Let T be a set of graphs. A graph G is called T-free if G contains no induced subgraphs isomorphic to any graph in T. If T = { K_(1, 3) }, a T-free graph is called claw-free. The concept of quasi-claw-free graphs was introduced by Ainouche. A graph G is called quasi-claw-free if it satisfies the property: d(x, y) = 2 = > there exists u ∈ N(x) ∩ N(y) such that N[u] is contained in N[x] ∪ N[y]. It is obvious that every claw-free graph is quasi-claw-free. Allan and Laskar have shown that every claw-free graph has equal domination and independent domination numbers. In this paper the result of Allan and Laskar is extended. In particular, It is shown that every S-free and quasi-claw-free graph has equal domination and independent domination numbers, where S is set of graphs such that every graph in S has at least two induced claws and it is specified explicitly in the paper.
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