Quasi-claw-free graphs (Q1377703)

A graph \(G\) is called quasi-claw-free if it satisfies that, for each pair \((x,y)\) of vertices \(x\) and \(y\) of \(G\) with \(d(x,y) = 2\), there is a vertex \(u \in N(x) \cap N(y)\) such that \(N(u) \subseteq N(x) \cup N(y)\). It is obvious that every claw-free (\(K_{1,3}\)-free) graph is quasi-claw-free, but, not vice versa. In this paper, a few known results for claw-free graphs are generalized for quasi-claw-free graphs.