guard games on graphs a team of mobile agents called guards tries to keep an intruder out of an assigned area by blocking all possible attacks in a graph model for this setting the guards and the intruder are located on the vertices of a graph and they move from node to node via connecting edges the area protected by the guards is an induced subgraph of the given graph we investigate the algorithmic aspects of the guarding problem which is to find the minimum number of guards sufficient to patrol the area we show that the guarding problem is pspace hard and provide a set of approximation algorithms all approximation algorithms are based on the study of a variant of the game where the intruder must reach the guarded area in a single step in order to win this variant of the game appears to be a 2 approximation for the guarding problem and for graphs without cycles of length 5 the minimum number of required guards in both games coincides we give a polynomial time algorithm for solving the one step guarding problem in graphs of bounded treewidth and complement this result by showing that the problem is w1 hard parameterized by the treewidth of the input graph we also show that the problem is fixed parameter tractable fpt parameterized by the treewidth and maximum degree of the input graph finally we turn our attention to a large class of sparse graphs including planar graphs and graphs of bounded genus namely apex minor free graphs we prove that the one step guarding problem is fpt and possess a ptas on apex minor free graphs

on the maximum weight minimal separator abstract given an undirected and connected graph g v e and two vertices s t v a vertex subset s that separates s and t is called an s t separator and an s t separator is called minimal if no proper subset of s separates s and t moreover we say that a set s is a minimal separator of g if s is a minimal s t separator for some s and t in this paper we consider finding a minimal s t separator with maximum weight on a vertex weighted graph we first prove that these problems are np hard on the other hand we give an o tw o tw time deterministic algorithm based on tree decompositions where o is the order notation omitting the polynomial factor of n moreover we improve the algorithm by using the rank based approach and the running time is o 38 2 tw finally we give an o 9 tw w 2 time randomized algorithm to determine whether there exists a minimal s t separator where w is its weight and tw is the treewidth of g

zeros of l functions attached to maass forms this paper considers psl 2 automorphic cusp forms the maass forms and their associated dirichlet series there are two major results the first uses real variable techniques to give an explicit reconstruction of the cusp form in terms of the dirichlet series this representation is valid in the entire upper half plane the second uses this representation to show that the dirichlet series has many zeros on its critical line this is the analogue of the hardy littlewood result for the riemann zeta function

approximation of minimum weight spanners for sparse graphs a t spanner of a graph g is its spanning subgraph s such that the distance between every pair of vertices in s is at most t times their distance in g the sparsestt spanner problem asks to find for a given graph g and an integer t a t spanner of g with the minimum number of edges the problem is known to be np hard for all t 2 and even more it is np hard to approximate it with ratio o logn for every t 2 for t 5 the problem remains np hard for planar graphs and the approximability status of the problem on planar graphs was open we resolve this open issue by showing that the sparsestt spanner problem admits the efficient polynomial time approximation scheme eptas for every t 1 our result holds for a much wider class of graphs namely the class of apex minor free graphs which contains the classes of planar and bounded genus graphs moreover it is possible to extend our results to weighted apex minor free graphs when the maximum edge weight is bounded by some constant

parameterized and approximation complexity of partial vc dimension abstract we introduce the problem partial vc dimension that asks given a hypergraph h x e and integers k and l whether one can select a set c x of k vertices of h such that the set e c e e of distinct hyperedge intersections with c has size at least l the sets e c define equivalence classes over e partial vc dimension is a generalization of vc dimension which corresponds to the case l 2 k and of distinguishing transversal which corresponds to the case l e the latter is also known as test cover in the dual hypergraph we also introduce the associated fixed cardinality maximization problem max partial vc dimension that aims at maximizing the number of equivalence classes induced by a solution set of k vertices we study the algorithmic complexity of partial vc dimension and max partial vc dimension both on general hypergraphs and on more restricted instances in particular neighborhood hypergraphs of graphs

existential second order logic over graphs charting the tractability frontier fagin s theorem the first important result of descriptive complexity asserts that a property of graphs is in np if and only if it is definable by an existential second order formula in this article we study the complexity of evaluating existential second order formulas that belong to prefix classses of existential second order logic where a prefix class is the collection of all existential second order formulas in prenex normal form such that the second order and the first order quantifiers obey a certain quantifier pattern we completely characterize the computational complexity of prefix classes of existential second order logic in three different contexts 1 over directed graphs 2 over undirected graphs with self loops and 3 over undirected graphs without self loops our main result is that in each of these three contexts a dichotomy holds that is to say each prefix class of existential second order logic either contains sentences that can express np complete problems or each of its sentences expresses a polynomial time solvable problem although the boundary of the dichotomy coincides for the first two cases it changes as one moves to undirected graphs without self loops the key difference is that a certain prefix class based on the well known ackermann class of first order logic contains sentences that can express np complete problems over graphs of the first two types but becomes tractable over undirected graphs without self loops moreover establishing the dichotomy over undirected graphs without self loops turns out to be a technically challenging problem that requires the use of sophisticated machinery from graph theory and combinatorics including results about graphs of bounded tree width and ramsey s theorem

treewidth computations ii lower bounds for several applications it is important to be able to compute the treewidth of a given graph and to find tree decompositions of small width reasonably fast good lower bounds on the treewidth of a graph can amongst others help to speed up branch and bound algorithms that compute the treewidth of a graph exactly a high lower bound for a specific graph instance can tell that a dynamic programming approach for solving a problem is infeasible for this instance this paper gives an overview of several recent methods that give lower bounds on the treewidth of graphs

generalizing constraint satisfaction on trees hybrid tractability and variable elimination the constraint satisfaction problem csp is a central generic problem in artificial intelligence considerable progress has been made in identifying properties which ensure tractability in such problems such as the property of being tree structured in this paper we introduce the broken triangle property which allows us to define a novel tractable class for this problem which significantly generalizes the class of problems with tree structure we show that the broken triangle property is conservative i e it is preserved under domain reduction and hence under arc consistency operations and that there is a polynomial time algorithm to determine an ordering of the variables for which the broken triangle property holds or to determine that no such ordering exists we also present a non conservative extension of the broken triangle property which is also sufficient to ensure tractability and can also be detected in polynomial time we show that both the broken triangle property and its extension can be used to eliminate variables and that both of these properties provide the basis for preprocessing procedures that yield unique closures orthogonal to value elimination by enforcement of consistency finally we also discuss the possibility of using the broken triangle property in variable ordering heuristics

a complexity dichotomy for matching cut in bipartite graphs of fixed diameter abstract in a graph a matching cut is an edge cut that is a matching matching cut is the problem of deciding whether or not a given graph has a matching cut which is known to be np complete even when restricted to bipartite graphs it has been proved that matching cut is polynomially solvable for graphs of diameter two in this paper we show that for any fixed integer d 3 matching cut is np complete in the class of graphs of diameter d this resolves an open problem posed by borowiecki and jesse jozefczyk 2008 6 we then show that for any fixed integer d 4 matching cut is np complete even when restricted to the class of bipartite graphs of diameter d complementing the hardness results we show that matching cut is polynomial time solvable in the class of bipartite graphs of diameter at most three and point out a new and simple polynomial time algorithm solving matching cut in graphs of diameter 2

weight functions for chebyshev quadrature la fonction poids 1 x 2 1 2 r x ou r x est une fonction rationnelle d ordre 1 1 est etudiee il s agit de determiner si elle admet des quadratures de tchebychev

towards fixed parameter tractable algorithms for abstract argumentation abstract argumentation frameworks have received a lot of interest in recent years most computational problems in this area are intractable but several tractable fragments have been identified in particular dunne showed that many problems can be solved in linear time for argumentation frameworks of bounded tree width however these tractability results which were obtained via courcelle s theorem do not directly lead to efficient algorithms the goal of this paper is to turn the theoretical tractability results into efficient algorithms and to explore the potential of directed notions of tree width for defining larger tractable fragments as a by product we will sharpen some known complexity results

simple zeros of the ramanujan τ dirichlet series it is known that l s has all of its non real zeros in the strip l l 2 t 6 t l simple zeros in the region 0 t t the methods of these papers however will probably not give any results for l s the levinson method requires a mean value theorem for the square of the modulus of s multiplied by a dirichlet polynomial the length of this polynomial is critical and it seems unlikely that such a theorem with a long enough polynomial could be proven at this time in the case of l as far as the method of 1 is concerned we relied on a factorization of the dedekind zeta function as a product of the

computing the numbers of independent sets and matchings of all sizes for graphs with bounded treewidth abstract in the theory and applications of graphs it is a basic problem to compute the numbers of independent sets and matchings of given sizes since the problem of computing the total number of independent sets and that of matchings of graphs is p complete it is unlikely to give efficient algorithms to find the numbers of independent sets and matchings of given sizes in this paper for graphs with order n and treewidth at most p we present two dynamic algorithms to compute the numbers of independent sets of all sizes with runtime o 2p pn3 and the numbers of matchings of all sizes with runtime o 22p pn3 respectively by the algorithms presented in this paper for graphs with small treewidths the numbers of independent sets and matchings of all possible sizes can be computed efficiently

on mean values of mollifiers and l functions associated to primitive cusp forms we study the second moment of the l function associated to a holomorphic primitive cusp form of even weight perturbed by a new family of mollifiers this family is a natural extension of the mollifers considered by conrey and by bui conrey and young as an application we improve the current lower bound on critical zeros of holomorphic primitive cusp forms

parallel algorithms with optimal speedup for bounded treewidth we describe the first parallel algorithm with optimal speedup for constructing minimum width tree decompositions of graphs of bounded treewidth on n vertex input graphs the algorithm works in o log n 2 time using o n operations on the erew pram we also give faster parallel algorithms with optimal speedup for the problem of deciding whether the treewidth of an input graph is bounded by a given constant and for a variety of problems on graphs of bounded treewidth including all decision problems expressible in monadic second order logic on n vertex input graphs the algorithms use o n operations together with o log n log n time on the erew pram or o log n time on the crcw pram

msol restricted contractibility to planar graphs abstract we study the computational complexity of graph planarization via edge contraction the problem contract asks whether there exists a set s of at most k edges that when contracted produces a planar graph we work with a more general problem called p restrictedcontract in which s in addition is required to satisfy a fixed msol formula p s g we give an fpt algorithm in time o n 2 f k which solves p restrictedcontract where n is number of vertices of the graph and p s g is i inclusion closed and ii inert contraction closed where inert edges are the edges non incident to any inclusion minimal solution s as a specific example we can solve the l subgraph contractibility problem in which the edges of the set s are required to form disjoint connected subgraphs of size at most l this problem can be solved in time o n 2 f k l using the general algorithm we also show that for l 2 the problem is np complete

compatibility of unrooted phylogenetic trees is fpt a collection of t1 t2 tk of unrooted leaf labelled phylogenetic trees all with different leaf sets is said to be compatible if there exists a tree t such that each tree ti can be obtained from t by deleting leaves and contracting edges determining compatibility is np hard and the fastest algorithm to date has worst case complexity of around nk time n being the number of leaves here we present an o nf k algorithm proving that compatibility of unrooted phylogenetic trees is fixed parameter tractable fpt with respect to the number k of trees

the recognizability of sets of graphs is a robust property once the set of finite graphs is equipped with an algebra structure arising from the definition of operations that generalize the concatenation of words one can define the notion of a recognizable set of graphs in terms of finite congruences applications to the construction of efficient algorithms and to the theory of context free sets of graphs follow naturally the class of recognizable sets depends on the signature of graph operations we consider three signatures related respectively to hyperedge replacement hr context free graph grammars to vertex replacement vr context free graph grammars and to modular decompositions of graphs we compare the corresponding classes of recognizable sets we show that they are robust in the sense that many variants of each signature where in particular operations are defined by quantifier free formulas a quite flexible framework yield the same notions of recognizability we prove that for graphs without large complete bipartite subgraphs hr recognizability and vr recognizability coincide the same combinatorial condition equates hr context free and vr context free sets of graphs inasmuch as possible results are formulated in the more general framework of relational structures

trees grids and mso decidability from graphs to matroids monadic second order mso logic has proved to be a useful tool in many areas of application reaching flom decidability and complexity to picture processing correctness of programs and parallel processes to characterize the structural borderline between decidability and undecidability is a classical research problem here this problem is related to questions in computational complexity especially to the model checking problem for which many tools developed in the area of decidability have proved to be useful for more than two decades it was conjectured in d seese the structure of the models of decidable monadic theories of graphs ann pure appl logic 53 1991 169 195 that decidability of monadic theories of countable structures implies that the theory can be reduced via interpretability to a theory of trees it is one of the main goals of this article to prove a variant of this conjecture for matroids representable over a finite field matroids can be viewed as a wide generalization of graphs and they seem to capture some second order properties in a more suitable way than graphs themselves cf the recent development in matroid structure theory j f geelen a h m gerards g p whittle branch width and well quasi ordering in matroids and graphs j combin theory ser b 84 2002 270 290 j f geelen a h m gerards n robertson g p whittle excluding a planar graph from a gf q representable matroid manuscript 2003 more exactly we prove for every finite field f that any class of f representable matroids with a decidable mso theory must have uniformly bounded branch width moreover we show that bounding the branch width of all matroids in general is not sufficient to obtain a decidable mso theory our paper gives a rather detailed introduction to these different subjects and shows that a blend of ideas and methods from logic together with structural matroid theory can lead to new tools and algorithms and can shed light on some old open problems

incremental list coloring of graphs parameterized by conservation incrementally k list coloring a graph means that a graph is given by adding vertices step by step and for each intermediate step we ask for a vertex coloring such that each vertex has one of the colors specified by its associated list containing some of in total k colors we introduce the conservative version of this problem by adding a further parameter c n specifying the maximum number of vertices to be recolored between two subsequent graphs differing by one vertex the conservation parameter c models the natural quest for a modest evolution of the coloring in the course of the incremental process instead of performing radical changes we show that even on bipartite graphs the problem is np hard for k 3 and w 1 hard for an unbounded number of colors when parameterized by c in contrast also on general graphs the problem becomes fixed parameter tractable with respect to the combined parameter k c we prove that the problem has an exponential size kernel with respect to k c and there is no polynomial size kernel unless np conp poly furthermore we investigate the parameterized complexity on various subclasses of perfect graphs we show fixed parameter tractability for the combined parameter treewidth and number k of colors finally we provide empirical findings on the practical relevance of our approach in terms of an effective graph coloring heuristic

minimal interval completion through graph exploration given an arbitrary graph g v e and an interval graph h v f with email protected f we say that h is an interval completion of g the graph h is called a minimal interval completion of g if for any sandwich graph h v f with email protected f f h is not an interval graph in this paper we give a o nm time algorithm computing a minimal interval completion of an arbitrary graph the output is an interval model of the completion

tree projections and structural decomposition methods minimality and game theoretic characterization tree projections provide a mathematical framework that encompasses all the various purely structural decomposition methods that have been proposed in the literature to single out classes of nearly acyclic hyper graphs such as the tree decomposition method which is the most powerful decomposition method on graphs and the generalized hypertree decomposition method which is its natural counterpart on arbitrary hypergraphs the paper analyzes this framework by focusing in particular on minimal tree projections that is on tree projections without useless redundancies first it is shown that minimal tree projections enjoy a number of properties that are usually required for normal form decompositions in various structural decomposition methods in particular they enjoy the same kind of connection properties as minimal tree decompositions of graphs with the result being tight in the light of the negative answer that is provided to the open question about whether they enjoy a slightly stronger notion of connection property defined to speed up the computation of hypertree decompositions second it is shown that tree projections admit a natural game theoretic characterization in terms of the robber and captain game in this game as for the robber and cops game characterizing tree decompositions the existence of winning strategies implies the existence of monotone ones as a special case the robber and captain game can be used to characterize the generalized hypertree decomposition method where such a game theoretic characterization was missing and asked for besides their theoretical interest these results have immediate algorithmic applications both for the general setting and for structural decomposition methods that can be recast in terms of tree projections

increasing the minimum degree of a graph by contractions abstract the degree contractibility problem is to test whether a given graph g can be modified to a graph of minimum degree at least d by using at most k contractions we prove the following three results first degree contractibility is np complete even when d 14 second it is fixed parameter tractable when parameterized by k and d third it is w 1 hard when parameterized by k we also study its variant where the input graph is weighted i e has some edge weighting and the contractions preserve these weights the weighted degree contractibility problem is to test if a weighted graph g can be contracted to a weighted graph of minimum weighted degree at least d by using at most k weighted contractions we show that this problem is np complete and that it is fixed parameter tractable when parameterized by k in addition we pinpoint a relationship with the problem of finding a minimal edge cut of maximum size in a graph and study the parameterized complexity of this problem and its variants

on weight functions admitting chebyshev quadrature in this paper we prove the existence of chebyshev quadrature for three new weight functions which are quite different from the two known examples given by ullman 15 and byrd and stalla 2 in particular we indicate a simple method to construct weight functions for which there exist infinitely many chebyshev quadrature rules

constant factor approximations of branch decomposition and largest grid minor of planar graphs in o n 1 time we give constant factor approximation algorithms for computing the optimal branch decompositions and largest grid minors of planar graphs for a planar graph g with n vertices let bw g be the branchwidth of g and gm g the largest integer g such that g has a gxg grid as a minor let c 1 be a fixed integer and a b arbitrary constants satisfying a c 1 and b 2c 1 we give an algorithm which constructs in o n 1 1 clogn time a branch decomposition of g with width at most abw g we also give an algorithm which constructs a gxg grid minor of g with g gm g b in o n 1 1 clogn time the constants hidden in the big o notations are proportional to c a c 1 and c b 2c 1 respectively

chebyshev quadrature rules for a new class of weight functions proof is given that the weight functions w x p i fr p x x x i x on 0 1 admit chebyshev quadratures for any fixed p 1 and every n for the particular cases when p i and p 2 the nodes are tabulated to ten decimal places for n point rules with n 2 4 6 8 and 12 numerical tables are also given for a coefficient in the expression of the error term

polynomial kernels for hitting forbidden minors under structural parameterizations abstract we investigate polynomial time preprocessing for the problem of hitting forbidden minors in a graph using the framework of kernelization for a fixed finite set of connected graphs f the f deletion problem is the following given a graph g and integer k is it possible to delete k vertices from g to ensure the resulting graph does not contain any graph from f as a minor earlier work by fomin lokshtanov misra and saurabh focs 12 showed that when f contains a planar graph an instance g k can be reduced in polynomial time to an equivalent one of size k o 1 in this work we focus on structural measures of the complexity of an instance with the aim of giving nontrivial preprocessing guarantees for instances whose solutions are large motivated by several impossibility results we parameterize the f deletion problem by the size of a vertex modulator whose removal results in a graph of constant treedepth we prove that for each set f of connected graphs and constant the f deletion problem parameterized by the size of a treedepth modulator has a polynomial kernel our kernelization is fully explicit and does not depend on protrusion reduction or well quasi ordering which are sources of algorithmic non constructivity in earlier works on f deletion our main technical contribution is to analyze how models of a forbidden minor in a graph g with modulator x interact with the various connected components of g x using the language of labeled minors we analyze the fragments of potential forbidden minor models that can remain after removing an optimal f deletion solution from a single connected component of g x by bounding the number of different types of behavior that can occur by a polynomial in x we obtain a polynomial kernel using a recursive preprocessing strategy our results extend earlier work for specific instances of f deletion such as vertex cover and feedback vertex set it also generalizes earlier preprocessing results for f deletion parameterized by a vertex cover which is a treedepth one modulator

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

a linear time algorithm for finding tree decompositions of small treewidth in this paper we give for constant k a linear time algorithm that given a graph g v e determines whether the treewidth of g is at most k and if so finds a tree decomposition of g with treewidth at most k a consequence is that every minor closed class of graphs that does not contain all planar graphs has a linear time recognition algorithm another consequence is that a similar result holds when we look instead for path decompositions with pathwidth at most some constant k

treewidth distance on phylogenetic trees in this article we study the treewidth of the display graph an auxiliary graph structure obtained from the fusion of phylogenetic i e evolutionary trees at their leaves earlier work has shown that the treewidth of the display graph is bounded if the trees are in some formal sense topologically similar here we further expand upon this relationship we analyse a number of reduction rules commonly used in the phylogenetics literature to obtain fixed parameter tractable algorithms in some cases the subtree reduction the reduction rules behave similarly with respect to treewidth while others the cluster reduction behave very differently and the behaviour of the chain reduction is particularly intriguing because of its link with graph separators and forbidden minors we also show that the gap between treewidth and tree bisection and reconnect tbr distance can be infinitely large and that unlike for example planar graphs the treewidth of the display graph can be as much as linear in its number of vertices a number of other auxiliary results are given we conclude with a discussion and list a number of open problems

three connected graphs whose maximum nullity is at most three for a graph g v e with vertex set v 1 2 n let s g be the set of all n n real valued symmetric matrices a which represent g the maximum nullity of a graph g denoted by m g is the largest possible nullity of any matrix a s g fiedler showed that a graph g has m g 1 if and only if g is a path johnson et al gave a characterization of all graphs g with m g 2 independently hogben and van der holst gave a characterization of all 2 connected graphs with m g 2 in this paper we show that k connected graphs g have m g k that k connected partial k graphs g have m g k and that for 3 connected graphs g m g 3 if and only if g is a partial 3 path

boundary classes for graph problems involving non local properties abstract we continue the study of boundary classes for np hard problems and focus on seven np hard graph problems involving non local properties hamiltonian cycle hamiltonian cycle through specified edge hamiltonian path feedback vertex set connected vertex cover connected dominating set and graph vc con dimension our main result is the determination of the first boundary class for feedback vertex set we also determine boundary classes for hamiltonian cycle through specified edge and hamiltonian path and give some insights on the structure of some boundary classes for the remaining problems

kernel bounds for path and cycle problems connectivity problems like k path and k disjoint paths relate to many important milestones in parameterized complexity namely the graph minors project color coding and the recent development of techniques for obtaining kernelization lower bounds this work explores the existence of polynomial kernels for various path and cycle problems by considering nonstandard parameterizations we show polynomial kernels when the parameters are a given vertex cover a modulator to a cluster graph or a promised max leaf number we obtain lower bounds via cross composition e g for hamiltonian cycle and related problems when parameterized by a modulator to an outerplanar graph

deciding first order properties of locally tree decomposable structures we introduce the concept of a class of graphs or more generally relational structures being locally tree decomposable there are numerous examples of locally tree decomposable classes among them the class of planar graphs and all classes of bounded valence or of bounded tree width we also consider a slightly more general concept of a class of structures having bounded local tree width we show that for each property of structures that is definable in first order logic and for each locally tree decomposable class c of structures there is a linear time algorithm deciding whether a given structure a c has property for classes c of bounded local tree width we show that for every k 1 there is an algorithm solving the same problem in time o n1 1 k where n is the cardinality of the input structure

default logic and bounded treewidth abstract in this paper we study reiter s propositional default logic when the treewidth of a certain graph representation semi primal graph of the input theory is bounded we establish a dynamic programming algorithm on tree decompositions that decides whether a theory has a consistent stable extension cons our algorithm can even be used to enumerate all generating defaults enumsd that lead to stable extensions we show that our algorithm decides cons in linear time in the input theory and triple exponential time in the treewidth further our algorithm solves enumsd with a pre computation step that is linear in the input theory and triple exponential in the treewidth followed by a linear delay to output solutions finally we take the exponential time hypothesis eth into consideration and establish lower bounds of bounded treewidth algorithms for cons

turan inequalities and zeros of dirichlet series associated with certain cusp forms the turan inequalities are a countably infinite set of conditions about the power series coefficients of certain entire functions which are necessary in order for the function to have only real zeros we give a one parameter family of generalized dirichlet series each with functional equation for which the turan inequalities hold for the associated 4 function normalized so that the critical line is the real axis for a discrete set of values of the parameter the dirichlet series has an euler product and is the l series associated to a modular form for these we expect the analogue of the riemann hypothesis to hold for the rest of the values of the parameter we do not expect an analogue of the riemann hypothesis we show for one particular value of the parameter that the dirichlet series in fact has zeros within the region of absolute convergence 1 riemann s integral representation for the function one approach to the study of the zeros of the riemann zeta function begins with the integral representation for riemann s 4 function the function is given by s is s l f s21f s 2 c s where 4 is riemann s zeta function and s a it is a complex variable the mellin transform of 4 involves the theta function h z e n z 2 n oo where e z exp 27riz we have s 2f 4 a00 v x xs 2 2 jox where u e 2 0 iu i yiku e n 2 n 1 using the transformation formula for h h l z z1 26 z received by the editors april 10 1992 1991 mathematics subject classification primary 1 1f66 secondary 30c1 5 research of the second author supported in part by the alfred p sloan foundation research of both authors supported in part by a grant from the nsf 1994 american mathematical society 0002 9947 94 1 00 25 per page

on the complexity of constrained nash equilibria in graphical games a widely accepted rational behavior for non cooperative players is based on the notion of nash equilibrium although the existence of a nash equilibrium is guaranteed in the mixed framework i e when players select their actions in a randomized manner in many real world applications the existence of any equilibrium is not enough rather it is often desirable to single out equilibria satisfying some additional requirements in order for instance to guarantee a minimum payoff to certain players which we call constrained nash equilibria in this paper a formal framework for specifying these kinds of requirement is introduced and investigated in the context of graphical games where a player p may directly be interested in some of the other players only called the neighbors of p this setting is very useful for modeling large population games where typically each player does not directly depend on all the players and representing her utility function extensively is either inconvenient or infeasible based on this framework the complexity of deciding the existence and of computing constrained equilibria is then investigated in the light of evidencing how the intrinsic difficulty of these tasks is affected by the requirements prescribed at the equilibrium and by the structure of players interactions the analysis is carried out for the setting of mixed strategies as well as for the setting of pure strategies i e when players are forced to deterministically choose the action to perform in particular for this latter case restrictions on players interactions and on constraints are identified that make the computation of nash equilibria an easy problem for which polynomial and highly parallelizable algorithms are presented

on the parameterized complexity of the geodesic hull number abstract several recent papers obtained complexity results regarding the geodesic hull number hn gd g of a graph g in this paper we prove that determining whether hn gd g k is w 2 hard parameterized by k in diameter two graphs and is w 1 hard parameterized by tw k where tw is the treewidth of g despite this for graphs with bounded treewidth tw we prove that hn gd g is computable in polynomial time o tw 2 tw 5 n 2 tw 5

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

efficient algorithms for center problems in cactus networks efficient algorithms for solving the center problems in weighted cactus networks are presented in particular we have proposed the following algorithms for the weighted cactus networks of size n an o nlogn time algorithm to solve the 1 center problem and an o nlog 3n time algorithm to solve the weighted continuous 2 center problem we have also provided improved solutions to the general p center problems in cactus networks the developed ideas are then applied to solve the obnoxious 1 center problem in weighted cactus networks

query evaluation via tree decompositions a number of efficient methods for evaluating first order and monadic second order queries on finite relational structures are based on tree decompositions of structures or queries we systematically study these methods in the first part of the article we consider arbitrary formulas on tree like structures we generalize a theorem of courcelle 1990 by showing that on structures of bounded tree width a monadic second order formula with free first and second order variables can be evaluated in time linear in the structure size plus the size of the output in the second part we study tree like formulas on arbitrary structures we generalize the notions of acyclicity and bounded tree width from conjunctive queries to arbitrary first order formulas in a straightforward way and analyze the complexity of evaluating formulas of these fragments moreover we show that the acyclic and bounded tree width fragments have the same expressive power as the well known guarded fragment and the finite variable fragments of first order logic respectively

dominating set is fixed parameter tractable in claw free graphs we show that the dominating set problem parameterized by solution size is fixed parameter tractable fpt in graphs that do not contain the claw k 1 3 the complete bipartite graph on four vertices where the two parts have one and three vertices respectively as an induced subgraph we present an algorithm that uses 2 o k 2 n o 1 time and polynomial space to decide whether a claw free graph on n vertices has a dominating set of size at most k note that this parameterization of dominating set is w 2 hard on the set of all graphs and thus is unlikely to have an fpt algorithm for graphs in general the most general class of graphs for which an fpt algorithm was previously known for this parameterization of dominating set is the class of k i j free graphs which exclude for some fixed i j n the complete bipartite graph k i j as a subgraph for i j 2 the class of claw free graphs and any class of k i j free graphs are not comparable with respect to set inclusion we thus extend the range of graphs over which this parameterization of dominating set is known to be fixed parameter tractable we also show that in some sense it is the presence of the claw that makes this parameterization of the dominating set problem hard more precisely we show that for any t 4 the dominating set problem parameterized by the solution size is w 2 hard in graphs that exclude the t claw k 1 t as an induced subgraph our arguments also imply that the related connected dominating set and dominating clique problems are w 2 hard in these graph classes finally we show that for any t n the clique problem parameterized by solution size which is w 1 hard on general graphs is fpt in t claw free graphs our results add to the small and growing collection of fpt results for graph classes defined by excluded subgraphs rather than by excluded minors

exact algorithms for treewidth and minimum fill in we show that the treewidth and the minimum fill in of an n vertex graph can be computed in time mathcal o 1 8899 n our results are based on combinatorial proofs that an n vertex graph has mathcal o 1 7087 n minimal separators and mathcal o 1 8135 n potential maximal cliques we also show that for the class of asteroidal triple free graphs the running time of our algorithms can be reduced to mathcal o 1 4142 n

a linear kernel for finding square roots of almost planar graphs abstract a graph h is a square root of a graph g if g can be obtained from h by the addition of edges between any two vertices in h that are at distance 2 from each other the square root problem is that of deciding whether a given graph admits a square root we consider this problem for planar graphs in the context of the distance from triviality framework for an integer k a planar k v graph or k apex graph is a graph that can be made planar by the removal of at most k vertices we prove that a generalization of square root in which some edges are prescribed to be either in or out of any solution has a kernel of size o k for planar k v graphs when parameterized by k our result is based on a new edge reduction rule which as we shall also show has a wider applicability for the square root problem