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

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

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

the complexity of coloring games on perfect graphs abstract in this paper we consider the following type of game two players must color the vertices of a given graph g v e in a prescribed order in such a way that no two adjacent vertices are colored with the same color in one variant the first player which is unable to move loses the game in another variant player 1 wins the game if and only if the game ends with all vertices colored in this paper we obtain several results on the complexity of the problem to decide whether there is a winning strategy for player 1 in a given game instance when g is restricted to split graphs interval graphs or bipartite graphs

classical locally finite tits chamber systems of rank 3 a chamber system v 9 j zi is over some index set z is a set of chambers together with partitions in z of v if ze z and c e v then bj is the join of all partitions fijzi e z and d c is the element of fij containing c d c is again a chamber system over j with fij restricted to z we say q is connected if d c furthermore 111 is the rank of v the most important examples of chamber systems are those obtained from groups if g is a group b a subgroup of g and xi ie z are subgroups of g containing b then 5 g b xi icl has as chambers the cosets bg g e g two such cosets bg bh being i adjacent if and only if xi g xih then g acts chamber transitively on with kernel b n bg and is connected if and only if g x 1 i e z furthermore each chamber system with transitive automorphism group g can be obtained in this way by setting b g and xi gd cr a rank 2 chamber system u hi fi is called a classical generalized ma gon mq 3 if is isomorphic to g b pi p where g is an essentially simple rank 2 group of lie type in the sense of cl b is a bore1 subgroup and p pj are the two maximal parabolic subgroups of g containing b here md is given by the relation vu wj wiw g 1 defined by the weyl group of g a rank 2 chamber system is a generalized digon if is isomorphic to v g pin p pi p where pi p are different proper subgroups of g satisfying g pipj pjpi a connected chamber system over z is called a classical tits chamber system if for each ce and i j c z di j c is either a generalized digon or a classical generalized m gon for some mi e n is focally finite if all d c are finite for motivation to consider such classical locally finite tits chamber systems see 15 and 16 if is a classical tits chamber system then the diagram d z is defined in 9 0021 8693 89 3 00

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

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

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

une generalisation de l inegalite de stein lorenzini soient k un corps de caracteristique quelconque k une cloture algebrique de k et p un polynome non constant de n 2 variables a coefficients dans k pour k on note n le nombre de facteurs irreductibles distincts f i d une decomposition de p sur k nous montrons l enonce suivant qui generalise des resultats de stein 1989 et lorenzini 1993 si le polynome p est non compose sur k alors n 1 est au plus egal a min ideg f i 1

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

parabolic systems the gf 3 case parabolic systems defined over gf q have been classified by timmesfeld for q 4 and by stroth for q 2 see references we deal with the case q 3

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

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

on the fibration method for zero cycles and rational points conjectures on the existence of zero cycles on arbitrary smooth projective varieties over number elds were proposed by colliot th el ene sansuc kato and saito in the 1980 s we prove that these conjectures are compatible with brations for brations into rationally connected varieties over a curve in particular they hold for the total space of families of homogeneous spaces of linear groups with connected geometric stabilisers we prove the analogous result for rational points conditionally on a conjecture on locally split values of polynomials which a recent work of matthiesen establishes in the case of linear polynomials over the rationals

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

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

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

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

irreducible values of polynomials abstract schinzel s hypothesis h is a general conjecture in number theory on prime values of polynomials that generalizes e g the twin prime conjecture and dirichlet s theorem on primes in arithmetic progression we prove a quantitative arithmetic analog of this conjecture for polynomial rings over pseudo algebraically closed fields this implies results over large finite fields via model theory a main tool in the proof is an irreducibility theorem a la hilbert

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

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