Distance-Dominating Cycles in Quasi Claw-Free Graphs.

We say that a graph G is quasi claw-free if every pair ( a₁, a₂) of vertices at distance 2 satisfies { u∈ N ( a₁)∩ N ( a₂) | N[ u]⊆ N[ a₁]∪ N [ a₂]}≠∅. A cycle C is m-dominating if every vertex of G is of distance at most m from C. We prove that if G is a κ-connected (κ≥2) quasi claw-free graph then either G has an m-dominating cycle or G has a set of at least κ+1 vertices such that the distance between every pair of them is at least 2m+3. [ABSTRACT FROM AUTHOR]