A graph G is called claw ― free if G has no induced subgraph isomorphic to K_(1, 3). A subset D of the vertex set V(G) is called a dominating set of a graph G if every vertex in V (G) is either contained in D or adjacent to some vertex of D. In this paper the following two theorems are proved. Theorem 1. Let G be a 2 ― connected claw ― free graph. If G has a 2 ― element dominating set { a, b } and both a and b are contained in a cycle of length r, where 3 ≤ r ≤ |V(G)| ― 1, in G, then both a and b are contained in a cycle of length r + 1 in G. Theorem 2. Let G be a 2 ― connected claw ― free graph. If a subset S of the vertex set V(G) contains a 2 ― element dominating set { a, b } of G, then there exists a cycle C in G such that V(C) ∩ S = S. Each of the two theorems above generalizes a theorem obtained by Ageev in [1].
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