Edge Contraction and Forbidden Induced Subgraphs

Given a family of graphs \(\mathcal {H}\) , a graph G is \(\mathcal {H}\) -free if any subset of V(G) does not induce a subgraph of G that is isomorphic to any graph in \(\mathcal {H}\) . We present sufficient and necessary conditions for a graph G such that G/e is \(\mathcal {H}\) -free for any edge e in E(G). Thereafter, we use these conditions to characterize \(2K_{2}\) -free, \(C_{4}\) -free, \(C_{5}\) -free, and split graphs.