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Two non-adjacent vertices x and y in a graph G form an even pair if every induced path between them has an even number of edges. For a given pair fx; yg in a graph G, we denote by G xy the graph obtained from G by contracting x and y. In 1982, Fonlupt and Uhry proved that if G is perfect then so is G xy. In 1987, Meyniel used this fact to prove that no… (More)

In this article, we define a new class of graphs, the fat-extended P 4-laden graphs, and we show a polynomial time algorithm to determine the Grundy number of the graphs in this class. This result implies that the Grundy number can be found in polynomial time for any graph of the following classes: P 4

An even pair in a graph is a pair of non-adjacent vertices such that every chord-less path between them has even length. A graph is called strict quasi-parity when every induced subgraph that is not a clique has an even pair, and it is called perfectly contractile when every induced subgraph can be turned into a clique through a sequence of even-pair… (More)

HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt età la diffusion… (More)

—Given an undirected graph G = (V, E) and a weight function w : V → R + , a vertex coloring of G is a partition of V into independent sets, or color classes. The weight of a vertex coloring of G is defined as the sum of the weights of its color classes, where the weight of a color class is the weight of a heaviest vertex belonging to it. In the W… (More)

A graph is a strict-quasi parity (SQP) graph if every induced subgraph that is not a clique contains a pair of vertices with no odd chordless path between them (an``even pair''). We present an O
n 3 algorithm for recognizing planar strict quasi-parity graphs, based on Wen-Lian Hsu's decomposition of planar (perfect) graphs and on the (non-algorithmic)… (More)

The Grundy number of a graph G is the largest number of colors used by any execution of the greedy algorithm to color G. The problem of determining the Grundy number of G is polynomial if G is a P 4-free graph and N P-hard if G is a P 5-free graph. In this article, we define a new class of graphs, the fat-extended P 4-laden graphs, and we show a polynomial… (More)