Independent Sets in Classes Related to Chair-Free Graphs

Abstract

The Maximum Weight Independent Set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. MWIS is known to be N P -complete in general, even under various restrictions. Let S i,j,k be the graph consisting of three induced paths of lengths i, j, k with a common initial vertex. The complexity of the MWIS problem for S 1,2,2 -free graphs, and for S 1,1,3 -free graphs are open. In this paper, we show that the MWIS problem can solved in polynomial time for (S 1,2,2 , S 1,1,3 , cochair)-free graphs, by analyzing the structure of the subclasses of this class of graphs. This extends some known results in the literature.

Maximum weight stable set in (P7, bull)-free graphs and (S1,2Maffray1, Pastor2 2018Discrete Mathematics6120View full textAdd to dashboardCite

“…Therefore the new frontier to explore is when the forbidden induced subgraph has six or more vertices. There are several results on the existence of a polynomial time algorithm for MWSS in subclasses of P 6 -free graphs [13,14,18,20,22,23]. Mosca [21] proved that MWSS is solvable in polynomial for the class of (P 7 , banner)-free graphs.…”

Independence and Efficient Domination on P6-free Graphs

“…For P 4 -free graphs a polynomialtime algorithm was given by Corneil et al [14] in 1981, and it took more than 30 years until a polynomialtime algorithm for the problem on P 5 -free graphs was discovered by Lokshtanov et al [25] in 2014. In the meanwhile, a substantial amount of work was devoted to Independent Set on subclasses of P 5 -free graphs [4,5,11,18,28,34], and some progress has been reported on subclasses of P 6 -free graphs [23,29,30,31].…”