Cover-incomparability graphs and chordal graphs

The problem of recognizing cover-incomparability graphs (i.e. the graphs obtained from posets as the edge-union of their covering and incomparability graph) was shown to be NP-complete in general [J. Maxová, P. Pavlíkova, A. Turzík, On the complexity of cover-incomparability graphs of posets, Order 26 (2009) 229-236], while it is for instance clearly polynomial within trees. In this paper we concentrate on (classes of) chordal graphs, and show that any cover-incomparability graph that is a chordal graph is an interval graph. We characterize the posets whose cover-incomparability graph is a block graph, and a split graph, respectively, and also characterize the cover-incomparability graphs among block and split graphs, respectively. The latter characterizations yield linear time algorithms for the recognition of block and split graphs, respectively, that are cover-incomparability graphs.     

Introducing subclasses of basic chordal graphs

Basic chordal graphs arose when comparing clique trees of chordal graphs and compatible trees of dually chordal graphs. They were defined as those chordal graphs whose clique trees are exactly the compatible trees of its clique graph.In this work, we consider some subclasses of basic chordal graphs, like hereditary basic chordal graphs, basic DV and basic RDV graphs, we characterize them and we find some other properties they have, mostly involving clique graphs.     

Brambles and independent packings in chordal graphs

An independent packing of triangles is a set of pairwise disjoint triangles, no two of which are joined by an edge. A triangle bramble is a set of triangles, every pair of which intersect or are joined by an edge. More generally, I consider independent packings and brambles of any specified connected graphs, not just triangles. I give a min-max theorem for the maximum number of graphs in an independent packing of any family of connected graphs in a chordal graph, and a dual min-max theorem for the maximum number of graphs in a bramble in a chordal graph.     

On the interval completion of chordal graphs

For a given graph G=(V,E), the interval completion problem of G is to find an edge set F such that the supergraph H=(V,E∪F) of G is an interval graph and |F| is minimum. It has been shown that it is equivalent to the minimum sum cut problem, the profile minimization problem and a kind of graph searching problems. Furthermore, it has applications in computational biology, archaeology, and clone fingerprinting. In this paper, we show that it is NP-complete on split graphs and propose an efficient algorithm on primitive starlike graphs.     

On 2-walks in chordal planar graphs

A 2-walk is a closed spanning trail which uses every vertex at most twice. A graph is said to be chordal if each cycle different from a 3-cycle has a chord. We prove that every chordal planar graph G with toughness has a 2-walk.     

Polarity of chordal graphs

Polar graphs are a common generalization of bipartite, cobipartite, and split graphs. They are defined by the existence of a certain partition of vertices, which is NP-complete to decide for general graphs. It has been recently proved that for cographs, the existence of such a partition can be characterized by finitely many forbidden subgraphs, and hence tested in polynomial time. In this paper we address the question of polarity of chordal graphs, arguing that this is in essence a question of colourability, and hence chordal graphs are a natural restriction. We observe that there is no finite forbidden subgraph characterization of polarity in chordal graphs; nevertheless we present a polynomial time algorithm for polarity of chordal graphs. We focus on a special case of polarity (called monopolarity) which turns out to be the central concept for our algorithms. For the case of monopolar graphs, we illustrate the structure of all minimal obstructions; it turns out that they can all be described by a certain graph grammar, permitting our monopolarity algorithm to be cast as a certifying algorithm.     

Locally connected spanning trees in strongly chordal graphs and proper circular-arc graphs

A locally connected spanning tree of a graph G is a spanning tree T of G such that the set of all neighbors of v in T induces a connected subgraph of G for every v∈V(G). The purpose of this paper is to give linear-time algorithms for finding locally connected spanning trees on strongly chordal graphs and proper circular-arc graphs, respectively.     

An implicit representation of chordal comparability graphs in linear time

Ma and Spinrad have shown that every transitive orientation of a chordal comparability graph is the intersection of four linear orders. That is, chordal comparability graphs are comparability graphs of posets of dimension four. Among other uses, this gives an implicit representation of a chordal comparability graph using O(n) integers so that, given two vertices, it can be determined in O(1) time whether they are adjacent, no matter how dense the graph is. We give a linear time algorithm for finding the four linear orders, improving on their bound of O(n²).     

Geodeticity of the contour of chordal graphs

A vertex v is a boundary vertex of a connected graph G if there exists a vertex u such that no neighbor of v is further away from u than v. Moreover, if no vertex in the whole graph V(G) is further away from u than v, then v is called an eccentric vertex of G. A vertex v belongs to the contour of G if no neighbor of v has an eccentricity greater than the eccentricity of v. Furthermore, if no vertex in the whole graph V(G) has an eccentricity greater than the eccentricity of v, then v is called a peripheral vertex of G. This paper is devoted to study these kinds of vertices for the family of chordal graphs. Our main contributions are, firstly, obtaining a realization theorem involving the cardinalities of the periphery, the contour, the eccentric subgraph and the boundary, and secondly, proving both that the contour of every chordal graph is geodetic and that this statement is not true for every perfect graph.     

Vertex partitions of chordal graphs

A k‐tree is a chordal graph with no (k + 2)‐clique. An ?‐tree‐partition of a graph G is a vertex partition of G into ‘bags,’ such that contracting each bag to a single vertex gives an ?‐tree (after deleting loops and replacing parallel edges by a single edge). We prove that for all k ≥ ? ≥ 0, every k‐tree has an ?‐tree‐partition in which each bag induces a connected ‐tree. An analogous result is proved for oriented k‐trees. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 167–172, 2006     

A linear time algorithm to list the minimal separators of chordal graphs

Kumar and Madhavan [Minimal vertex separators of chordal graphs, Discrete Appl. Math. 89 (1998) 155-168] gave a linear time algorithm to list all the minimal separators of a chordal graph. In this paper we give another linear time algorithm for the same purpose. While the algorithm of Kumar and Madhavan requires that a specific type of PEO, namely the MCS PEO is computed first, our algorithm works with any PEO. This is interesting when we consider the fact that there are other popular methods such as Lex BFS to compute a PEO for a given chordal graph.     

Intersection models of weakly chordal graphs

We first present new structural properties of a two-pair in various graphs. A two-pair is used in a well-known characterization of weakly chordal graphs. Based on these properties, we prove the main theorem: a graph G is a weakly chordal ()-free graph if and only if G is an edge intersection graph of subtrees on a tree with maximum degree 4. This characterizes the so called [4, 4, 2] graphs. The proof of the theorem constructively finds the representation. Thus, we obtain an algorithm to construct an edge intersection model of subtrees on a tree with maximum degree 4 for such a given graph. This is a recognition algorithm for [4, 4, 2] graphs.     

Chordal bipartite, strongly chordal, and strongly chordal bipartite graphs

Robert E. Jamison characterized chordal graphs by the edge set of every k-cycle being the symmetric difference of k−2 triangles. Strongly chordal (and chordal bipartite) graphs can be similarly characterized in terms of the distribution of triangles (respectively, quadrilaterals). These results motivate a definition of ‘strongly chordal bipartite graphs’, forming a class intermediate between bipartite interval graphs and chordal bipartite graphs.     

弦图的补图的最小填充

从图论观点讲,最小填充问题就是在一个图G中添加边集F,使得图G的母图G F是一个弦图而且所添边的边数| F|是最小的,其中最小值| F|称为图G的填充数,表示为f( G) .对一般图来说,最小填充问题是NP-困难的,但是对一些特殊图类来说,这个问题是在多项式时间内可解的.本文给出了弦图的补图-G的填充数f(-G) .     

On the strong chromatic index and maximum induced matching of tree-cographs,permutation graphs and chordal bipartite graphs

We show that there exist linear-time algorithms that compute the strong chromatic index and a maximum induced matching of tree-cographs when the decomposition tree is a part of the input. We also show that there exist efficient algorithms for the strong chromatic index of (bipartite) permutation graphs and of chordal bipartite graphs.     

Two methods for the generation of chordal graphs

In this paper two methods for automatic generation of connected chordal graphs are proposed: the first one is based on new results concerning the dynamic maintenance of chordality under edge insertions; the second is based on expansion/merging of maximal cliques. Theoretical and experimental results are presented. In both methods, chordality is preserved along the whole generation process. L. Markenzon’s research is partially supported by grant 301068/2003-8, CNPq, Brazil.