Showing papers on "K-tree published in 2009"
TL;DR: The notion of the transition graph T(G) whose vertices are maximal cliques of G and arcs are transitions between cliques is introduced and it is shown that under some specific numbering, the transition graphs has a hamiltonian path for chordal and comparability graphs.
78 citations
TL;DR: This paper presents a novel approach for partial clique enumeration, which is based on a continuous formulation of the clique problem developed in the 1960s by Motzkin and Straus, and is able to avoid extracting the same clique multiple times.
34 citations
20 Jun 2009
TL;DR: It is proved in this paper that the pathwidth problem is NP-hard for particular subclasses of chordal graphs, and it is deduced that the problem remains hard for weighted trees.
Abstract: The pathwidth of a graph G is the minimum clique number of H minus one, over all interval supergraphs H of G . We prove in this paper that the pathwidth problem is NP-hard for particular subclasses of chordal graphs, and we deduce that the problem remains hard for weighted trees. We also discuss subclasses of chordal graphs for which the problem is polynomial.
16 citations
TL;DR: This note proves the following Andrade-Boros-Gurvich conjecture: A graph is almost CIS if and only if it is a split graph with a unique split partition.
Abstract: A graph $G$ is called CIS if each maximal clique intersects each maximal stable set in $G$ and is called almost CIS if it has a unique disjoint pair $(C,S)$ consisting of a maximal clique $C$ and a maximal stable set $S$. While it is still unknown if there exists a good structural characterization of all CIS graphs, in this note we prove the following Andrade-Boros-Gurvich conjecture: A graph is almost CIS if and only if it is a split graph with a unique split partition.
14 citations
TL;DR: It is shown that if T(G)>=@t(G-S)@w( G-S)+1k-2, for any subset S of V(G), with k>=3, then G has a k-tree.
13 citations
TL;DR: A conjecture of Ganley and Heath is disproved by showing that when $k\geq3$, there are $k-trees that do not embed in $k$ pages, and an algorithm is presented that produces$k$-page embeddings for £k-Trees in a special class.
Abstract: A $p$-page embedding of a graph $G$ is a vertex-ordering $\pi$ of $V(G)$ (along the “spine” of a book) and an assignment of edges to $p$ half-planes (called “pages”) such that no page contains crossing edges (alternating endpoints) relative to $\pi$ The pagenumber of $G$ is the least $p$ such that $G$ has a $p$-page embedding We disprove a conjecture of Ganley and Heath by showing that when $k\geq3$, there are $k$-trees that do not embed in $k$ pages We also present an algorithm that produces $k$-page embeddings for $k$-trees in a special class
11 citations
TL;DR: It is proved that the recognition problem for coordinated graphs is NP-hard, and it isNP-complete even when restricted to the class of {gem, C4, odd hole}-free graphs with maximum degree four, maximum clique size three and at most three cliques sharing a common vertex.
Abstract: A graph G is coordinated if the minimum number of colors that can be assigned to the cliques of H in such a way that no two cliques with non-empty intersection receive the same color is equal to the maximum number of cliques of H with a common vertex, for every induced subgraph H of G. In previous works, polynomial time algorithms were found for recognizing coordinated graphs within some classes of graphs. In this paper we prove that the recognition problem for coordinated graphs is NP-hard, and it is NP-complete even when restricted to the class of {gem, C4, odd hole}-free graphs with maximum degree four, maximum clique size three and at most three cliques sharing a common vertex.
8 citations
Posted Content•
TL;DR: It is shown that for any chordal graph the authors can construct in linear time a simplicial elimination scheme starting with a pending maximal clique attached via a minimal separator maximal under inclusion among all minimal separators.
Abstract: We present here some results on particular elimination schemes for chordal graphs, namely we show that for any chordal graph we can construct in linear time a simplicial elimination scheme starting with a pending maximal clique attached via a minimal separator maximal (resp. minimal) under inclusion among all minimal separators.
8 citations
TL;DR: The structure of an unoriented graph R d on the set of reflexive polytopes of a fixed dimension d is suggested, and an explicit finite list of quivers is presented giving all d -dimensional reflexive flow poly topes up to lattice isomorphism.
8 citations
Proceedings Article•
IBM1
TL;DR: A novel preprocessing technique to reduce the graph size before enumerating the large maximal cliques, which is of great practical interest since enumerating maximal clique enumeration is a computationally hard problem and the execution time increases rapidly with the input size.
Abstract: Identifying communities in social networks is a problem of great interest. One popular type of community is where every member of the community knows all others, which can be viewed as a clique in the graph representing the social network. In several real life situations, nding small cliques may not be interesting as they are large in number and low in information content. Hence, in this paper, we propose a variant of maximal clique enumeration problem where we try to enumerate only large maximal cliques. We describe a novel preprocessing technique to reduce the graph size before enumerating the large maximal cliques. This is of great practical interest since enumerating maximal cliques is a computationally hard problem and the execution time increases rapidly with the input size. We also present a new maximal clique enumeration algorithm SELMaC2, which exploits the constraint on minimum size of the desired maximal cliques. We present experimental results on several real life social networks. Our results show that the preprocessing methods achieve signican t reduction in the graph size. Also our algorithm has fewer intermediate steps and is faster than the competing algorithms adapted from the literature by incorporating the minimum size criterion. Our results also show the scalability of our approach.
8 citations
26 Jun 2009
TL;DR: Given a graph of n vertices and whose maximum degree is Δ, it is proved that if Δ is less than or equal to 2.493dlg n (d≥1: a constant), then the maximum clique problem is solvable in the polynomial time of O(n2+d).
Abstract: The maximum clique problem is known to be a typical NP-complete problem, and hence it is believed to be impossible to solve it in polynomial-time. So, it is important to know a reasonable sufficient condition under which the maximum clique problem can be proved to be polynomial-time solvable. In this paper, given a graph of n vertices and whose maximum degree is Δ, we prove that if Δ is less than or equal to 2.493dlg n (d≥1: a constant), then the maximum clique problem is solvable in the polynomial time of O(n2+d). The proof is based on a very simple algorithm which is obtained from an algorithm CLIQUES that generates all maximal cliques in a depth-first way in O(3n/3)-time (which is published in Theoretical Computer Science 363, 2006, as "The worstcase time complexity for generating all maximal cliques and computational experiments" by E. Tomita et al.). The proof itself is very simple.
TL;DR: The main difficulty found here, was the case C n ( 1, 2, 4 ) which is clique divergent, but no previously known technique could be used to prove it.
TL;DR: It is shown how to find in time O ( k n ) an optimal colouring, amaximum independent set, a maximum clique, and an optimal clique cover of an n-vertex chordal graph G with directed vertex leafage k if a representation of G is given.
Journal Article•
TL;DR: An algorithm enumerating all minimal covers using the ⊂-minimal elements of the interval order, as well as an independence Metropolis sampler, and characterized maximal removable sets, which are the complements of minimal covers, are produced.
Abstract: We address the problem of determining all sets which form minimal covers of maximal cliques for interval graphs. We produce an algorithm enumerating all minimal covers using the ⊂-minimal elements of the interval order, as well as an independence Metropolis sampler. We characterize maximal removable sets, which are the complements of minimal covers, and produce a distinct algorithm to enumerate them. We use this last characterization to provide bounds on the maximum number of minimal covers for an interval order with a given number of maximal cliques, and present some simulation results on the number of minimal covers in different settings. This work was supported in part by the National Sciences and Engineering Research Council of Canada, the Fonds quebecois de la recherche sur la nature et les technologies and the New Zealand Marsden Fund.
TL;DR: This study will undertake a similar study but by considering minimal separators and their properties, and finds new characterizations of dually chordal graphs.
TL;DR: It is proved that every chordal planar graph G with toughness t(G)>34 has a 2-walk, which is a closed spanning trail which uses every vertex at most twice.
01 Jan 2009
TL;DR: In this article, Wang et al. introduced the concept of essential maximal cliques, which are maximal clique containing an edge that is not in any other clique, and studied the class of graphs for which the set of all essential cliques of a graph can form an edge maximal-clique covering of minimum size.
Abstract: An edge maximal clique covering of a graph is a set of maximal cliques that contains every edge. In 1990, W. D. Wallis and G.-H. Zhang [On maximal clique irreducible graphs, Journal of Combinatorial Mathematics and Combinatorial Computing (JCMCC) 8 (1990), 187-193] introduced the concept of essential maximal cliques, which are maximal cliques containing an edge that is not in any other maximal clique. They studied the graphs for which the set of all maximal cliques forms an edge maximal clique covering of minimum size, namely maximal clique irreducible graphs. In 2003, T.-M. Wang [On characterizing weakly maximal clique irreducible graphs, Congressus Numerantium 163 (2003), 177-188] introduced and studied the class of graphs for which the set of all essential maximal cliques forms an edge maximal clique covering of minimum size, namely weakly maximal clique irreducible graphs. On the other hand, the Helly property of the set of maximal cliques is related to the maximal clique irreducibility. A graph is clique-Helly if the set of all maximal cliques satisfies the Helly property, and it is called hereditary clique-Helly if every induced sub-graph is clique-Helly. In 1993, E. Prisner showed that hereditary clique-Helly graphs and hereditary maximal clique irreducible graphs are the