contain z 1 and z 2 respectively this problem is known as the 2 disjoint connected subgraphs problem it is already np complete for the class of n vertex graphs g v e in which z 1 and z 2 each contain a connected set that dominates all vertices in email protected z email protected z 2 we present an o 1 2051 n time algorithm that solves it for this graph class as a consequence we can also solve this problem in o 1 2051 n time for the classes of n vertex p 6 free graphs and split graphs this is an improvement upon a recent o 1 5790 n time algorithm for these two classes our approach translates the problem to a generalized version of hypergraph 2 coloring and combines inclusion exclusion with measure and conquer

qualitative and quantitative analysis of stochastic processes based on measured data i theory and applications to synthetic data analysis of stochastic processes governed by the langevin equation is discussed the analysis is based on a general method for non parametric estimation of deterministic and random terms of the langevin equation directly from given data separate estimation of the terms corresponds to decomposition of process dynamics into deterministic and random components such decomposition provides a basis for qualitative and quantitative analysis of process dynamics in part i the following analysis possibilities are described and illustrated using various synthetic datasets 1 qualitative inspection of the estimated terms presented as fields 2 reconstruction of the deterministic and stochastic evolution of the process and 3 approximation of the deterministic term by an analytical function and quantitative treatment of the equations obtained in part ii these analysis possibilities are applied to experimental datasets from metal cutting and laser beam welding

recognizing hyperelliptic graphs in polynomial time abstract based on analogies between algebraic curves and graphs baker and norine introduced divisorial gonality a graph parameter for multigraphs related to treewidth multigraph algorithms and number theory various equivalent definitions of the gonality of an algebraic curve translate to different notions of gonality for graphs called stable gonality and stable divisorial gonality we consider so called hyperelliptic graphs multigraphs of gonality 2 in any meaning of graph gonality and provide a safe and complete set of reduction rules for such multigraphs this results in an algorithm to recognize hyperelliptic graphs in time o m n log n where n is the number of vertices and m the number of edges of the multigraph a corollary is that we can decide with the same runtime whether a two edge connected graph g admits an involution such that the quotient g is a tree

a sufficient condition for backtrack bounded search backtrack search is often used to solve constraint satisfaction problems a relationship involving the structure of the constraints is described that provides a bound on the backtracking required to advance deeper into the backtrack tree this analysis leads to upper bounds on the effort required for solution of a class of constraint satisfaction problems the solutions involve a combination of relaxation preprocessing and backtrack search the bounds are expressed in terms of the structure of the constraint connections specifically the effort is shown to have a bound exponential in the size of the largest biconnected component of the constraint graph as opposed to the size of the graph as a whole

lex bfs and partition refinement with applications to transitive orientation interval graph recognition and consecutive ones testing by making use of lexicographic breadth rst search lex bfs and partition renement with pivots we obtain very simple algorithms for some well known problems in graph theory we give a o n mlogn algorithm for transitive orientation of a comparability graph and simple linear algorithms to recognize interval graphs convex graphs y semichordal graphs and matrices that have the consecutive ones property previous approaches to these problems used dicult preprocessing steps such as computing pq trees or modular decomposition the algorithms we give are easy to understand and straightforward to prove they do not make use of sophisticated data structures and the complexity analysis is straightforward c 2000 elsevier science b v all rights reserved

search for an immobile entity on a network we consider the problem of searching for a single uniformly distributed immobile entity on an undirected network this problem differs from edge covering problems e g the chinese postman problem cpp since the objective here is not to find the minimum length tour that covers all the edges at least once but instead to minimize the expected time to find the entity we introduce a heuristic algorithm to deal with the search process given that the entity is equally likely to be at any point on the network computational results are presented

monotony properties of connected visible graph searching search games are attractive for their correspondence with classical width parameters for instance the invisible search number a k a node search number of a graph is equal to its pathwidth plus 1 and the visible search number of a graph is equal to its treewidth plus 1 the connected variants of these games ask for search strategies that are connected i e at every step of the strategy the searched part of the graph induces a connected subgraph we focus on monotone search strategies i e strategies for which every node is searched exactly once the monotone connected visible search number of an n node graph is at most o logn times its visible search number first we prove that this logarithmic bound is tight precisely we prove that there is an infinite family of graphs for which the ratio monotone connected visible search number over visible search number is w logn second we prove that as opposed to the non connected variant of visible graph searching recontamination helps for connected visible search precisely we prove that for any k 4 there exists a graph with connected visible search number at most k and monotone connected visible search number k

finding hidden independent sets in interval graphs we design efficient competitive algorithms for discovering hidden information using few queries specifically consider a game in a given set of intervals and their implied interval graph g in which our goal is to discover an unknown independent set x by making the fewest queries of the form is point p covered by an interval in x our interest in this problem stems from two applications experimental gene discovery with pcr technology and the game of battleship in a 1 dimensional setting we provide adaptive algorithms for both the verification scenario given an independent set is it x and the discovery scenario find x without any information under some assumptions these algorithms use an asymptotically optimal number of queries in every instance

all structured programs have small tree width and good register allocation the register allocation problem for an imperative program is often modelled as the coloring problem of the interference graph of the control flow graph of the program the interference graph of a flow graph g is the intersection graph of some connected subgraphs of g these connected subgraphs represent the lives or life times of variables so the coloring problem models that two variables with overlapping life times should be in different registers for general programs with unrestricted gotos the interference graph can be any graph and hence we cannot in general color within a factor o n from optimality unless np p

chatter onset in non regenerative cutting a numerical study a non linear model of non regenerative cutting is analyzed when damped the model has a stable fixed point for small values of chip thickness as chip thickness is increased the fixed point becomes unstable via a sub critical hopf bifurcation this transition corresponds to the onset of chatter when dynamic noise is added to the system chatter onset can occur in the sub critical parameter region although chatter free cutting is predicted by the stability analysis of the fixed point a noisy bifurcation diagram is compared qualitatively to a diagram obtained experimentally in regenerative cutting similar features regarding the onset of chatter are observed in both diagrams

static search games played over graphs and general metric spaces we define a general game which forms a basis for modelling situations of static search and concealment over regions with spatial structure the game involves two players the searching player and the concealing player and is played over a metric space each player simultaneously chooses to deploy at a point in the space the searching player receiving a payoff of 1 if his opponent lies within a predetermined radius r of his position the concealing player receiving a payoff of 1 otherwise the concepts of dominance and equivalence of strategies are examined in the context of this game before focusing on the more specific case of the game played over a graph methods are presented to simplify the analysis of such games both by means of the iterated elimination of dominated strategies and through consideration of automorphisms of the graph lower and upper bounds on the value of the game are presented and optimal mixed strategies are calculated for games played over a particular family of graphs

jumping robbers in digraphs abstract in cops and robber games a team of cops tries to capture a robber on a given graph the cops occupy some vertices of the graph and the robber occupies one vertex given concrete rules how the players can move and when the robber is captured the minimum number of cops needed to capture the robber is a natural parameter of the graph richerby and thilikos 25 generalise the game to a game where the cops must capture multiple robbers they show that the number of cops needed to capture r robbers is at most k log r where k is the minimum number of cops who can capture one robber furthermore this bound is tight moreover capturing every new robber may demand some new cops and infinitely many robbers demand as many cops as one invisible robber we generalise the results of richerby and thilikos in two aspects we give a linear bound on the number of additional cops in the game 1 on directed graphs and 2 where the robbers have more power every robber can leave his vertex and jump to another robber we show that our bound is tight and that the reason for a worse lower bound is not the directedness of the graphs but the ability of the robbers to jump we also prove that adding new robbers induces a hierarchy of the cop numbers for a fixed graph which is strict on some graphs and converges to the minimum number of cops needed to capture an invisible robber as in 25

a two person game on graphs where each player tries to encircle his opponent s men we present results on a combinatorial game which was proposed to one of the authors by ingo althofer personal communication let g be an undirected finite graph without loops and multiple edges and let k be a positive integer with k 12 g 1 there are two players called white and black both having k men of their color in turn beginning with white the players position their men one at a time on unoccupied vertices of g when all men are placed the players take turns moving a man of their color along an edge to an unoccupied adjacent vertex again beginning with white a player wins if his opponent cannot carry out his next move since none of his men has an unoccupied neighbor if the game does not stop then the outcome is a draw we always assume that both players play optimal among other questions we deal with the following ones 1 1 is it true that for all g and k white cannot win the game 2 2 does there exist a tree t and a positive integer k for which the outcome is a draw let g denote the covering number of g i e g is the minimum number of vertices covering all edges of g we prove that black wins the game if g k we use this result to show that white never wins the game if gis bipartite thus providing a partial answer to the first question we answer the second question in the affirmative by constructing an infinite series of trees for which the outcome is a draw for some k moreover we present results on extremal problems arising in the context of the game we also completely solve the cases when g is a path or a cycle further we completely settle the case k 2 in the proofs of our results matchings and cycles in graphs play a predominant role

fugitive search games on graphs and related parameters abstract the goal of a fugitive search game on a graph is to trap a fugitive that hides on the vertices of the graph by systematically placing searchers on the vertices the fugitive is assumed to have complete knowledge of the graph and of the searchers moves but is restricted to move only along paths whose vertices are not guarded by searchers the search number of the graph is the least number of searchers necessary to trap the fugitive variants of the fugitive search game have been related to important graph parameters like treewidth and pathwidth in this paper we introduce a class of fugitive search games where the searchers do not see the fugitive and the fugitive can only move just before a searcher is placed on the vertex it occupies letting the fugitive s speed i e the maximum number of edges the fugitive can traverse at a single move vary we get different games we show that if the speed of the fugitive is unbounded then the search number minus 1 is equal to the treewidth of the graph while if the speed is 1 then the search number minus 1 is equal to a polynomially computable graph parameter which is called width or alternatively linkage and is studied in the context of the constraint satisfaction and boolean satisfiability problems we also show that in the above two cases the search number remains the same even if we consider only search strategies that at every step further restrict the fugitive s possible resorts this monotonicity property is usually expressed as recontamination does not help moreover we give an equivalent characterization of the search number for any given fugitive speed in terms of an elimination ordering of the vertices of the graph using this characterization we show that for any graph if the length of any chordless cycle is bounded by a constant s s 3 then the treewidth of the graph plus 1 is equal to the search number for fugitive speed s 2

monotonicity of non deterministic graph searching in graph searching a team of searchers are aiming at capturing a fugitive moving in a graph in the initial variant called invisible graph searching the searchers do not know the position of the fugitive until they catch it in another variant the searchers permanently know the position of the fugitive i e the fugitive is visible this latter variant is called visible graph searching a search strategy that catches any fugitive in such a way that the part of the graph reachable by the fugitive never grows is called monotone a priori monotone strategies may require more searchers than general strategies to catch any fugitive this is however not the case for visible and invisible graph searching two important consequences of the monotonicity of visible and invisible graph searching are 1 the decision problem corresponding to the computation of the smallest number of searchers required to clear a graph is in np and 2 computing optimal search strategies is simplified by taking into account that there exist some that never backtrack fomin et al f v fomin p fraigniaud n nisse nondeterministic graph searching from pathwidth to treewidth in proceedings of the 30th international symposium on mathematical foundations of computer science mfcs 05 2005 pp 364 375 introduced an important graph searching variant called non deterministic graph searching that unifies visible and invisible graph searching in this variant the fugitive is invisible and the searchers can query an oracle that permanently knows the current position of the fugitive the question of the monotonicity of non deterministic graph searching was however left open in this paper we prove that non deterministic graph searching is monotone in particular this result is a unified proof of monotonicity for visible and invisible graph searching as a consequence the decision problem corresponding to non deterministic graph searching belongs to np moreover the exact algorithms designed by fomin et al do compute optimal non deterministic search strategies

recontamination does not help to search a graph this paper is concerned with a game on graphs called graph searching the object of this game is to clear all edges of a contaminated graph clearing is achieved by moving searchers a kind of token along the edges of the graph according to clearing rules certain search strategies cause edges that have been cleared to become contaminated again megiddo et al 9 conjectured that every graph can be searched using a minimum number of searchers without this recontamination occurring that is without clearing any edge twice in this paper this conjecture is proved this places the graph searching problem in np completing the proof by megiddo et al that the graph searching problem is np complete furthermore by eliminating the need to consider recontamination this result simplifies the analysis of searcher requirements with respect to other properties of graphs

fast edge searching and fast searching on graphs given a graph g v e in which a fugitive hides on vertices or along edges graph searching problems are usually to find the minimum number of searchers required to capture the fugitive in this paper we consider the problem of finding the minimum number of steps to capture the fugitive we introduce the fast edge searching problem in the edge search model which is the problem of finding the minimum number of steps called the fast edge search time to capture the fugitive we establish relations between the fast edge searching and the fast searching that is the problem of finding the minimum number of searchers to capture the fugitive in the fast search model while the family of graphs whose fast search number is at most k is not minor closed for any positive integer k 2 we show that the family of graphs whose fast edge search time is at most k is minor closed we establish relations between the fast fast edge searching and the node searching these relations allow us to transform the problem of computing node search numbers to the problem of computing fast edge search numbers or fast search numbers using these relations we prove that the problem of deciding given a graph g and an integer k whether the fast edge search number of g is less than or equal to k is np complete and it remains np complete for eulerian graphs we also prove that the problem of determining whether the fast edge search number of g is half of the number of odd vertices in g is np complete and it remains np complete for planar graphs with maximum degree 4 we present a linear time approximation algorithm for the fast edge search time that always delivers solutions of at most 1 v 1 e 1 times the optimal value this algorithm also gives us a tight upper bound on the fast search number of graphs we also show a lower bound on the fast search number using the minimum degree and the number of odd vertices

the complexity of zero visibility cops and robber in this work we deal with the computational complexity aspects of the zero visibility cops and robber game we provide an algorithm that computes the zero visibility copnumber of a tree in linear time and show that the corresponding decision problem is np complete even for the class of starlike graphs

min cut is np complete for edge weighted trees abstract we show that the min cut linear arrangement problem min cut is np complete for trees with polynomial size edge weights and derive from this the np completeness of min cut for planar graphs with maximum vertex degree 3 this is used to show the np completeness of search number vertex separation progressive black white pebble demand and topological bandwidth for planar graphs with maximum vertex degree 3

complexity and monotonicity results for domination games abstract in this paper we study domination games a class of games introduced by fomin kratsch and muller in 8 domination games are a variant of the well known graph searching games also called cops and robber games where a number of cops tries to capture a robber hiding on the vertices of a graph variants of these games are often used to provide a game theoretic characterization of important graph parameters such as pathwidth treewidth and hypertreewidth we are primarily interested in questions concerning the complexity and monotonicity of these games we show that dominating games are computationally much harder than standard cops and robber games and establish strong non monotonicity results for various notions of monotonicity that arise naturally in the context of domination games answering a question of 8 we show that there are graphs where the shortest winning strategy for a minimal number of cops must necessarily be of exponential length

approximation algorithms for digraph width parameters several problems that are np hard on general graphs are efficiently solvable on graphs with bounded treewidth efforts have been made to generalize treewidth and the related notion of pathwidth to digraphs directed treewidth dag width and kelly width are some such notions which generalize treewidth whereas directed pathwidth generalizes pathwidth each of these digraph width measures have an associated decomposition structure in this paper we present approximation algorithms for all these digraph width parameters in particular we give an o log n approximation algorithm for directed treewidth and an o log 3 2 n approximation algorithm for directed pathwidth dag width and kelly width our algorithms construct the corresponding decompositions whose widths agree with the above mentioned approximation factors

connected graph searching in the graph searching game the opponents are a set of searchers and a fugitive in a graph the searchers try to capture the fugitive by applying some sequence of moves that include placement removal or sliding of a searcher along an edge the fugitive tries to avoid capture by moving along unguarded paths the search number of a graph is the minimum number of searchers required to guarantee the capture of the fugitive in this paper we initiate the study of this game under the natural restriction of connectivity where we demand that in each step of the search the locations of the graph that are clean i e non accessible to the fugitive remain connected we give evidence that many of the standard mathematical tools used so far in classic graph searching fail under the connectivity requirement we also settle the question on the price of connectivity that is how many searchers more are required for searching a graph when the connectivity demand is imposed we make estimations of the price of connectivity on general graphs and we provide tight bounds for the case of trees in particular for an n vertex graph the ratio between the connected searching number and the non connected one is o logn while for trees this ratio is always at most 2 we also conjecture that this constant ratio upper bound for trees holds also for all graphs our combinatorial results imply a complete characterization of connected graph searching on trees it is based on a forbidden graph characterization of the connected search number we prove that the connected search game is monotone for trees i e restricting search strategies to only those where the clean territories increase monotonically does not require more searchers a consequence of our results is that the connected search number can be computed in polynomial time on trees moreover we show how to make this algorithm distributed finally we reveal connections of this parameter to other invariants on trees such as the horton strahler number

a comparison of two variations of a pebble game on graphs abstract the number of pebbles used in the black black white pebble game corresponds to the storage requirement of the deterministic non deterministic evaluation of a straight line program suppose a distinguished vertex of a directed acyclic graph can be pebbled with k pebbles in the black white pebble game then it can be pebbled with k 1 2k k 1 1 pebbles in the black pebble game

qualitative and quantitative analysis of stochastic processes based on measured data ii applications to experimental data analysis of stochastic processes governed by the langevin equation is discussed the analysis is based on a general method for non parametric estimation of deterministic and random terms of the langevin equation directly from given data separate estimation of the terms corresponds to the decomposition of process dynamics into deterministic and random components part i of the paper presented several possibilities for qualitative and quantitative analysis of process dynamics based on such decomposition in part ii some of these analysis possibilities are applied to experimental datasets from metal cutting and laser beam welding

nonconstructive tools for proving polynomial time decidability recent advances in graph theory and graph algorithms dramatically alter the traditional view of concrete complexity theory in which a decision problem is generally shown to be in p by producing an efficient algorithm to solve an optimization version of the problem nonconstructive tools are now available for classifying problems as decidable in polynomial time by guaranteeing only the existence of polynomial time decision algorithms in this paper these new methods are employed to prove membership in p for a number of problems whose complexities are not otherwise known powerful consequences of these techniques are pointed out and their utility is illustrated a type of partially ordered set that supports this general approach is defined and explored

on search games that include ambush we present a stochastic game that models ambush search in a finite region q which has area but no other structure the searcher can search a unit area of q in unit time or adopt an ambush mode for a certain period the searcher captures the hider when the searched region contains the hider s location or if the hider moves while the searcher is in ambush mode the payoff in this zero sum game is the capture time our game is motivated by the still unsolved princess and monster game on a star graph with a large number of leaves

computing the vertex separation of unicyclic graphs we describe an o n log n algorithm for the computation of the vertex separation of unicyclic graphs the algorithm also computes a linear layout with optimal vertex separation in the same time bound pathwidth node search number and vertex separation are different ways of defining the same notion path decompositions and search strategies can be derived from linear layouts the algorithm applies existing linear time techniques for the computation of the vertex separation of trees together with corresponding optimal layouts we reformulate the earlier work on the linear time computation of optimal layouts a polynomial time algorithm for the problem on unicyclic graphs can be inferred from existing more general methods for graphs of fixed treewidth since unicyclic graphs have treewidth two but the time complexity of the resulting method would seem to be an inordinately high order polynomial our algorithm we claim is practical the addition of one edge to a tree does seem to require a considerably more elaborate algorithm

tree clustering for constraint networks research note abstract the paper offers a systematic way of regrouping constraints into hierarchical structures capable of supporting search without backtracking the method involves the formation and preprocessing of an acyclic database that permits a large variety of queries and local perturbations to be processed swiftly either by sequential backtrack free procedures or by distributed constraint propagation processes

a partial k arboretum of graphs with bounded treewidth the notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies this survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class also some mutual relations between such classes are discussed

a polynomial algorithm for the min cut linear arrangement of trees an algorithm is presented that finds a min cut linear arrangement of a tree in o n log n time an extension of the algorithm determines the number of pebbles needed to play the black and white pebble game on a tree

digraph measures kelly decompositions games and orderings we consider various well known equivalent complexity measures for graphs such as elimination orderings k trees and cops and robber games and study their natural translations to digraphs we show that on digraphs the translations of these measures are also equivalent and induce a natural connectivity measure we introduce a decomposition for digraphs and an associated width kelly width which is equivalent to the aforementioned measure we demonstrate its usefulness by exhibiting potential applications including polynomial time algorithms for np complete problems on graphs of bounded kelly width and complexity analysis of asymmetric matrix factorization finally we compare the new width to other known decompositions of digraphs

mixed searching and proper path width abstract this paper considers a mixed search game which is a natural generalization of edge search and node search games extensively studied so far we establish a relationship between the mixed search number of a graph g and the proper path width of g introduced by the authors in a previous paper we also prove complexity results

nonstationary langevin equation statistical properties and application to explain effects observed in cardiological time series using the langevin equation we develop the model of a stochastic process subject to a given time dependent regulatory mechanism the effects of this nonstationarity on the statistical properties of the time series i e on global and conditional probability densities and on the moments of the distribution are derived application of these results on simple model trends allows one to approximate cardiological data and thus to explain effects recently observed in the reconstruction of the deterministic part of the langevin equation for time series of heart rate

polynomial time algorithms for the min cut problem on degree restricted trees polynomial algorithms are described that solve the min cut linear arrangement problem on degree restricted trees for example the cutwidth or folding number of an arbitrary degree d tree can be found in o n log n d 2 steps this has applications to integrated circuit layout in particular the layout of weinberger arrays 41 this also yields an algorithm for determining the black white pebble demand of degree three trees we also show that for degree three trees cutwidth is identical to search number and give a forbidden subgraph characterization of degree three trees having cutwidth k

the searchlight problem for road networks we consider the problem of searching for a mobile intruder hiding in a road network given as the union of two or more lines or two or more line segments in the plane some of the intersections of the road network are occupied by stationary guards equipped with a number of searchlights each of which can emit a single ray of light in any direction along the lines or line segments it is on the goal is to detect the intruder that is to illuminate its location guards may alter the direction in which they aim a searchlight but need to switch it off for some finite time interval to effect the change in contrast the intruder may move with arbitrary speed along the network but cannot pass guards and exploit this time interval to recontaminate previously illuminated sections of the network for various classes of road networks characterized by the number n of lines or line segments comprising it and the number g n 1 of possible locations of guards fixed in advance and guaranteed to give complete coverage we present several upper and lower bounds on the worst case number of searchlights each placed at one of the guard positions required to successfully search a given road network in particular we prove the following results 1 min 2 g 1 n 2 searchlights are sometimes necessary and min 7 3 g n 1 are always sufficient for searching a road network given as the union of n lines 2 g log n g searchlights are sometimes necessary and o g 2 log n searchlights are always sufficient for searching a road network given as the union of n line segments and3 at most one searchlight per guard position and hence a total of at most g searchlights is always sufficient for searching a road network given as the union of axis aligned lines or line segments the proofs of the upper bounds induce algorithms for generating a search schedule for detecting the intruder using the claimed number of searchlights the problem of searching for an intruder hiding in a road network is introduced upper and lower bounds on the worst case number of searchlights are presented the proofs of the upper bounds induce algorithms for generating a search schedule

on minimum cost edge searching we consider the problem of finding edge search strategies of minimum cost the cost of a search strategy is the sum of searchers used in the clearing steps of the search one of the natural questions is whether it is possible to find a search strategy that minimizes both the cost and the number of searchers used to clear a given graph g we call such a strategy ideal we prove by an example that ideal search strategies do not exist in general on the other hand we provide a formula for the cost of clearing complete graphs from our construction it follows that an ideal search strategy of a complete graph does exist and can be calculated efficiently for general graphs g we give a polynomial time o logn approximation algorithm for finding minimum cost search strategies we also prove that recontamination does not help to obtain minimum cost edge search strategies

graphs of linear clique width at most 3 abstract a graph has linear clique width at most k if it has a clique width expression using at most k labels such that every disjoint union operation has an operand which is a single vertex graph we give the first characterisation of graphs of linear clique width at most 3 and we give the first polynomial time recognition algorithm for graphs of linear clique width at most 3 in addition we present new characterisations of graphs of linear clique width at most 2 we also give a layout characterisation of graphs of bounded linear clique width a similar characterisation was independently shown by gurski and by lozin and rautenbach

an annotated bibliography on guaranteed graph searching graph searching encompasses a wide variety of combinatorial problems related to the problem of capturing a fugitive residing in a graph using the minimum number of searchers in this annotated bibliography we give an elementary classification of problems and results related to graph searching and provide a source of bibliographical references on this field

pathwidth bandwidth and completion problems to proper interval graphs with small cliques we study two related problems motivated by molecular biology given a graph g and a constant k does there exist a supergraph g of g that is a unit interval graph and has clique size at most k given a graph g and a proper k coloring c of g does there exist a supergraph g of g that is properly colored by c and is a unit interval graph we show that those problems are polynomial for fixed k on the other hand we prove that the first problem is equivalent to deciding if the bandwidth of g is at most k 1 hence it is np hard and w t hard for all t we also show that the second problem is w 1 hard this implies that for fixed k both of the problems are unlikely to have an o n alpha algorithm where alpha is a constant independent of k a central tool in our study is a new graph theoretic parameter closely related to pathwidth an unexpected useful consequence is the equivalence of this parameter to the bandwidth of the graph

exclusive graph searching vs pathwidth in graph searching a team of searchers aims at capturing an invisible fugitive moving arbitrarily fast in a graph equivalently the searchers try to clear a contaminated network the problem is to compute the minimum number of searchers required to accomplish this task several variants of graph searching have been studied mainly because of their close relationship with the pathwidth of a graph in this paper we study the complexity of the exclusive graph searching variant we show that the problem is np hard in planar graphs and it can be solved in linear time in the class of cographs we also show that monotone exclusive graph searching is np complete in split graphs where pathwidth is known to be solvable in polynomial time moreover we prove that monotone exclusive graph searching is in p in a subclass of star like graphs where pathwidth is known to be np hard

fast mixed searching and related problems on graphs abstract in this paper we introduce the fast mixed search model which is a combination of the fast search model and the mixed search model given a graph g in which a fugitive hides on vertices or along edges the fast mixed search problem is to find the minimum number of searchers to capture the fugitive in the fast mixed search model we establish a relation between the fast mixed search problem and the induced path cover problem we also consider relations between the fast mixed search problem and other graph search problems we prove that fast mixed searching from leaves is np complete and it remains np complete for planar graphs with maximum degree 3 we also prove that fast mixed searching between leaves is np complete and it remains np complete for graphs with maximum degree 4 we present linear time algorithms for computing the fast mixed search number and optimal search strategies of some classes of graphs including trees cacti and proper interval graphs

computing the chromatic number using graph decompositions via matrix rank abstract computing the smallest number q such that the vertices of a given graph can be properly q colored known as the chromatic number is one of the oldest and most fundamental problems in combinatorial optimization the q coloring problem has been studied intensively using the framework of parameterized algorithmics resulting in a very good understanding of the best possible algorithms for several parameterizations based on the structure of the graph for example algorithms are known to solve the problem on graphs of treewidth tw in time o q tw while a running time of o q e tw is impossible assuming the strong exponential time hypothesis seth while there is an abundance of work for parameterizations based on decompositions of the graph by vertex separators almost nothing is known about parameterizations based on edge separators we fill this gap by studying q coloring parameterized by cutwidth and parameterized by pathwidth in bounded degree graphs our research uncovers interesting new ways to exploit small edge separators we present two algorithms for q coloring parameterized by cutwidth ctw a deterministic one that runs in time o 2 ctw where is the square matrix multiplication exponent and a randomized one with runtime o 2 ctw in sharp contrast to earlier work the running time is independent of q the dependence on cutwidth is optimal we prove that even 3 coloring cannot be solved in o 2 e ctw time assuming seth our algorithms rely on a new rank bound for a matrix that describes compatible colorings combined with a simple communication protocol for evaluating a product of two polynomials this also yields an o d 2 1 pw time randomized algorithm for q coloring on graphs of pathwidth pw and maximum degree d such a runtime was first obtained by bjorklund but only for graphs with few proper colorings we also prove that this result is optimal in the sense that no o d 2 1 e pw time algorithm exists assuming seth

on the complexity of the positive semidefinite zero forcing number abstract the positive semidefinite zero forcing number of a graph is a graph parameter that arises from a non traditional type of graph colouring and is related to a more conventional version of zero forcing we establish a relation between the zero forcing and the fast mixed searching which implies some np completeness results for the zero forcing problem relationships between positive semidefinite zero forcing sets and clique coverings are well understood for chordal graphs building upon constructions associated with optimal tree covers and forest covers we present a linear time algorithm for computing the positive semidefinite zero forcing number of chordal graphs we also prove that it is np complete to determine whether a graph has a positive semidefinite zero forcing set with an additional property

exponentially stable nonlinear systems have polynomial lyapunov functions on bounded regions this paper presents a proof that existence of a polynomial lyapunov function is necessary and sufficient for exponential stability of a sufficiently smooth nonlinear vector field on a bounded set the main result states that if there exists an n times continuously differentiable lyapunov function which proves exponential stability on a bounded subset of rn then there exists a polynomial lyapunov function which proves exponential stability on the same region such a continuous lyapunov function will exist if for example the vector field is at least n times continuously differentiable the proof is based on a generalization of the weierstrass approximation theorem to differentiable functions in several variables specifically polynomials can be used to approximate a differentiable function using the sobolev norm w 1 infin to any desired accuracy this approximation result is combined with the second order taylor series expansion to show that polynomial lyapunov functions can approximate continuous lyapunov functions arbitrarily well on bounded sets the investigation is motivated by the use of polynomial optimization algorithms to construct polynomial lyapunov functions

the linkage of a graph the linkage of a graph is defined to be the maximum min degree of any of its subgraphs it is known that the linkage of a graph is equal to its width for an arbitrary linear ordering of the vertices of the graph consider the maximum with respect to any vertex v of the number of vertices connected with v and preceding it in the ordering the width of the graph is the minimum of these maxima over all possible linear orderings width has been used in artificial intelligence in the context of constraint satisfaction problems csps a more general notion is defined by considering not the number of vertices preceding and connected with v but rather the least number of vertices preceding and connected with any cluster of at most j consecutive vertices extending to the right up to v j is a given integer the graph parameter thus defined is called j width no efficient algorithm was known for computing the j width in this paper we introduce a graph parameter depending on j that refers to the subgraphs of the graph and generalizes the notion of linkage we prove the min max theorem that this graph parameter which we call j linkage is equal to j width and we then give a polynomial time algorithm for computing it for constant j we also find tight lower and upper bounds for the j linkage equivalently the j width of graphs with given numbers of vertices and edges it is interesting to note that a lower bound for the width of a graph had been found by erdos as we show however that bound is not tight moreover we prove that our lower bound for width is also a tight lower bound for treewidth pathwidth and bandwidth graph parameters that may be arbitrarily larger than width finally we show that computing the j linkage is a p complete problem whereas we prove that approximating it is a threshold problem it is in nc for approximation factors 1 2

brief paper on the conservatism of the sum of squares method for analysis of time delayed systems in this paper we present a converse lyapunov result which shows that using polynomial lyapunov functionals to prove the stability of linear time delay systems is not conservative this result motivates the sum of squares approach to stability analysis of linear time delay systems this main result is based on an extension of the weierstrass approximation theorem to show that linear varieties of polynomials can be used to approximate linear varieties of the space of continuous functions

monotonicity in digraph search problems in this paper we study the monotonicity and complexity of five digraph search problems directed searching mixed directed searching internal directed searching internal strong searching and internal weak searching in the first three search problems both searchers and intruder must follow the edge directions when they move along edges in the internal strong search problem the intruder must move in the edge directions but searchers need not in the internal weak search problem searchers must move in the edge directions but the intruder need not there are three actions for searchers in the first two search problems placing removing and sliding and there are only two actions for searchers in the last three internal search problems placing and sliding note that the internal strong searching is a strong version of the internal directed searching the internal weak searching is a weak version of the internal directed searching and the internal edge searching is an analogy of the internal directed searching on undirected graphs we prove that the first three problems are monotonic and the last two problems are non monotonic respectively it is interesting that the internal directed searching is monotonic while the internal strong searching the internal weak searching and the internal edge searching are all non monotonic we also show that the first four problems are np complete and the last problem is np hard we solve the open problem on whether a non monotonic searching problem can be np complete

csp duality and trees of bounded pathwidth we study non uniform constraint satisfaction problems definable in monadic datalog stratified by the use of non linearity we show how such problems can be described in terms of homomorphism dualities involving trees of bounded pathwidth and in algebraic terms for this we introduce a new parameter for trees that closely approximates pathwidth and can be characterised via a hypergraph searching game