Abstract
In this paper, we define a graph G as semi- P 4-sparse if G does not contain as induced subgraph a P 5, a P 5 or the complement of a fork, where a fork is the tree of order 5 with 3 pendent vertices. This new class of graphs contains strictly the class of P 4-sparse graphs. Using modular decomposition we first propose a linear recognition algorithm for semi- P 4-sparse graphs and next, we show that with very little work, we can extend the linear algorithms of Chvátal et al. (1987) concerning the class of perfect graphs that are P 5, P 5 and C 5-free, for finding an optimal coloring, a largest clique and a largest stable set of a semi- P 4-sparse graph G. Finally, we characterize by forbidden configurations the closure by substitution-composition (the inverse operation of modular decomposition) of semi- P 4-sparse graphs, composition operation of graphs that (among other properties) preserves perfection.
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