Intersection models of weakly chordal graphs

Matemática Contemporânea, 2015

Algorithms for detecting particular induced subgraphs have been the focus of much research recently, mostly related to their connection to many classes of graphs defined by forbidden induced subgraphs. In this context, Chudnovsky and Seymour proposed a useful tool, called three-in-a-tree algorithm which solves the following problem in polynomial time: given a graph with three prescribed vertices, test if there is an induced tree containing these vertices. In our paper we deal with a generalization of this problem, known as k-in-a-tree problem. For the case where k is part of the input, the problem is known to be NP-complete. For fixed k, the complexity of this problem is open for k ≥ 4, although there are polynomial time algorithms for restricted cases, such as claw-free graphs and graphs with girth at least k. In this paper we give a O(nm 2 ) time algorithm for the k-in-a-tree problem for chordal graphs, even in the case where k is part of the input. Furthermore, the algorithm outputs an induced tree containing the k prescribed vertices when there is such tree.

Electronic Notes in Discrete Mathematics, 2018

Chordal graphs are the graphs in which every cycle of length at least four has a chord. A set S is a vertex separator for vertices a and b if the removal of S of the graph separates a and b into distinct connected components. A graph G is chordal if and only if every minimal vertex separator is a clique. We study subclasses of chordal graphs defined by restrictions imposed on the intersections of its minimal separator cliques. Our goal is to characterize them by forbidden induced subgraphs. Some of these classes have already been studied such as chordal graphs in which two minimal separators have no empty intersection if and only if they are equal. Those graphs are known as strictly chordal graphs and they were first introduced as block duplicate graphs by Golumbic and Peled [5], they were also considered in [7] and [2], showing that strictly chordal graphs are exactly the (gem, dart)-free graphs.

Journal of Combinatorial Theory, Series B, 1986

The intersection graph for a family of sets is obtained by associating each set with a vertex of the graph and joining two vertices by an edge exactly when their corresponding sets have a nonempty intersection. Intersection graphs arise naturally in many contexts, such as scheduling conflicting events, and have been widely studied. We present a unilied framework for studying several classes of intersection graphs arising from families of paths in a tree. Four distinct classes of graphs arise by considering paths to be the sets of vertices or the edges making up the path, and by allowing the underlying tree to be undirected or directed; in the latter case only directed paths are allowed. Two further classes are obtained by requiring the directed tree to be rooted. We introduce other classes of graphs as well. The rooted directed vertex path graphs have been studied by Gavril: the vertex path graphs have been studied by Gavril and Renz; the edge path graphs have been studied by Golumbic and Jamison, Lobb, Syslo. and Tarjan. The main results are a characterization of these graphs in terms of their "clique tree"representations and a unified recognition algorithm. The algorithm repeatedly separates an arbitrary graph by a (maximal) clique separator, checks the form of the resultant nondecomposable "atoms," and finally checks that each separation step is valid. In all cases, the first two steps can be performed in polynomial time. In all but one case, the final step is based on a certain two-coloring condition and so can be done efficiently; in the other case the recognition problem can be shown to be NP-complete since a certain three-coloring condition is needed. The strength of this unified approach is that it clarifies and unities virtually all of the important known results for these graphs and provides substantial new results as well. For example, the exact intersecting relationships among these graphs. and between these graphs and chordal and perfect graphs fall out easily as corollaries. A number of other results, such as bounds on the number of (maximal) cliques, related optimization problems on these graphs, etc., are presented along with open problems.

Parallel Processing Letters, 2004

We prove algorithmic characterizations of weakly chordal graphs, which lead to efficient parallel algorithms for recognizing P5-free and [Formula: see text]-free weakly chordal graphs. For an input graph on n vertices and m edges, our algorithms run in O( log 2n) time and require O(m2/ log n) processors on the EREW PRAM model of computation. The proposed recognition algorithms efficiently detect P5 s and [Formula: see text] in weakly chordal graphs in O( log n) time with O(m2/ log n) processors on the EREW PRAM. Additionally, we show how the algorithms can be augmented to provide a certificate for the existence of a P5 (or a [Formula: see text]) in case the input graph is not P5-free (respectively, [Formula: see text]-free) weakly chordal.

Discrete Applied Mathematics, 2014

Chordal graphs and their clique graphs (called dually chordal graphs) possess characteristic tree representations, namely, the clique tree and the compatible tree, respectively. The following problem is studied: given a chordal graph G, determine if the clique trees of G are exactly the compatible trees of the clique graph of G. This leads to a new subclass of chordal graphs, basic chordal graphs, which is here characterized. The question is also approached backwards: given a dually chordal graph G, we find all the basic chordal graphs with clique graph equal to G. This approach leads to the possibility of considering several properties of clique trees of chordal graphs and finding their counterparts in compatible trees of dually chordal graphs.

Discrete Mathematics, 2012

Every chordal graph G admits a representation as the intersection graph of a family of subtrees of a tree. A classic way of finding such an intersection model is to look for a maximum spanning tree of the valuated clique graph of G. Similar techniques have been applied to find intersection models of chordal graph subclasses as interval graphs and path graphs. In this work, we extend those methods to be applied beyond chordal graphs: we prove that a graph G can be represented as the intersection of a Helly separating family of graphs belonging to a given class if and only if there exists a spanning subgraph of the clique graph of G satisfying a particular condition. Moreover, such a spanning subgraph is characterized by its weight in the valuated clique graph of G. The specific case of Helly circular-arc graphs is treated. We show that the canonical intersection models of those graphs correspond to the maximum spanning cycles of the valuated clique graph.

Discrete Applied Mathematics, 2000

In this paper those graphs are studied for which a so-called strong ordering of the vertex set exists. This class of graphs, called strongly orderable graphs, generalizes the strongly chordal graphs and the chordal bipartite graphs in a quite natural way. We consider two characteristic elimination orderings for strongly orderable graphs, one on the vertex set and the second on the edge set, and prove that these graphs can be recognized in O(|V | + |E|)|V | time. Moreover, a special strong ordering of a strongly orderable graph can be produced in the same time bound. We present variations of greedy algorithms that compute a minimum coloring, a maximum clique, a minimum clique partition and a maximum independent set of a strongly orderable graph in linear time if such a special strong ordering is given.

Discrete Mathematics, 1998

In this paper, we present a simple charactrization of strongly chordal graphs. A chordal graph is strongly chordal if and only if every cycle on six or more vertices has an induced triangle with exactly two edges of the triangle as the chords of the cycle. (~) 1998 Elsevier Science B.V. All rights reserved

A graph G is a (P1 , P2)-intersection graph of a graph if there exists a host graph H and a family of 1-paths and 2-paths in H such that G is the intersection graph of these paths. The (P1 , P2)-intersection graphs of a tree and a path are characterized by forbidden subgraphs. Also the (P1 , P2 , P3)-intersection graphs of a path are characterized by forbidden subgraphs.

SIAM Journal on Discrete Mathematics, 1998

Recently in several papers, graphs with maximum neighborhood orderings were characterized and turned out to be algorithmically useful. This paper gives a unified framework for characterizations of those graphs in terms of neighborhood and clique hypergraphs which have the Helly property and whose line graph is chordal. These graphs are dual (in the sense of hypergraphs) to chordal graphs. By using the hypergraph approach in a systematical way new results are obtained, some of the old results are generalized, and some of the proofs are simplified.