The edge-closure concept for claw-free graphs and the stability of forbidden subgraphs

Ryjáček introduced a closure concept for claw-free graphs based on local completion of a locally connected vertex. Connected graphs AA, for which the class of (C,A)(C,A)-free graphs is stable under the closure, were completely characterized. In this paper, we introduce a variation of the closure concept based on local completion of a locally connected edge of a claw-free graph. The closure is uniquely determined and preserves the value of the circumference of a graph. We show that the class of (C,A)(C,A)-free graphs is stable under the edge-closure if A∈{H,Pi,Ni,j,k}A∈{H,Pi,Ni,j,k}.