on the descending central sequence of absolute galois groups let p be an odd prime number and f a field containing a primitive p th root of unity we prove a new restriction on the group theoretic structure of the absolute galois group g f of f namely the third subgroup g f 3 in the descending p central sequence of g f is the intersection of all open normal subgroups n such that g f n is 1 bbb z p 2 or the extra special group m p 3 of order p 3 and exponent p 2

optimal approximation of sparse hessians and its equivalence to a graph coloring problem we consider the problem of approximating the hessian matrix of a smooth non linear function using a minimum number of gradient evaluations particularly in the case that the hessian has a known fixed sparsity pattern we study the class of direct methods for this problem and propose two new ways of classifying direct methods examples are given that show the relationships among optimal methods from each class the problem of finding a non overlapping direct cover is shown to be equivalent to a generalized graph coloring problem the distance 2 graph coloring problem a theorem is proved showing that the general distance k graph coloring problem is np complete for all fixedk 2 and hence that the optimal non overlapping direct cover problem is also np complete some worst case bounds on the performance of a simple coloring heuristic are given an appendix proves a well known folklore result which gives lower bounds on the number of gradient evaluations needed in any possible approximation method

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

optimal estimation of jacobian and hessian matrices that arise in finite difference calculations in this paper the problem of estimating jacobian and hessian matrices arising in the finite difference approximation of partial differential equations is considered using the notion of computational molecule or stencil schemes are developed that require the minimum number of differences to estimate these matrices a procedure applicable to more complicated structures is also given 1 introduction in the past few years there has been a growing interest in studying efficient ways to obtain good estimates of sparse jacobian or hessian matrices two basic approaches have been followed in the first approach it is assumed that one has an estimate of the jacobian hessian which one wishes to revise given arbi trary changes in the variables and the corresponding changes in the functions gradients this has led to the development of several sparse quasi newton methods see 11 14 12 for example in the second approach it is assumed that one can specify the changes in the variables one then forms an estimate of the jacobian hessian matrix from the observed changes in the functions gradients when the dimension of the matrix is large and the evaluation of each function gradient vector is expensive it is necessary to take the sparsity structure of the matrix into account in order to make this approach efficient several methods for doing this for arbitrary sparsity structures have already appeared in the literature 2 5 7 10 13 in this paper we consider the second approach for cases where the sparsity structure arises from finite difference approximations to partial differential equa tions both symmetric and unsymmetric matrices are considered although the algorithms presented do not generalize to every pattern generated by finite dif ferences they cover the most useful structures and in some cases extend existing methods to more complex situations in the next section we describe the method of curtis powell and reid 5 henceforth referred to as the cpr method for estimating sparse jacobian matrices we then show how the so called computational molecule or stencil 1 of the finite difference operator associated with the jacobian matrix under study can be used to determine the groupings i e function differencing directions used by the

restrictions of graph partition problems let j be a set of n jobs all with job processing time or job length one and let m be a set of k machines incompatibilities between jobs are described by an undirected graph g j e with vertex set j if g contains an edge between two jobs j and j j j we demand that these two jobs can not be executed by the same machine an assignment of the jobs to the machines which satisfies the incompatibility relation is called a schedule the processing time of a machine is given by the number of assigned jobs since each job has processing time one

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

graphical methods for efficient likelihood inference in gaussian covariance models in graphical modelling a bi directed graph encodes marginal independences among random variables that are identified with the vertices of the graph we show how to transform a bi directed graph into a maximal ancestral graph that i represents the same independence structure as the original bi directed graph and ii minimizes the number of arrowheads among all ancestral graphs satisfying i here the number of arrowheads of an ancestral graph is the number of directed edges plus twice the number of bi directed edges in gaussian models this construction can be used for more efficient iterative maximization of the likelihood function and to determine when maximum likelihood estimates are equal to empirical counterparts

induced subgraph isomorphism the complexity of the subgraph isomorphism problem where the pattern graph is of fixed size is well known to depend on the topology of the pattern graph here we present two results which in contrast provide evidence that no topology of an induced subgraph of fixed size can be substantially easier to detect or count than an independent set of related size we show that any fixed pattern graph having a maximum independent set of size k that is disjoint from other maximum independent sets is not easier to detect as an induced subgraph than an independent set of size k it follows in particular that an induced path on 2 k 1 vertices is not easier to detect than an independent set on k vertices and that an induced cycle on 2k vertices is not easier to detect than an independent set on k vertices in view of linear time upper bounds on the detection of induced path of length two and three our lower bound is tight similar corollaries hold for the detection of induced complete bipartite graphs and an induced paw and its generalizations we show also that for an arbitrary pattern graph h on k vertices with no isolated vertices there is a simple subdivision of h resulting from splitting each edge into a path of length four and attaching a distinct path of length three at each vertex of degree one that is not easier to detect or count than an independent set on k vertices respectively next we show that the so called diamond and its generalizations on k vertices are not easier to detect as induced subgraphs than an independent set on three vertices or an independent set on k vertices respectively for c 4 we give a weaker evidence of its hardness in terms of an independent set on three vertices finally we derive several results relating the complexity of the edge colored variant of induced subgraph isomorphism to that of the standard variant

recognizing circle graphs in polynomial time the main result of this paper is an 0 v x e time algorithm for deciding whether a given graph is a circle graph that is the intersection graph of a set of chords on a circle the algorithm utilizes two new graph theoretic results regarding necessary induced subgraphs of graphs having neither articulation points nor similar pairs of vertices furthermore as a substep of the algorithm it is shown how to find in 0 v x e time a decomposition of a graph into prime graphs thereby improving on a result of cunningham

copolarity of isometric actions we introduce a new integral invariant for isometric actions of compact lie groups the copolarity roughly speaking it measures how far from being polar the action is we generalize some results about polar actions in this context in particular we develop some of the structural theory of copolarity k representations we classify the irreducible representations of copolarity one and we relate the copolarity of an isometric action to the concept of variational completeness in the sense of bott and sainelson

on the double roman domination of graphs abstract a double roman dominating function of a graph g is a labeling f v g 0 1 2 3 such that if f v 0 then the vertex v must have at least two neighbors labeled 2 under f or one neighbor with f w 3 and if f v 1 then v must have at least one neighbor with f w 2 the double roman domination number dr g of g is the minimum value of v v g f v over such functions in this paper we firstly give some bounds of the double roman domination numbers of graphs with given minimum degree and graphs of diameter 2 and further we get that the double roman domination numbers of almost all graphs are at most n then we obtain sharp upper and lower bounds for d r g d r g moreover a linear time algorithm for the double roman domination number of a cograph is given and a characterization of the double roman cographs is provided those results partially answer two open problems posed by beeler et al 2016

note on the complexity of deciding the rainbow vertex connectedness for bipartite graphs a path in an edge colored graph is said to be a rainbow path if no two edges on the path share the same color an edge colored graph is strongly rainbow connected if there exists a rainbow geodesic path between every pair of vertices the strong rainbow connection number of g denoted by scr g respectively rc g is the smallest number of colors that are needed in order to make g strongly rainbow connected a vertex colored graph g is rainbow vertex connected if any pair of vertices in g are connected by a path whose internal vertices have distinct colors the rainbow vertex connection number of a connected graph g denoted by rvc g is the smallest number of colors that are needed in order to make g rainbow vertex connected though for a general graph g it is np complete to decide whether rc g 2 or rvc g 2 in this paper we show that the problem becomes easy when g is a bipartite graph whereas deciding whether rc g 3 or rvc g 3 is still np complete even when g is a bipartite graph moreover it is known that deciding whether a given edge vertex colored with an unbound number of colors graph is rainbow vertex connected is np complete we will prove that it is still np complete even when the edge vertex colored graph is bipartite we also show that a few np hard problems on rainbow connection are indeed np complete

a characterization of c fields via galois groups abstract it is known that those fields f which are of class c in the sense of r ware can be characterized by their witt ring structure as well as by the behavior of their binary quadratic forms and the structure of their galois 2 extensions in this paper we show that c fields which have at least 8 square classes are characterized by the absence of a certain group dc of order 16 as a galois group over them moreover we show that the level of such a field is determined by the presence or absence of the cyclic group of order 4 and the dihedral group of order 8 as galois groups over it these results provide an interesting continuation of the work of ware in the appendix we give an explicit construction of field extensions with galois group dc

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

proper connection number and connected dominating sets the proper connection number pc g of a connected graph g is defined as the minimum number of colors needed to color its edges so that every pair of distinct vertices of g is connected by at least one path in g such that no two adjacent edges of the path are colored the same and such a path is called a proper path in this paper we show that for every connected graph with diameter 2 and minimum degree at least 2 its proper connection number is at most 3 and the bound is sharp then we give an upper bound 3 n 1 1 of the proper connection number for every connected graph of order n 4 and minimum degree we also show that for every connected graph g the proper connection number pc g is upper bounded by pc g d 2 where d is a connected two way two step dominating set of g bounds of the form pc g 4 or pc g 2 for many special graph classes follow as easy corollaries from this result which include connected interval graphs asteroidal triple free graphs circular arc graphs threshold graphs and chain graphs all with minimum degree at least 2 furthermore we get the sharp upper bound 3 for the proper connection numbers of interval graphs and circular arc graphs through analyzing their structures

incremental modular decomposition modular decomposition is a form of graph decomposition that has been discovered independently by researchers in graph theory game theory network theory and other areas this paper reduces the time needed to find the modular decomposition of a graph from o n 3 to o n 2 together with a new algorithm for transitive orientation given in 21 this leads to fast new algorithms for a number of problems in graph recognition and isomorphism including recognition of comparability graphs and permutation graphs the new algorithm works by inserting each vertex successively into the decomposition tree using o n time to insert each vertex

estimation of sparse jacobian matrices and graph coloring blems given a mapping with a sparse jacobian matrix the problem of minimizing the number of function evaluations needed to estimate the jacobian matrix by differences is investigated this problem can be attacked as a graph coloring problem and this approach leads to very efficient algorithms the behavior of these algorithms is studied and in particular it is proved that two of the algorithms are optimal for band graphs numerical evidence is presented which indicates that these two algorithms are nearly optimal on practical problems

on switching classes nlc width cliquewidth and treewidth in this paper we consider a connection between switching of undirected graphs and the notions of nlc width cliquewidth and treewidth in particular we show that the nlc widths and the cliquewidths of two graphs in a switching class are at most a constant factor apart 2 for the former 4 for the latter a similar result can be shown not to hold for treewidth it is easy to find a switching classes in which the distance between the lowest treewidth and the highest is dependent on the number of vertices of the graph we also show that for nlc width every width between the lowest and the highest of the switching class is attained by some graph in that switching class we prove that this also holds for treewidth

acyclic and star coloring of p4 reducible and p4 sparse graphs an acyclic coloring of a graph g is a proper vertex coloring such that g contains no bicolored cycles the more restricted notion of star coloring of g is an acyclic coloring in which each path of length 3 is not bicolored in this paper we mainly study on the acyclic and star coloring of p4 reducible and p4 sparse graphs moreover we list polynomial time algorithms for giving an optimal acyclic or star coloring of a p4 reducible or p4 sparse graph

strong triadic closure in cographs and graphs of low maximum degree abstract the maxstc problem is an assignment of the edges with two types of labels namely strong and weak that maximizes the number of strong edges such that any two vertices that have a common neighbor with a strong edge are adjacent the cluster deletion problem seeks for the minimum number of edge removals of a given graph such that the remaining graph is a disjoint union of cliques both problems are known to be np hard and an optimal solution for the cluster deletion problem provides a feasible solution for the maxstc problem however not necessarily an optimal one in this work we conduct the first systematic study that reveals graph families for which the optimal solutions for maxstc and cluster deletion coincide we first show that maxstc coincides with cluster deletion on cographs and thus maxstc is solvable in polynomial time on cographs as a side result we give an interesting computational characterization of the maximum independent set on the cartesian product of two cographs furthermore we address the influence of the low degree bounds to the complexity of the maxstc problem we show that this problem is polynomial time solvable on graphs of maximum degree three whereas maxstc becomes np complete on graphs of maximum degree four the proof of the latter result implies that there is no subexponential time algorithm for maxstc unless the exponential time hypothesis fails

polar coordinates induced by actions of compact lie groups let g be a connected lie subgroup of the real orthogonal group 0 n for the action of g on r we construct linear subspaces a that intersect all orbits we determine for which g there exists such an a meeting all the g orbits orthogo nally groups that act transitively on spheres are obvious examples with few exceptions all possible g arise as the isotropy subgroups of riemannian symmetric spaces introduction let g be a compact lie group acting on a real vectorspace v and let k be a g invariant inner product on v having a linear cross section a c v of

the topology of isoparametric submanifolds it has been known since a famous paper of bott and samelson that using morse theory the homology and cohomology of certain homogeneous spaces can be computed algorithmically from dynkin diagram and multiplicity data l conlon and j dadok noted that these spaces are the orbits of the isotropy representations of symmetric spaces recently the theory of isoparametric hypersurfaces has been generalized to a theory of isoparametric submanifolds of arbitrary codimension in euclidean space and these same orbits turn out to be exactly the homogeneous examples even the nonhomogeneous examples have associated to them weyl groups with dynkin diagrams marked with multiplicities we extend and simplify the bott samelson method to compute the homology and cohomology of isoparametric submanifolds from their marked dynkin diagrams

on extended p 4 reducible and extended p 4 sparse graphs abstract a graph g was defined in 16 as p 4 reducible if no vertex in g belongs to more than one chordless path on four vertices or p 4 a graph g is defined in 15 as p 4 sparse if no set of five vertices induces more than one p 4 in g p 4 sparse graphs generalize both p 4 reducible and the well known class of p 4 free graphs or cographs in an extended abstract in 11 the first author introduced a method using the modular decomposition tree of a graph as the framework for the resolution of algorithmic problems this method was applied to the study of p 4 sparse and extended p 4 sparse graphs in this paper we begin by presenting the complete information about the method used in 11 we propose a unique tree representation of p 4 sparse and a unique tree representation of p 4 reducible graphs leading to a simple linear recognition algorithm for both classes of graphs in this way we simplify and unify the solutions for these problems presented in 16 19 the tree representation of an n vertex p 4 sparse or a p 4 reducible graph is the key for obtaining o n time algorithms for the weighted version of classical optimization problems solved in 20 these problems are np complete on general graphs finally by relaxing the restriction concerning the exclusion of the c 5 cycles from p 4 sparse and p 4 reducible graphs we introduce the class of the extended p 4 sparse and the class of the extended p 4 reducible graphs we then show that a minimal amount of additional work suffices for extending most of our algorithms to these new classes of graphs

vertex disjoint paths on clique width bounded graphs we show that the l vertex disjoint paths problem between l pairs of vertices can be solved in linear time for co graphs but is np complete for graphs of clique width at most 6 and nlc width at most 4 the np completeness follows from the fact that the line graph of a graph of tree width k has clique width at most 2k 2 and nlc width at most k 2 and a result by nishizeki et al the edge disjoint paths problem is np complete for series parallel graphs discrete appl math 115 2001 177 186 the vertex disjoint paths problem is the first graph problem shown to be np complete on graphs of bounded clique width but solvable in linear time on co graphs and graphs of bounded tree width additionally we show that the r vertex disjoint paths problem between each of l pairs of vertices can be solved in polynomial time for co graphs if l is given to the input and for graphs of bounded clique width if l is fixed

taut submanifolds and foliations we give an equivalent description of taut submanifolds of complete riemannian manifolds as exactly those submanifolds whose normal exponential map has integrable fibers it turns out that every taut submanifold is also mathbb z 2 taut so that tautness is essentially the same as mathbb z 2 tautness in the case where the normal exponential map of a submanifold has integrable fibers we explicitely construct generalized bott samelson cycles for the critical points of the energy functionals on the path spaces which generically represent a basis for the mathbb z 2 cohomology we also consider singular riemannian foliations all of whose leaves are taut and discuss some of their main features using our characterization of taut submanifolds we are able to show that tautness of a singular riemannian foliation is actually a property of the quotient

coupon coloring of cographs abstract coupon coloring is a new coloring which has many applications a k coupon coloring of a graph g is a k coloring of g by colors k 1 2 k such that the neighborhood of every vertex of g contains vertices of all colors from k the maximum integer k for which a k coupon coloring exists is called the coupon coloring number of g and it is denoted by c g in this paper we studied the coupon coloring of cographs which are graphs that can be generated from the single vertex graph k 1 by complementation and disjoint union and have applications in many interesting problems we use the cotree representation of a cograph to give a polynomial time algorithm to color the vertices of a cograph and then prove that this coloring is a coupon coloring with maximum colors hence get the coupon coloring numbers of the cograph

complexity of independent set reconfigurability problems we study problems of reconfigurability of independent sets in graphs we consider three different models token jumping token sliding and token addition and removal and analyze relationships between them we prove that independent set reconfigurability in perfect graphs under any of the three models generalizes the shortest path reconfigurability problem in general graphs and is therefore pspace complete on the positive side we give polynomial results for even hole free graphs and p 4 free graphs

partitioning group correction cholesky techniques for large scale sparse unconstrained optimization this paper studies a successive partitioning group correction cholesky algorithm and its modified algorithm for solving large scale sparse unconstrained optimization problems these methods employ an initial cholesky factorization of their approximation hessians and then correct the partitioning group s of their diagonal factor and lower triangular factor directly and successively at each step iterations are generated using forward and backward substitution employing the update factorizations a self correcting property a q superlinear convergence result and an r convergence rate estimate show that the two methods both have good local convergence properties the numerical results show that the two methods especially the modified algorithm may be competitive with some current used algorithms

the efficient computation of sparse jacobian matrices using automatic differentiation this paper is concerned with the efficient computation of sparse jacobian matrices of nonlinear vector maps using automatic differentiation ad specifically we propose the use of a graph coloring technique bicoloring to exploit the sparsity of the jacobian matrix j and thereby allow for the efficient determination of j using ad software we analyze both a direct scheme and a substitution process we discuss the results of numerical experiments indicating significant practical potential of this approach

on the computation of the trace form of some galois extensions abstract we investigate the trace form trl k l k x trl kx2of a finite galois extensionl k in particular we study 2 extensions of degree 16 using some reduction theorems these results yield a classification of nearly all trace forms of galois extensions of degree 31 finally we study the trace form of a cyclotomic extension and of its maximal real subfield

on nested split graphs whose second largest eigenvalue is less than 1 abstract we determine all nested split graphs i e graphs having no induced subgraphs equal to 2 k 2 p 4 or c 4 having the second largest eigenvalue less than 1 and give some data and observations regarding obtained results

an optimal path cover algorithm for cographs abstract the class of cographs or complement reducible graphs arises naturally in many different areas of applied mathematics and computer science in this paper we present an optimal algorithm for determining a minimum path cover for a cograph g in case g has a hamiltonian path cycle our algorithm exhibits the path cycle as well

on clique separators nearly chordal graphs and the maximum weight stable set problem clique separators in graphs are a helpful tool used by tarjan as a divide and conquer approach for solving various graph problems such as the maximum weight stable set mws problem maximum clique graph coloring and minimum fill in but few examples of graph classes having clique separators are known we use this method to solve mws in polynomial time for two classes where the unweighted maximum stable set ms problem is solvable in polynomial time by augmenting techniques but the complexity of the mws problem was open another example namely a result by alekseev for the mws problem on a subclass of p 5 free graphs obtained by clique separators can be improved by our techniques we also combine clique separators with decomposition by homogeneous sets in graphs and use the following notion a graph is email protected if for each of its vertices the subgraph induced by the set of its nonneighbors has property p we deal with the cases email protected chordal perfect this also simplifies a result obtained by a method called struction

on line algorithms for orders partially ordered sets appear as a basic tool in computer science and are particularly accurate for modeling dynamic behavior of complex systems motivated by considerations on the diagnosis of distributed computations a new kind of algorithm on posets has been developed and is now widely considered in this paper we present this on line algorithmics on posets and we survey the main results obtained under this approach

a linear time recognition algorithm for p 4 reducible graphs p4 reducible graphs are precisely the graphs none of whose vertices belong to more than one chordless path with three edges as it turns out the class of p4 reducible graphs strictly contains the well known class of cographs a remarkable property of p4 reducible graphs is their unique tree representation up to isomorphism in this paper we present a linear time algorithm to recognize p4 reducible graphs and to construct their corresponding tree representation

on the relationship between clique width and treewidth treewidth is generally regarded as one of the most useful parameterizations of a graph s construction clique width is a similar parameterization that shares one of the powerful properties of treewidth namely if a graph is of bounded treewidth or clique width then there is a polynomial time algorithm for any graph problem expressible in monadic second order logic using quantifiers on vertices in the case of clique width you must assume a clique width parse expression is given in studying the relationship between treewidth and clique width courcelle and olariu discrete appl math 101 2000 pp 77 114 showed that any graph of bounded treewidth is also of bounded clique width in particular for any graph g with treewidth k the clique width of g is at most 4 2k 1 1 in this paper we improve this result by showing that the clique width of g is at most 3 2k 1 and more importantly that there is an exponential lower bound on this relationship in particular for any k there is a graph g with treewidth equal to k where the clique width of g is at least 2 lfloor k 2 rfloor 1

on atomic structure of p5 free subclasses and maximum weight independent set problem graph decompositions play a crucial role in structural graph theory and in designing efficient graph algorithms among them clique separator decomposition a decomposition tree of the graph whose leaves have no clique separator so called atoms used by tarjan for solving various optimization problems recently received much attention in this note we first derive the atomic structure of two subclasses of p 5 free graphs where p 5 is a chordless path on five vertices these results enable us to provide efficient solutions for the maximum weight independent set problem in these classes of graphs

on finding rainbow and colorful paths in the colorful path problem we are given a graph g v e and an arbitrary vertex coloring function c v k the goal is to find a colorful path i e a path on k vertices that visits each color this problem has been introduced in the classical work of alon et al 1995 1 and the authors proposed a dynamic programming algorithm that runs in time 2 k n o 1 and uses o 2 k space since then the only progress obtained is reducing the space size to a polynomial at the cost of using randomization in this work we show that a progress in time complexity is unlikely if colorful path can be solved in time 2 e k n o 1 then set cover admits a 2 e n n m o 1 time algorithm the same applies to other versions of the problem when edges are colored instead of vertices or we ask for a walk instead of a path or when the requested path walk has specified endpoints we study also a second very related problem in rainbowst connectivity we are given a k edge colored graph and two vertices s and t the goal is to decide whether there is a rainbow path between s and t that is a path on which no color repeats in its vertex variant rainbow vertexst connectivity the input graph is k vertex colored and a rainbow path is defined analogously uchizawa et al 2011 14 show that both variants can be solved in 2 k n o 1 time and exponential space we show that the space size can be reduced to a polynomial while keeping the same running time in contrast to the polynomial space algorithm for colorful path our algorithm for finding rainbow paths is deterministic

restrictions of graph partition problems part i in this paper partition problems into k independent sets or cliques of bounded size k are analyzed for several classes of graphs we prove the computational complexity of both problems restricted to cographs split graphs bipartite graphs and interval graphs given general or constant k and k it is shown that the assignment problem for operations in a branching flow graph to processors each with a limit on the number of executable operations equals the first problem restricted to cographs in addition a job assignment problem given intervals for each job and k machines each executing at most k jobs equals the first problem restricted to interval graphs it is shown that both problem are np complete

injective coloring of some graph operations an injective coloring of a graph g is a vertex coloring such that any two vertices with a common vertex receive distinct colors the injective chromatic number i g of a graph g is the least k such that there is an injective k coloring graph operations are important methods for constructing new graphs and they play key roles in the design and analysis of networks in this study we give some sharp bounds or exact values of graph operations including the cartesian product direct product lexicographic product union join and disjunction of graphs

a geometric proof of the berger holonomy theorem we give a geometric proof of the berger holonomy theorem the proof uses euclidean submanifold geometry of orbits and gives a link between riemannian holonomy groups and normal holonomy groups

a fully dynamic algorithm for the recognition of p4 sparse graphs in this paper we solve the dynamic recognition problem for the class of p 4 sparse graphs the objective is to handle edge vertex additions and deletions to recognize if each such modification yields a p 4 sparse graph and if yes to update a representation of the graph our approach relies on maintaining the modular decomposition tree of the graph which we use for solving the recognition problem we establish properties for each modification to yield a p 4 sparse graph and obtain a fully dynamic recognition algorithm which handles edge modifications in o 1 time and vertex modifications in o d time for a vertex of degree d thus our algorithm implies an optimal edges only dynamic algorithm and a new optimal incremental algorithm for p 4 sparse graphs

handsome proof nets perfect matchings and cographs we first extract the combinatorial result behind various proofs of the sequentialisation theorem for multiplicative proof nets this result is an inductive characterisation of graphs with a unique perfect matching extending these techniques we give a definition of multiplicative proof nets in which commutativity but also associativity of the multiplicative connectives is interpreted as equality this is done by representing a sequent by a cograph and axioms by a perfect matching the main advantage of this presentation is aesthetic any such graph without any further requirement is a proof structure and the correctness criterion also is a natural graph theoretical property a direct and purely graph theoretical proof of these results is available as a research report in which more details can be found c retore handsome proof nets r b graphs perfect matchings and series parallel graphs rapport de recherche rr 3652 inria march 1999 http www inria fr rrrt publications eng html

finding a potential community in networks abstract an independent 2 clique of a graph is a subset of vertices that is an independent set and such that any two vertices inside have a common neighbor outside in this paper we study the complexity of finding an independent 2 clique of maximum size in several graph classes and we compare it with the complexity of maximum independent set we prove that this problem is np hard on apex graphs apx hard on line graphs not n 1 2 approximable on bipartite graphs and not n 1 approximable on split graphs while it is polynomial time solvable on graphs of bounded degree and their complements graphs of bounded treewidth planar graphs c 3 c 6 free graphs threshold graphs interval graphs and cographs

distributed memory parallel algorithms for distance 2 coloring and their application to derivative computation the distance 2 graph coloring problem aims at partitioning the vertex set of a graph into the fewest sets consisting of vertices pairwise at distance greater than two from each other its applications include derivative computation in numerical optimization and channel assignment in radio networks we present efficient distributed memory parallel heuristic algorithms for this nphard problem as well as for two related problems used in the computation of jacobians and hessians parallel speedup is achieved through graph partitioning speculative iterative coloring and a bsplike organization of parallel computation results from experiments conducted on a pc cluster employing up to 96 processors and using large size real world as well as synthetically generated test graphs show that the algorithms are scalable in terms of quality of solution the algorithms perform remarkably well the number of colors used by the parallel algorithms was observed to be very close to the number used by the sequential counterparts which in turn are quite often near optimal moreover the experimental results show that the parallel distance 2 coloring algorithm compares favorably with the alternative approach of solving the distance 2 coloring problem on a graph g by first constructing the square graph g2 and then applying a parallel distance 1 coloring algorithm on g2 implementations of the algorithms are made available via the zoltan load balancing library

distributed memory parallel algorithms for distance 2 coloring and related problems in derivative computation the distance 2 graph coloring problem aims at partitioning the vertex set of a graph into the fewest sets consisting of vertices pairwise at distance greater than 2 from each other its applications include derivative computation in numerical optimization and channel assignment in radio networks we present efficient distributed memory parallel heuristic algorithms for this np hard problem as well as for two related problems used in the computation of jacobians and hessians parallel speedup is achieved through graph partitioning speculative iterative coloring and a bulk synchronous parallel like organization of parallel computation results from experiments conducted on a pc cluster employing up to 96 processors and using large size real world as well as synthetically generated test graphs show that the algorithms are scalable in terms of quality of solution the algorithms perform remarkably well the numbers of colors used by the parallel algorithms are observed to be very close to the numbers used by their sequential counterparts which in turn are quite often near optimal moreover the experimental results show that the parallel distance 2 coloring algorithm compares favorably with the alternative approach of solving the distance 2 coloring problem on a graph mathcal g by first constructing the square graph mathcal g 2 and then applying a parallel distance 1 coloring algorithm on mathcal g 2 implementations of the algorithms are made available via the zoltan toolkit

on maximum independent set of categorical product and ultimate categorical ratios of graphs we first present polynomial algorithms to compute the independence number of the categorical product for two cographs or two splitgraphs respectively then we prove that computing the maximum independent set of the categorical product of a planar graph of maximum degree three and a k 4 is np complete the ultimate categorical independence ratio of a graph g is defined as lim k g k n k the ultimate categorical independence ratio can be computed in polynomial time for cographs splitgraphs permutation graphs interval graphs and graphs of bounded treewidth also we present an o 3 n 3 time exact exponential algorithm for the ultimate categorical independence ratio of general graphs we further present a ptas for the ultimate categorical independence ratio of planar graphs lastly we show that the ultimate categorical independent domination ratio for complete multipartite graphs is zero except when the graph is complete bipartite with color classes of equal size in which case it is 1 2

edge colouring of regular graphs of large degree we consider the following conjecture letgbe ak regular simple graph with an even numbernof vertices ifk n 2thengisk edge colourable we show that this conjecture is true for graphs that are join of two graphs and we provide a polynomial time algorithm for finding a k edge colouring of these graphs

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

new acyclic and star coloring algorithms with application to computing hessians acyclic and star coloring problems are specialized vertex coloring problems that arise in the efficient computation of hessians using automatic differentiation or finite differencing when both sparsity and symmetry are exploited we present an algorithmic paradigm for finding heuristic solutions for these two np hard problems the underlying common technique is the exploitation of the structure of two colored induced subgraphs for a graph g on n vertices and m edges the time complexity of our star coloring algorithm is o n overline d 2 where overline d k a generalization of vertex degree denotes the average number of distinct paths of length at most k edges starting at a vertex in g the time complexity of our acyclic coloring algorithm is larger by a multiplicative factor involving the inverse of ackermann s function the space complexity of both algorithms is o m to the best of our knowledge our work is the first practical algorithm for the acyclic coloring problem for the star coloring problem our algorithm uses fewer colors and is considerably faster than a previously known o n overline d 3 time algorithm computational results from experiments on various large size test graphs demonstrate that the algorithms are fast and produce highly effective solutions the use of these algorithms in hessian computation is expected to reduce overall runtime drastically

a simple paradigm for graph recognition application to cographs and distance hereditary graphs an easy way for graph recognition algorithms is to use a two step process first compute a characteristic feature as if the graph belongs to that class second check whether the computed characteristic feature as if the graph belongs to that class second check whether the computed separating them may yield new and much more easily understood algorithms in this paper we apply that paradigm to the cograph and distance hereditary graph recognition problems

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

the complexity of determining the rainbow vertex connection of a graph a vertex colored graph is rainbow vertex connected if any two vertices are connected by a path whose internal vertices have distinct colors which was introduced by krivelevich and yuster the rainbow vertex connection of a connected graph g denoted by rvc g is the smallest number of colors that are needed in order to make g rainbow vertex connected in this paper we study the complexity of determining the rainbow vertex connection of a graph and prove that computing rvc g is np hard moreover we show that it is already np complete to decide whether rvc g 2 we also prove that the following problem is np complete given a vertex colored graph g check whether the given coloring makes g rainbow vertex connected

note on the perfect roman domination number of graphs abstract in this note we list some propositions of the perfect roman domination number of graphs and give the characterization of graphs g with special value of the perfect roman domination furthermore r p f r p f is given at the last a linear time algorithm is shown to compute the perfect roman domination number of a cograph

estimation of sparse hessian matrices and graph coloring problems large scale optimization problems often require an approximation to the hessian matrix if the hessian matrix is sparse then estimation by differences of gradients is attractive because the number of required differences is usually small compared to the dimension of the problem the problem of estimating hessian matrices by diferences can be phrased as follows given the sparsity structure of a symmetric matrix a obtain vectors d 1 d 2 ldots d p such that ad 1 ad 2 ldots ad p determine a uniquely with p as small as possible we approach this problem from a graph theoretic point of view and show that both direct and indirect approaches to this problem have a natural graph coloring interpretation the complexity of the problem is analyzed and efficient practical heuristic procedures are developed numerical results illustrate the differences between the various approaches

locally identifying coloring of graphs with few p4s abstract a lid coloring locally identifying coloring of a graph is a proper coloring such that for any edge uv if u and v have distinct closed neighborhoods then the set of colors used on vertices of the closed neighborhoods of u and v are distinct the lid chromatic number is the minimum number of colors used in a lid coloring in this paper we prove a relation between lid coloring and a variation called strong lid coloring with this we obtain linear time algorithms to calculate the lid chromatic number for some classes of graphs with few p 4 s such as cographs p 4 sparse graphs and q q 4 graphs we also prove that the lid chromatic number is o n 1 e inapproximable in polynomial time for every e 0 unless p np

contraction and deletion blockers for perfect graphs and h free graphs abstract we study the following problem for given integers d k and graph g can we reduce some fixed graph parameter of g by at least d via at most k graph operations from some fixed set s as parameters we take the chromatic number clique number and independence number and as operations we choose edge contraction ec and vertex deletion vd we determine the complexity of this problem for s ec and s vd and for a number of subclasses of perfect graphs we use these results to determine the complexity of the problem for s ec and s vd and restricted to h free graphs

some observations on maximum weight stable sets in certain p5 free graphs abstract the maximum weight stable set problem mws is the weighted version of the maximum stable set problem ms which is np hard the class of p5 free graphs i e graphs with no induced path of five vertices is the unique minimal class defined by forbidding a single connected subgraph for which the computational complexity of ms is an open question at the same time it is known that ms can be efficiently solved for p 5 f free graphs where f is any graph of five vertices different to a c5 in this paper we introduce some observations on p5 free graphs and apply them to introduce certain subclasses of such graphs for which one can efficiently solve mws that extends or improves some known results and implies together with other known results that mws can be efficiently solved for p 5 f free graphs where f is any graph of five vertices different to a c5

a theorem on permutation graphs with applications abstract in this paper a theorem that characterizes a relation between cographs and permutation graphs is introduced this theorem has an application to the recognition of a cograph by the aid of this theorem sequential and parallel algorithms to determine if a given permutation graph is a cograph can be derived

further hardness results on the rainbow vertex connection number of graphs a vertex colored graph g is rainbow vertex connected if any pair of vertices in g are connected by a path whose internal vertices have distinct colors which was introduced by krivelevich and yuster the rainbow vertex connection number of a connected graph g denoted by rvc g is the smallest number of colors that are needed in order to make g rainbow vertex connected in a previous paper we showed that it is np complete to decide whether a given graph g has rvc g 2 in this paper we show that for every integer k 2 deciding whether rvc g k is np hard we also show that for any fixed integer k 2 this problem belongs to np class and so it becomes np complete

what color is your jacobian graph coloring for computing derivatives graph coloring has been employed since the 1980s to efficiently compute sparse jacobian and hessian matrices using either finite differences or automatic differentiation several coloring problems occur in this context depending on whether the matrix is a jacobian or a hessian and on the specifics of the computational techniques employed we consider eight variant vertex coloring problems here this article begins with a gentle introduction to the problem of computing a sparse jacobian followed by an overview of the historical development of the research area then we present a unifying framework for the graph models of the variant matrix estimation problems the framework is based upon the viewpoint that a partition of a matrix into structurally orthogonal groups of columns corresponds to distance 2 coloring an appropriate graph representation the unified framework helps integrate earlier work and leads to fresh insights enables the design of more efficient algorithms for many problems leads to new algorithms for others and eases the task of building graph models for new problems we report computational results on two of the coloring problems to support our claims most of the methods for these problems treat a column or a row of a matrix as an atomic entity and partition the columns or rows or both a brief review of methods that do not fit these criteria is provided we also discuss results in discrete mathematics and theoretical computer science that intersect with the topics considered here

a general theory of canonical forms si g est un groupe de lie compact et m une g variete riemannienne d orbites principales de codimension k alors une section ou une forme canonique pour m est une sous variete fermee lisse a k dimensions de m qui rencontre toutes les orbites de m orthogonalement

on algorithms for p 5 gem free graphs a graph is p5 gem free when it does not contain p5 an induced path with five vertices or a gem a graph formed by making an universal vertex adjacent to each of the four vertices of the induced path p4 as an induced subgraph we present o n2 time recognition algorithms for chordal gem free graphs and for p5 gem free graphs using a characterization of p5 gem free graphs by their prime graphs with respect to modular decomposition and their modular decomposition trees a brandstadt d kratsch on the structure of p5 gem free graphs discrete appl math 145 2005 155 166 we give linear time algorithms for the following np complete problems on p5 gem free graphs minimum coloring maximum weight stable set maximum weight clique and minimum clique cover