. We apply Lekkerkerker and Boland's recognition algorithm for triangulated graphs to the class of weakly triangulated graphs. This yields a new characterization of weakly triangulated graphs, as well as a new O(m 2 ) recognition algorithm which, unlike the previous ones, is not based on the notion of a 2-pair, but rather on the structural properties of the minimal separators of the graph. It also gives the strongest relationship to the class of triangulated graphs that has been established so far. CR Classification: G.2.2, F.2.2 Key words: Weakly triangulated graphs, graph recognition, graph characterization, minimal separators, triangulated graphs. 1. Introduction Weakly triangulated graphs were introduced by Hayward [11] as a natural extension of the perfect class of triangulated graphs. A graph is triangulated, or chordal, if it does not contain a chordless cycle on four or more vertices. A graph is weakly triangulated if neither the graph nor its complement contains a chordl...

We use the notion of potential maximal clique to characterize the maximal cliques appearing in minimal triangulations of a graph. We show that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fill-in are polynomially tractable for these graphs. We prove that for all classes of graphs for which polynomial algorithms computing the treewidth and the minimum fill-in exist, we can list their potential maximal cliques in polynomial time. Our approach unies these algorithms. Finally we show how to compute in polynomial time the potential maximal cliques of weakly triangulated graphs, for which the treewidth and the minimum fill-in problems were open.

Single nucleotide polymorphisms (SNPs) are the most frequent form of human genetic variation. They are of fundamental importance for a variety of applications including medical diagnostic and drug design. They also provide the highest-resolution genomic fingerprint for tracking disease genes. This paper is devoted to algorithmic problems related to computational SNPs validation based on genome assembly of diploid organisms. In diploid genomes, there are two copies of each chromosome. A description

In this report, we first studied the characteristics of probe interval graphs. Then we analyzed an O(n³) algorithm, which recongnizes probe interval graphs and gives a possible solution. In addition, the algorithm was illustrated by two examples step by step. Based on our analysis, we provided a design for future implementation of this algorithm.

this paper, we define the P 4 -tidy graphs, a new class of graphs strictly containing the previous considered classes (excepted the class defined by Fouquet & Giakoumakis in [6]). We show that the modular decomposition tree T (G) of a graph G

We extend Dirac's characterization by the minimal separators of a triangulated graph to a new characterization for weakly triangulated graphs, and use this to interpret the known properties of weakly triangulated graphs as an extension of the corresponding properties of triangulated graphs. Our new insight, applied to a result of Rose on triangulated graphs, enables us to bound the number of minimal separators of a weakly triangulated graph.

. In the first part of this paper, a new structure for chordal graph is introduced, namely the clique graph. This structure is shown to be optimal with regard to the set of clique trees. The greedy aspect of the recognition algorithms of chordal graphs is studied. A new greedy algorithm that generalizes both Maximal cardinality Search (MCS) and Lexicographic Breadth first search is presented. The trace of an execution of MCS is defined and used in two linear time and space algorithms: one builds a clique tree of a chordal graph and the other is a simple recognition procedure of chordal graphs. Introduction Since chordal graphs have no chordless cycle of length more than 3, they can be considered as a generalization of trees. In [9], chordal graphs have been considered as the intersection graphs of subtrees of a tree. Chordal graphs are often represented by a clique tree (see [9, 19]). This is a structure which translates most of the information contained in a chordal graph. The struct...

In this paper, we study the problems of detecting holes and antiholes in general undirected graphs and present algorithms for them, which, for a graph on n vertices and m edges, run in O(n + m²) time and require O(nm) space; we thus provide a solution to the open problem posed by Hayward, Spinrad, and Sritharan in [12] asking for an O(n^4)- time algorithm for finding holes in arbitrary graphs. The key element of the algorithms is a special type of depthfirst search traversal which proceeds along P4 s (i.e., chordless paths on four vertices) of the input graph. We also describe a different approach which allows us to detect antiholes in graphs that do not contain chordless cycles on 5 vertices in O(n + m²) time requiring O(n +m) space. Our algorithms are simple and can be easily used in practice. Additionally, we show how our detection algorithms can be augmented so that they return a hole or an antihole whenever such a structure is detected in the input graph; the augmentation takes O(n +m) time and space.

. We show that Meyniel weakly triangulated graphs are co-perfectly orderable (equivalently, that P 5 -free weakly triangulated graphs are perfectly orderable). Our proof is algorithmic, and relies on a notion concerning separating sets, a property of weakly triangulated graphs, and several properties of Meyniel weakly triangulated graphs. Key Words. P 5 -free graph, weakly triangulated graph, perfectly orderable graph, Meyniel graph 1. Introduction. Graph coloring is in general a hard problem. As such, it is interesting to investigate restrictions under which a graph's chromatic number might be easily determined. One such investigation was initiated in 1984 by Chv'atal [C1], who proposed to study those graphs on which a certain efficient coloring algorithm always returned an optimal coloring. The algorithm is this: given a linear order of the vertices of a graph, color the vertices in order, assigning to each the smallest positive integer (each integer representing a color) not assi...

For a chordal graph G = (V, E), we study the problem of whether a new vertex u � ∈ V and a given set of edges between u and vertices in V can be added to G so that the resulting graph remains chordal. We show how to resolve this efficiently, and at the same time, if the answer is no, specify a maximal subset of the proposed edges that can be added along with u, or conversely, a minimal set of extra edges that can be added in addition to the given set, so that the resulting graph is chordal. In order to do this, we give a new characterization of chordal graphs and, for each potential new edge uv, a characterization of the set of edges incident to u that also must be added to G along with uv. We propose a data structure that can compute and add each such set in O(n) time. Based on these results, we present an algorithm that computes both a minimal triangulation and a maximal chordal subgraph of an arbitrary input graph in O(nm) time, using a totally new vertex incremental approach. In contrast to previous algorithms, our process is on-line in that each new vertex is added without reconsidering any choice made at previous steps, and without requiring any knowledge of the vertices that might be added subsequently. 1