Claw-free graphs with equal 2-domination and domination numbers

Abstract

For a graph G a subset D of the vertex set of G is a k-dominating set if every vertex not in D has at least k neighbors in D. The k-domination number k(G) is the minimum cardinality among the k-dominating sets of G. Note that the 1-domination number 1(G) is the usual domination number (G). Fink and Jacobson showed in 1985 that the inequality ?k(G)?(G)+k?2 is valid for every connected graph G. In this paper, we concentrate on the case k = 2, where k can be equal to ?, and we characterize all claw-free graphs and all line graphs G with ?(G) = ?2(G).