Showing papers on "Split graph published in 2004"
Book•
01 Mar 2004
834 citations
TL;DR: In this article, a covariance selection model is defined in terms of the Markov properties, i.e. conditional independences associated with G, which in turn are equivalent to specified zeros among the set of pairwise partial correlation coefficients.
Abstract: ggA multivariate Gaussian graphical Markov model for an undirected graph G, also called a covariance selection model or concentration graph model, is defined in terms of the Markov properties, i.e. conditional independences associated with G, which in turn are equivalent to specified zeros among the set of pairwise partial correlation coefficients. By means of Fisher's z-transformation and Sidak's correlation inequality, conservative simultaneous confidence intervals for the entire set of partial correlations can be obtained, leading to a simple method for model selection that controls the overall error rate for incorrect edge inclusion. The simultaneous p-values corresponding to the partial correlations are partitioned into three disjoint sets, a significant set S, an indeterminate set I and a nonsignificant set N. Our model selection method selects two graphs, a graph G SI whose edges correspond to the set S∪I, and a more conservative graph G S whose edges correspond to S only. Similar considerations apply to covariance graph models, which are defined in terms of marginal independence rather than conditional independence. The method is applied to some well-known examples and to simulated data.
213 citations
01 Jan 2004
TL;DR: Cameron et al. as mentioned in this paper introduced the concept of graph Laplacians and showed that they can be used to construct strong regular graphs and distance-transitive graphs, including Cayley graphs.
Abstract: Foreword Peter J. Cameron Introduction 1. Eigenvalues of graphs Michael Doob 2. Graphs and matrices Richard A. Brualdi and Bryan L. Shader 3. Spectral graph theory Dragos Cvetkovic and Peter Rowlinson 4. Graph Laplacians Bojan Mohar 5. Automorphism groups Peter J. Cameron 6. Cayley graphs Brian Alspach 7. Finite symmetric graphs Cheryle E. Praeger 8. Strongly regular graphs Peter J. Cameron 9. Distance-transitive graphs Arjeh M. Cohen 10. Computing with graphs and groups Leonard H. Soicher.
212 citations
TL;DR: Improved algorithms for graph colouring and computing the chromatic number of a graph are constructed by giving tight upper bounds on the number of maximal independent sets of size k in graphs with n vertices.
137 citations
Book•
01 Jan 2004TL;DR: In this article, the authors discuss the three principal techniques that have been employed, namely, forbidden subgraphs, root systems and star complements, and bring together the major results in the area, including the recent construction of all maximal exceptional graphs.
Abstract: Line graphs have the property that their least eigenvalue is greater than or equal to –2, a property shared by generalized line graphs and a finite number of so-called exceptional graphs. This book deals with all these families of graphs in the context of their spectral properties. The authors discuss the three principal techniques that have been employed, namely 'forbidden subgraphs', 'root systems' and 'star complements'. They bring together the major results in the area, including the recent construction of all the maximal exceptional graphs. Technical descriptions of these graphs are included in the appendices, while the bibliography provides over 250 references. This will be an important resource for all researchers with an interest in algebraic graph theory.
136 citations
TL;DR: In this article, the complements of interval bigraphs are precisely those circular arc graphs of clique covering number two, which admit a representation without two arcs covering the whole circle.
Abstract: We prove that the complements of interval bigraphs are precisely those circular arc graphs of clique covering number two, which admit a representation without two arcs covering the whole circle. We give another characterization of interval bigraphs, in terms of a vertex ordering, that we hope may prove helpful in finding a more efficient recognition algorithm than presently known. We use these results to show equality, amongst bipartite graphs, of several classes of structured graphs (proper interval bigraphs, complements of proper circular arc graphs, asteroidal-triple-free graphs, permutation graphs, and co-comparability graphs). Our results verify a conjecture of Lundgren and disprove a conjecture of Muller. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 313–327, 2004
116 citations
Journal Article•
TL;DR: It is shown that there exists a O(kn2 +k2 + 23.8161k)=O(n2) algorithm for n–kGraph Coloring for each fixed k, and a similar near linear-time algorithm that produces a 3k kernel for Vertex Cover is presented, which is the crown reduction rule.
Abstract: This paper examines a parameterized problem that we refer to as n - k GRAPH COLORING, i.e., the problem of determining whether a graph G with n vertices can be colored using n - k colors. As the main result of this paper, we show that there exists a O(kn 2 + k 2 + 2 3.8161k ) = O(n 2 ) algorithm for n - k GRAPH COLORING for each fixed k. The core technique behind this new parameterized algorithm is kernalization via maximum (and certain maximal) matchings. The core technical content of this paper is a near linear-time kernelization algorithm for n-k CLIQUE COVERING. The near linear-time kernelization algorithm that we present for n - k CLIQUE COVERING produces a linear size (3k - 3) kernel in O(k(n + m)) steps on graphs with n vertices and m edges. The algorithm takes an instance (G, k) of CLIQUE COVERING that asks whether a graph G can be covered using |V| - k cliques and reduces it to the problem of determining whether a graph G' = (V',E') of size < 3k - 3 can be covered using |V'|-k' cliques. We also present a similar near linear-time algorithm that produces a 3k kernel for VERTEX COVER. This second kernelization algorithm is the crown reduction rule.
115 citations
TL;DR: This paper proves that for K1,3-free graphs the problem of finding the largest independent set can be solved in a polynomial time, and that this is also true for the wider class of fork- free graphs.
97 citations
TL;DR: In this article, it was shown that the number of forbidden subgraphs is reduced to 4 if the graph is connected and 5 if it is an infinite field such that charF � ≤ 2i f and only if the complement of G is the join of G and a graph that is the union of at most two cliques and any number of complete bipartite graphs.
Abstract: Let F be a field, G =( V, E) be an undirected graph on n vertices, and let S(F, G) be the set of all symmetric n × n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. For example, if G is a path, S(F, G )co nsists of the symmetric irreducible tridiagonal matrices. Let mr(F, G) be the minimum rank over all matrices in S(F, G). Then mr(F, G) = 1 if and only if G is the union of a clique with at least 2 vertices and an independent set. If F is an infinite field such that charF � , then mr(F, G) ≤ 2i f and only if the complement of G is the join of a clique and a graph that is the union of at most two cliques and any number of complete bipartite graphs. A similar result is obtained in the case that F is an infinite field with char F = 2. Furthermore, in each case, such graphs are characterized as those for which 6 specific graphs do not occur as induced subgraphs. The number of forbidden subgraphs is reduced to 4 if the graph is connected. Finally, similar criteria is obtained for the minimum rank of a Hermitian matrix to be less than or equal to two. The complement is the join of a clique and a graph that is the union of any number of cliques and any number of complete bipartite graphs. The number of forbidden subgraphs is now 5, or in the connected case, 3.
97 citations
TL;DR: In this article, the Fourier transform of functions on regular graphs was studied and the authors showed that the independent sets induced by the base graph are the only maximum-size independent sets, and that any independent set of size close to the maximum is close to some independent sets of maximum size.
Abstract: We consider powers of regular graphs defined by the weak graph product and give a characterization of maximum-size independent sets for a wide family of base graphs which includes, among others, complete graphs, line graphs of regular graphs which contain a perfect matching and Kneser graphs. In many cases this also characterizes the optimal colorings of these products. We show that the independent sets induced by the base graph are the only maximum-size independent sets. Furthermore we give a qualitative stability statement: any independent set of size close to the maximum is close to some independent set of maximum size. Our approach is based on Fourier analysis on Abelian groups and on Spectral Techniques. To this end we develop some basic lemmas regarding the Fourier transform of functions on
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94 citations
TL;DR: It is shown that if G is a polygon-circle graph, then so is [ L ( G )] 2 , and the same holds for asteroidal triple-free and interval-filament graphs, and it follows that the induced matching problem is polytime-solvable in these classes.
15 Feb 2004
TL;DR: In this paper, it was shown that 3-colorability can be decided in polynomial time for the class of P6-free graphs with bounded dominating subgraphs.
Abstract: In this paper, we study a chromatic aspect for the class of P6-free graphs. Here, the focus of our interest are graph classes (defined in terms of forbidden induced subgraphs) for which the question of 3-colorability can be decided in polynomial time and, if so, a proper 3-coloring can be determined also in polynomial time. Note that the 3-colorability decision problem is a well-known NP-complete problem, even for special graph classes, e.g. for triangle- and K1,5-free graphs (Discrete Math. 162 (1-3) (1996) 313-317). Therefore, it is unlikely that there exists a polynomial algorithm deciding whether there exists a 3-coloring of a given graph in general. Our approach is based on an encoding of the problem with Boolean formulas making use of the existence of bounded dominating subgraphs. Together with a structural analysis of the nonperfect K4-free members of the graph class in consideration we obtain our main result that 3-colorability can be decided in polynomial time for the class of P6-free graphs.
TL;DR: It is shown that "almost" all perfect graphs are 3-clique-colorable, and exact bounds and polynomial algorithms that find the clique-chromatic number for some classes of graphs are shown and NP-completeness results for some others are proved.
Abstract: In this paper we are concerned with the so-called clique-colorations of a graph, that is, colorations of the vertices so that no maximal clique is monochromatic. On one hand, it is known to be NP-complete to decide whether a perfect graph is 2-clique-colorable, or whether a triangle-free graph is 3-clique-colorable; on the other hand, there is no example of a perfect graph where more than three colors would be necessary. We first exhibit some simple recursive methods to clique-color graphs and then relate the chromatic number, the domination number, and the maximum cardinality of a stable set to the clique-chromatic number. We show exact bounds and polynomial algorithms that find the clique-chromatic number for some classes of graphs and prove NP-completeness results for some others, trying to find the boundary between the two. For instance, while it is NP-complete to decide whether a graph of maximum degree 3 is 2-clique-colorable, K1,3-free graphs without an odd hole turn out to be always 2-clique-colorable by a polynomial algorithm. Finally, we show that "almost" all perfect graphs are 3-clique-colorable.
TL;DR: The main result is a forbidden subgraph characterization of chordal (k,l)-graphs, and the algorithm and the characterization extend to the 'list' (or 'pre-colouring extension') version of the split partition problem.
TL;DR: In this paper, the authors showed that all alternating group graphs are pan-connected, that is, every two vertices x and y in the graph are connected by a path of length k for each k satisfying dlx, yr ≤ k ≤ vVv - 1, where d lx, y denotes the distance between X and y, and vvv is the number of vertices in the network.
Abstract: Jwo et al. lNetworks 23 l1993r 315–326r introduced the alternating group graph as an interconnection network topology for computing systems. They showed that the proposed structure has many advantages over n-cubes and star graphs. For example, all alternating group graphs are hamiltonian-connected li.e., every pair of vertices in the graph are connected by a hamiltonian pathr and pancyclic li.e., the graph can embed cycles with arbitrary length with dilation 1r. In this article, we give a stronger result: all alternating group graphs are panconnected, that is, every two vertices x and y in the graph are connected by a path of length k for each k satisfying dlx, yr ≤ k ≤ vVv - 1, where dlx, yr denotes the distance between x and y, and vVv is the number of vertices in the graph. Moreover, we show that the r-dimensional alternating group graph AGr, r ≥ 4, is lr - 3r-vertex fault-tolerant Hamiltonian-connected and lr - 2r-vertex fault-tolerant hamiltonian. The latter result can be viewed as complementary to the recent work of Lo and Chen lIEEE Trans. Parallel and Distributed Systems 12 l2001r 209–222r, which studies the fault-tolerant hamiltonicity in faulty arrangement graphs. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44l4r, 302–310 2004
08 Jun 2004
TL;DR: It is proved that the geometric thickness of graphs whose maximum degree is no more than four is two, and an embedding algorithm for graphs with maximum degree three that uses an n x n grid and a more complex algorithm for embedding a graph withmaximum degree four.
Abstract: We prove that the geometric thickness of graphs whose maximum degree is no more than four is two. All of our algorithms run in O(n) time, where n is the number of vertices in the graph. In our proofs, we present an embedding algorithm for graphs with maximum degree three that uses an n x n grid and a more complex algorithm for embedding a graph with maximum degree four. We also show a variation using orthogonal edges for maximum degree four graphs that also uses an n x n grid. The results have implications in graph theory, graph drawing, and VLSI design.
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TL;DR: It is shown that careful insertion orders can produce graphs with desirable properties including low spanning ratio, logarithmic maximum degree and logarithsmic diameter.
Abstract: Let V be a set of n points in R2. The θ-graph of V is a geometric graph with vertex set V that has been studied extensively and which has several nice properties. We introduce a new variant of θ-graphs which we call ordered θ-graphs. These are graphs that are built incrementally by inserting the vertices one by one so that the resulting graph depends on the insertion order. We show that careful insertion orders can produce graphs with desirable properties including low spanning ratio, logarithmic maximum degree and logarithmic diameter.
12 Jul 2004
TL;DR: It is proved that the same rules, applied to any graph G of genus g, reduce the k-dominating set problem to a kernel of size O(k+g), i.e. linear kernel, and drastically the best so far combinatorial bound to the branchwidth of a graph in terms of its minimum dominating set and its genus is improved.
Abstract: Preprocessing by data reduction is a simple but powerful technique used for practically solving different network problems. A number of empirical studies shows that a set of reduction rules for solving Dominating Set problems introduced by Alber, Fellows & Niedermeier leads efficiently to optimal solutions for many realistic networks. Despite of the encouraging experiments, the only class of graphs with proven performance guarantee of reductions rules was the class of planar graphs. However it was conjectured in that similar reduction rules can be proved to be efficient for more general graph classes like graphs of bounded genus. In this paper we (i) prove that the same rules, applied to any graph G of genus g, reduce the k-dominating set problem to a kernel of size O(k+g), i.e. linear kernel. This resolves a basic open question on the potential of kernel reduction for graph domination. (ii) Using such a kernel we improve the best so far algorithm for k-dominating set on graphs of genus ≤ g from \(2^{O(g\sqrt{k}+g^{2})}n^{O(1)}\) to \(2^{O(\sqrt{gk}+g)}+n^{O(1)}\). (iii) Applying tools from the topological graph theory, we improve drastically the best so far combinatorial bound to the branchwidth of a graph in terms of its minimum dominating set and its genus. Our new bound provides further exponential speed-up of our algorithm for the k-dominating set and we prove that the same speed-up applies for a wide category of parameterized graph problems such as k-vertex cover, k-edge dominating set, k-vertex feedback set, k-clique transversal number and several variants of the k-dominating set problem. A consequence of our results is that the non-parameterized versions of all these problems can be solved in subexponential time when their inputs have sublinear genus.
TL;DR: It is pointed out that every chordal graph of diameter at most t - 1 admits a tree t-spanner whenever t ≥ 2 is even (respectively, t ≥ 3 is odd), and such a tree spanner can be constructed in linear time.
21 Jun 2004
TL;DR: In this article, a polynomial-time approximation scheme for the maximum weight independent set problem in unit disk graphs is presented, which is robust in the sense that it accepts any graph as input and either returns a (1+e)-approximate independent set or a certificate showing that the input graph is no unit disk graph.
Abstract: A unit disk graph is the intersection graph of unit disks in the euclidean plane. We present a polynomial-time approximation scheme for the maximum weight independent set problem in unit disk graphs. In contrast to previously known approximation schemes, our approach does not require a geometric representation (specifying the coordinates of the disk centers).
The approximation algorithm presented is robust in the sense that it accepts any graph as input and either returns a (1+e)-approximate independent set or a certificate showing that the input graph is no unit disk graph. The algorithm can easily be extended to other families of intersection graphs of geometric objects.
Journal Article•
TL;DR: A polynomial-time approximation scheme for the maximum weight independent set problem in unit disk graphs is presented and is robust in the sense that it accepts any graph as input and either returns a (1+e)-approximate independent set or a certificate showing that the input graph is no unit disk graph.
Abstract: A unit disk graph is the intersection graph of unit disks in the euclidean plane. We present a polynomial-time approximation scheme for the maximum weight independent set problem in unit disk graphs. In contrast to previously known approximation schemes, our approach does not require a geometric representation (specifying the coordinates of the disk centers). The approximation algorithm presented is robust in the sense that it accepts any graph as input and either returns a (1 + e)-approximate independent set or a certificate showing that the input graph is no unit disk graph. The algorithm can easily be extended to other families of intersection graphs of geometric objects.
TL;DR: It is shown that the clique-width of gem- and co-gem-free graphs is at most 16.5%, which is fundamental for the recently introduced concept ofClique- width of graphs which extends the famous concept of treewidth.
Abstract: The P4 is the induced path of four vertices. The gem consists of a P4 with an additional universal vertex being completely adjacent to the P4, and the co-gem is its complement graph. Gem- and co-gem-free graphs generalize the popular class of cographs (i. e. P4-free graphs). The tree structure and algebraic generation of cographs has been crucial for several concepts of graph decomposition such as modular and homogeneous decomposition. Moreover, it is fundamental for the recently introduced concept of clique-width of graphs which extends the famous concept of treewidth. It is well-known that the cographs are exactly those graphs of clique-width at most 2. In this paper, we show that the clique-width of gem- and co-gem-free graphs is at most 16.
20 Dec 2004
TL;DR: An open problem in sweeping is the existence of a graph with connected sweep number strictly less than monotonicconnected sweep number solved by constructing a graph W exhibiting exactly this property.
Abstract: An open problem in sweeping is the existence of a graph with connected sweep number strictly less than monotonic connected sweep number We solve this problem by constructing a graph W exhibiting exactly this property Further, we will examine a new method of constructing graphs that makes proving all such inequalities easier, and offer some new lower bounds on sweep numbers.
TL;DR: For every k there is a k-chromatic graph Gk with k-coloring where the neighbors of each color-class form an independent set as mentioned in this paper, and Gk exhibits remarkable similarity to Kneser graphs.
Abstract: We prove that for every k there is a k-chromatic graph with a k-coloring where the neighbors of each color-class form an independent set. This answers a question raised by N. J. A. Harvey and U. S. R. Murty [4]. In fact we find the smallest graph Gk with the required property for every k. The graph Gk exhibits remarkable similarity to Kneser graphs. The proof that Gk is k-chromatic relies on Lovasz's theorem about the chromatic number of graphs with highly connected neighborhood complexes. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 1–14, 2004
TL;DR: This paper investigates the power of local computations on graphs, by considering a classical problem in distributed algorithms, the recognition problem, and characterize the graph classes that are recognizable with or without structural knowledge.
Abstract: This paper investigates the power of local computations on graphs, by considering a classical problem in distributed algorithms, the recognition problem. Formally, we want to compute some topological information on a network of processes, possibly using additional knowledge about the structure of the underlying graph. We propose the notion of recognition with structural knowledge by means of local computations. Then we characterize the graph classes that are recognizable with or without structural knowledge. The characterizations use graph coverings and a distributed enumeration algorithm proposed by A. Mazurkiewicz. Several applications are presented, in particular concerning minor-closed classes of graphs.
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TL;DR: It is shown that chordal probe graphs may contain neither an odd-length chordless cycle nor the complement of a chordless cycles, hence they are perfect graphs.
TL;DR: This work provides an alternative description of these graphs as modified incidence graph of an affine plane, which leads to a complete determination of their automorphism groups.
TL;DR: The structure and bounded clique width of this graph class allows to solve several algorithmic problems on this class in linear time, among them the problems Maximum Weight Stable Set (MWS), Maximum Weight Clique, Domination, Steiner Tree and in general every algorithmic problem which is expressible in a certain kind of Monadic Second-Order Logic.
TL;DR: It is proved that coefficients have to be consecutive in any facet with exactly two different non-zero coefficients (assuming they are relatively prime) in the stable set polytope.
Abstract: We deal with non-rank facets of the stable set polytope of claw-free graphs. We extend results of Giles and Trotter [7] by (i) showing that for any nonnegative integer a there exists a circulant graph whose stable set polytope has a facet-inducing inequality with (a,a+1)-valued coefficients (rank facets have only coefficients 0, 1), and (ii) providing new facets of the stable set polytope with up to five different non-zero coefficients for claw-free graphs. We prove that coefficients have to be consecutive in any facet with exactly two different non-zero coefficients (assuming they are relatively prime). Last but not least, we present a complete description of the stable set polytope for graphs with stability number 2, already observed by Cook [3] and Shepherd [18].