Showing papers on "K-tree published in 2012"
TL;DR: It is conjecture that a maximal k-degenerate graph is class two if and only if it is overfull, and it is proved that this is the case in some special cases.
Abstract: A graph is k-degenerate if its vertices can be successively deleted so that when deleted, each has degree at most k. These graphs were introduced by Lick and White in 1970 and have been studied in several subsequent papers. We present sharp bounds on the diameter of maximal k-degenerate graphs and characterize the extremal graphs for the upper bound. We present a simple characterization of the degree sequences of these graphs and consider related results. Considering edge coloring, we conjecture that a maximal k-degenerate graph is class two if and only if it is overfull, and prove this in some special cases. We present some results on decompositions and arboricity of maximal k-degenerate graphs and provide two characterizations of the subclass of k-trees as maximal k-degenerate graphs. Finally, we define and prove a formula for the Ramsey core numbers.
11 citations
Proceedings Article•
31 Jan 2012TL;DR: It is proved that graphs excluding a fixed odd-minor have O(n2) cliques (which is tight), and a O( n3) time algorithm for listing all cliques is concluded.
Abstract: This paper is about: (1) bounds on the number of cliques in a graph in a particular class, and (2) algorithms for listing all cliques in a graph. We present a simple algorithm that lists all cliques in an n-vertex graph in O(n) time per clique. For O(1)-degenerate graphs, such as graphs excluding a fixed minor, we describe a O(n) time algorithm for listing all cliques. We prove that graphs excluding a fixed odd-minor have O(n2) cliques (which is tight), and conclude a O(n3) time algorithm for listing all cliques.
6 citations
10 Sep 2012
TL;DR: This paper develops an iterative approach where each thread can asynchronously update a subset of edges that are dynamically assigned to it per iteration and implements the algorithm on two different multithreaded architectures - Cray XMT, a massively multith readed platform, and AMD Magny-Cours, a shared memory multicore platform.
Abstract: Chordal graphs are triangulated graphs where any cycle larger than three is bisected by a chord. Many combinatorial optimization problems such as computing the size of the maximum clique and the chromatic number are NP-hard on general graphs but have polynomial time solutions on chordal graphs. In this paper, we present a novel multithreaded algorithm to extract a maximal chordal sub graph from a general graph. We develop an iterative approach where each thread can asynchronously update a subset of edges that are dynamically assigned to it per iteration and implement our algorithm on two different multithreaded architectures -- Cray XMT, a massively multithreaded platform, and AMD Magny-Cours, a shared memory multicore platform. In addition to the proof of correctness, we present the performance of our algorithm using a test set of synthetical graphs with up to half-a-billion edges and real world networks from gene correlation studies and demonstrate that our algorithm achieves high scalability for all inputs on both types of architectures.
4 citations
TL;DR: This work generalizes the idea of the clique cover number by defining the rank sum of a cover to be the sum of the minimum ranks of the graphs in the cover, and obtains a combinatorial solution to the minimum rank problem for an outerplanar graph.
4 citations
Posted Content•
TL;DR: In this paper, it was shown that the local clique cover number of every claw-free graph is at most a constant factor smaller than the maximum degree of the graph and a universal constant.
Abstract: A k-clique covering of a simple graph G, is an edge covering of G by its cliques such that each vertex is contained in at most k cliques The smallest k for which G admits a k-clique covering is called local clique cover number of G and is denoted by $lcc(G)$ Local clique cover number can be viewed as the local counterpart of the clique cover number which is equal to the minimum total number of cliques covering all edges In this paper, several aspects of the problem are studied and its relationships to other well-known problems are discussed Moreover, the local clique cover number of claw-free graphs and its subclasses are notably investigated In particular, it is proved that local clique cover number of every claw-free graph is at most $c\Delta / \log\Delta$, where $\Delta$ is the maximum degree of the graph and $c$ is a universal constant It is also shown that the bound is tight, up to a constant factor Furthermore, it is established that local clique number of the linear interval graphs is bounded by $\log\Delta + 1/2 \log \log\Delta + O(1)$ Finally, as a by-product, a new Bollobas-type inequality is obtained for the intersecting pairs of set systems
4 citations
17 Jun 2012
TL;DR: It is proved that even for chordal graphs, it is NP-complete to decide if there is a cross-cycle of a given cardinality, and hence, if a particular homology group of the independence complex is nontrivial.
Abstract: A cross-cycle in a flag simplicial complex K is an induced subcomplex that is isomorphic to the boundary of a cross-polytope and that contains a maximal face of K. A cross-cycle is an efficient way to define a non-zero class in the homology of K. For an independence complex of a graph G, a cross-cycle is equivalent to a combinatorial object: induced matching containing a maximal independent set. We study the complexity of finding cross-cycles in independence complexes. We show that in general this problem is NP-complete when input is a graph whose independence complex we consider. We then focus on the class of chordal graphs, where, as we show, cross-cycles detect all of homology of the independence complex. As our main result, we present a polynomial time algorithm for detecting a cross-cycle in the independence complex of a chordal graph. Our algorithm is based on the geometric intersection representation of chordal graphs and has an efficient implementation.As a corollary, we obtain polynomial time algorithms for such topological properties as contractibility or simple-connectedness of independence complexes of chordal graphs. These problems are undecidable for general independence complexes.We further prove that even for chordal graphs, it is NP-complete to decide if there is a cross-cycle of a given cardinality, and hence, if a particular homology group of the independence complex is nontrivial. As a corollary we obtain that computing homology groups of arbitrary simplicial complexes given as a list of facets is NP-hard.
4 citations
01 Jan 2012
TL;DR: The relationship between volumes of flow polytopes associated to signed graphs and the Kostant partition function has been studied in detail by Baldoni and Vergne using techniques of residues.
Abstract: We establish the relationship between volumes of flow polytopes associated to signed graphs and the Kostant partition function. A special case of this relationship, namely, when the graphs are signless, has been studied in detail by Baldoni and Vergne using techniques of residues. In contrast with their approach, we provide combina- torial proofs inspired by the work of Postnikov and Stanley on flow polytopes. As an application of our results we study a distinguished family of flow polytopes: the Chan-Robbins-Yuen polytopes. Inspired by their beautiful volume formula Q n 2 k=0 Cat(k) for the typeAn case, whereCat(k) is thek th Catalan number, we introduce typeC n+1 and Dn+1 Chan-Robbins-Yuen polytopes along with intriguing conjectures about their volumes. R´
3 citations
TL;DR: In this article, all minimal clique separators of all four standard products: Cartesian, the strong, the direct, and the lexicographic, as well as all maximal atoms of the Cartesian and the strong products were described.
Abstract: We describe in present work all minimal clique separators of all four standard products: the Cartesian, the strong, the direct, and the lexicographic, as well as all maximal atoms of the Cartesian, the strong and the lexicographic product, while we only partially describe maximal atoms of the direct product. In most cases a product have no clique separator and is thus the hole product the maximal atom.
2 citations
TL;DR: A study of the structural properties, some already known and some new, of the clique trees of a chordal graph and the characteristics of the sets that induce subtrees of every clique tree is undertaken.
Abstract: Chordal graphs have characteristic tree representations, the clique trees. The problems of finding one or enumerating them have already been solved in a satisfactory way. In this paper, the following related problem is studied: given a family Open image in new window of trees, all having the same vertex set V, determine whether there exists a chordal graph whose set of clique trees equals Open image in new window. For that purpose, we undertake a study of the structural properties, some already known and some new, of the clique trees of a chordal graph and the characteristics of the sets that induce subtrees of every clique tree. Some necessary and sufficient conditions and examples of how they can be applied are found, eventually establishing that a positive or negative answer to the problem can be obtained in polynomial time. If affirmative, a graph whose set of clique trees equals Open image in new window is also obtained. Finally, all the chordal graphs with set of clique trees equal to Open image in new window are characterized.
2 citations
TL;DR: It is shown that there exists a vertex critical graph which admits a family of k completely independent critical cliques for any k, where k ≥ 2, and its relationship with the double-critical conjecture that the only vertex doublecritical graph is the complete graph is shown.
Abstract: For natural numbers k and n, where 2 ≤ k ≤ n, the vertices of a graph are labeled using the elements of the k-fold Cartesian product In× In×· · ·× In. Two particular graph constructions will be given and the graphs so constructed are called generalized matrix graphs. Properties of generalized matrix graphs are determined and their application to completely independent critical cliques is investigated. It is shown that there exists a vertex critical graph which admits a family of k completely independent critical cliques for any k, where k ≥ 2. Some attention is given to this application and its relationship with the double-critical conjecture that the only vertex doublecritical graph is the complete graph.
2 citations
TL;DR: In this article, the authors generalized chordal graphs to the graphs that have clique representations that are 2-trees, or even series-parallel graphs (partial 2-tree) or maximal outerplanar graphs.
Abstract: Much of the theory—and the applicability—of chordal graphs is based on their being the graphs that have clique trees. Chordal graphs can be generalized to the graphs that have clique representations that are 2-trees, or even series-parallel graphs (partial 2-trees) or outerplanar or maximal outerplanar graphs. The resulting graph classes can be characterized by forbidding contractions of a few induced subgraphs. There is also a plausible application of such graphs to systems biology.
TL;DR: A necessary and sufficient condition for every minimal vertex separator to be contained in the closed neighborhood of a vertex is found and two major characterizations of dually chordal graphs are proved.
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TL;DR: In this paper, the authors classify k-orbit polytopes and determine when a korbit polytope is i-transitive, and then provide an explicit classification of three-orbit polynomials.
Abstract: The orbit graph of a k-orbit polytope is a graph on k nodes that shows how the flag orbits are related by flag adjacency. Using orbit graphs, we classify k-orbit polytopes and determine when a k-orbit polytope is i-transitive. We then provide an explicit classification of three-orbit polytopes, and we describe a generating set for their automorphism groups.
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TL;DR: In this article, the Hansen polytopes of split graphs were analyzed and shown to correspond to threshold graphs, and they were shown to have 3^d+16 nonempty faces.
Abstract: We analyze a remarkable class of centrally symmetric polytopes, the Hansen polytopes of split graphs. We confirm Kalai's 3^d-conjecture for such polytopes (they all have at least 3^d nonempty faces) and show that the Hanner polytopes among them (which have exactly 3^d nonempty faces) correspond to threshold graphs. Our study produces a new family of Hansen polytopes that have only 3^d+16 nonempty faces.
15 Mar 2012
TL;DR: In this article, a new clustering approach was presented, using gene interaction graphs to model this data, and decomposing the graphs by means of clique minimal separators, where clique separators are a clique whose removal increases the number of connected components of the graph.
Abstract: The study of gene interactions is an important research area in biology and grouping genes with similar expression profiles to clusters is a first step towards a better understanding of their functional relationships. In Kaba et al. 2007, a new clustering approach was presented, using gene interaction graphs to model this data, and decomposing the graphs by means of clique minimal separators. A clique separator is a clique whose removal increases the number of connected components of the graph; the decomposition is obtained by repeatedly copying a clique separator into the components it defines, until only subgraphs with no clique separators are left: these subgraphs will be our clusters. The advantage of our approach is that this decomposition can be computed efficiently, is unique, and yields overlapping clusters. For that, the similarity between each pair of genes is estimated by a distance function, then a family of gene interaction graphs is constructed by choosing several thresholds, where an edge is added between two genes if their distance is below the threshold. Hereby, both the choice of the distance function and of the threshold influences the construction of the gene interaction graphs. In Kaba et al. 2007, several criteria are developed to select thresholds in an appropriate way. Here we discuss the impact of the choice of the distance function; our results suggest that this choice does not effect the final decomposition of the gene interaction graphs into clusters.
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TL;DR: It will be shown that a dually chordal graph is 3-colourable if and only if it is perfect and has no clique of size four and that it is NP-complete in case of four colours and solvable in linear time with a simple algorithm in cases of three colours.
Abstract: A graph G is dually chordal if there is a spanning tree T of G such that any maximal clique of G induces a subtree in T. This paper investigates the Colourability problem on dually chordal graphs. It will show that it is NP-complete in case of four colours and solvable in linear time with a simple algorithm in case of three colours. In addition, it will be shown that a dually chordal graph is 3-colourable if and only if it is perfect and has no clique of size four.