Geodesic
On $P_5$-free locally split graphs
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 4, pp. 1063-1090

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In this paper we study a graph which contains no induced path of five vertices which is known as the P_5-free graph. We prove that every prime P_5-free locally split graph has either a bounded number of vertices, or is a subclass of a (2, 1) split graph, or is a split graph. Then we show that the Minimum Coloring problem (MC) and the maximum independent set problem (MIS) for P_5-free locally split graphs can be both solved in polynomial time.
Keywords: $SP_{5}$-free graphs, modular decomposition, recognition, maximum independent set, minimum coloring 
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Issaadi, Hayat; Ait Haddadene, Hacene; Kheddouci, Hamamache. On $P_5$-free locally split graphs. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 4, pp. 1063-1090. http://geodesic.mathdoc.fr/item/DMGT_2023_43_4_a10/