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.     

Independent domination in chordal graphs

A graph chordal if it does not contain any cycle of length greater than three as an induced subgraph. A set of S of vertices of a graph G = (V,E) is independent if not two vertices in S are adjacent, and is dominating if every vertex in V?S is adjacent to some vertex in S. We present a linear algorithm to locate a minimum weight independent dominating set in a chordal graph with 0–1 vertex weights.     

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.     

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.     

Obstructions to chordal circular-arc graphs of small independence number

A blocking quadruple (BQ) is a quadruple of vertices of a graph such that any two vertices of the quadruple either miss (have no neighbours on) some path connecting the remaining two vertices of the quadruple, or are connected by some path missed by the remaining two vertices. This is akin to the notion of asteroidal triple used in the classical characterization of interval graphs by Lekkerkerker and Boland [Klee, V., What are the intersection graphs of arcs in a circle?, American Mathematical Monthly 76 (1976), pp. 810–813.].In this note, we first observe that blocking quadruples are obstructions for circular-arc graphs. We then focus on chordal graphs, and study the relationship between the structure of chordal graphs and the presence/absence of blocking quadruples.Our contribution is two-fold. Firstly, we provide a forbidden induced subgraph characterization of chordal graphs without blocking quadruples. In particular, we observe that all the forbidden subgraphs are variants of the subgraphs forbidden for interval graphs [Klee, V., What are the intersection graphs of arcs in a circle?, American Mathematical Monthly 76 (1976), pp. 810–813.]. Secondly, we show that the absence of blocking quadruples is sufficient to guarantee that a chordal graph with no independent set of size five is a circular-arc graph. In our proof we use a novel geometric approach, constructing a circular-arc representation by traversing around a carefully chosen clique tree.     

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.     

Strong clique trees,neighborhood trees,and strongly chordal graphs

Maximal complete subgraphs and clique trees are basic to both the theory and applications of chordal graphs. A simple notion of strong clique tree extends this structure to strongly chordal graphs. Replacing maximal complete subgraphs with open or closed vertex neighborhoods discloses new relationships between chordal and strongly chordal graphs and the previously studied families of chordal bipartite graphs, clique graphs of chordal graphs (dually chordal graphs), and incidence graphs of biacyclic hypergraphs. © 2000 John Wiley & Sons, Inc. J. Graph Theory 33: 151–160, 2000     

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.     

A chordal preconditioner for large-scale optimization

We propose an automatic preconditioning scheme for large sparse numerical optimization. The strategy is based on an examination of the sparsity pattern of the Hessian matrix: using a graph-theoretic heuristic, a block-diagonal approximation to the Hessian matrix is induced. The blocks are submatrices of the Hessian matrix; furthermore, each block is chordal. That is, under a positive definiteness assumption, the Cholesky factorization can be applied to each block without creating any new nonzeros (fill). Therefore the preconditioner is space efficient. We conduct a number of numerical experiments to determine the effectiveness of the preconditioner in the context of a linear conjugate-gradient algorithm for optimization.     

More than one tough chordal planar graphs are Hamiltonian

We prove the result stated in the title. Furthermore, it is proved that for any ϵ > 0, there is a 1-tough chordal planar graph G_ϵ such that the length of a longest cycle of G_ϵ is less than ϵ|V(G_ϵ)|. © 1999 John Wiley & Sons, Inc. J. Graph Theory 32: 405–410, 1999     

Strictly chordal graphs are leaf powers

A fundamental problem in computational biology is the phylogeny reconstruction for a set of specific organisms. One of the graph theoretical approaches is to construct a similarity graph on the set of organisms where adjacency indicates evolutionary closeness, and then to reconstruct a phylogeny by computing a tree interconnecting the organisms such that leaves in the tree are labeled by the organisms and every organism appears as a leaf in the tree. The similarity graph is simple and undirected. For any pair of adjacent organisms in the similarity graph, their distance in the output tree, which is measured by the number of edges on the path connecting them, must be less than some pre-specified bound. This is known as the problem of recognizing leaf powers and computing leaf roots. Graphs that are leaf powers are known to be chordal. It is shown in this paper that all strictly chordal graphs are leaf powers and a linear time algorithm is presented to compute a leaf root for any given strictly chordal graph. An intermediate root-and-power problem, the Steiner root problem, is also examined.     

Two new sufficient conditions for Hamilton-connected graphs

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The clique-separator graph for chordal graphs

We present a new representation of a chordal graph called the clique-separator graph, whose nodes are the maximal cliques and minimal vertex separators of the graph. We present structural properties of the clique-separator graph and additional properties when the chordal graph is an interval graph, proper interval graph, or split graph. We also characterize proper interval graphs and split graphs in terms of the clique-separator graph. We present an algorithm that constructs the clique-separator graph of a chordal graph in O(n³) time and of an interval graph in O(n²) time, where n is the number of vertices in the graph.     

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.     

On signpost systems and connected graphs

By a signpost system we mean an ordered pair (W, P), where W is a finite nonempty set, P W × W × W and the following statements hold: if (u, v, w) P, then (v, u, u) P and (v, u, w) P, for all u, v, w W; if u v; then there exists r W such that (u, r, v) P, for all u, v W. We say that a signpost system (W, P) is smooth if the folowing statement holds for all u, v, x, y, z W: if (u, v, x), (u, v, z), (x, y, z) P, then (u, v, y) P. We say thay a signpost system (W, P) is simple if the following statement holds for all u, v, x, y W: if (u, v, x), (x, y, v) P, then (u, v, y), (x, y, u) P.By the underlying graph of a signpost system (W, P) we mean the graph G with V(G) = W and such that the following statement holds for all distinct u, v W: u and v are adjacent in G if and only if (u, v, v) P. The main result of this paper is as follows: If G is a graph, then the following three statements are equivalent: G is connected; G is the underlying graph of a simple smooth signpost system; G is the underlying graph of a smooth signpost system.Research was supported by Grant Agency of the Czech Republic, grant No. 401/01/0218.     

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.     

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.