Abstract
The class of 2 K 2 -free graphs includes several interesting subclasses such as split, pseudo-split, threshold graphs, complements to chordal, interval or trivially perfect graphs. The fundamental property of 2 K 2 -free graphs is that they contain polynomially many maximal independent sets. As a consequence, several important problems that are NP-hard in general graphs, such as 3-colorability, maximum weight independent set (WIS), minimum weight independent dominating set (WID), become polynomial-time solvable when restricted to the class of 2 K 2 -free graphs. In the present paper, we extend 2 K 2 -free graphs to larger classes with polynomial-time solvable WIS or WID. In particular, we show that WIS can be solved in polynomial time for ( K 2 + K 1 , 3 ) -free graphs and WID for ( K 2 + K 1 , 2 ) -free graphs. The latter result is in contrast with the fact that independent domination is NP-hard in the class of 2 K 1 , 2 -free graphs, which has been recently proven by Zverovich.
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