On the Grundy number of graphs with few P4's

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 time algorithm to determine the Grundy number of any graph in this class. Our class intersects the class of P 5-free graphs and strictly contains the class of P 4-free graphs. More precisely, our result implies that the Grundy number can be computed in polynomial time for any graph of the following classes: P 4