Removable and forced subgraphs of graphs

Abstract

A connected graph G is H-removable if H is a subgraph of G, G has at least |V(H)|+2 vertices and G−V(H′) has a perfect matching for every subgraph H′ of G isomorphic to H. So a connected graph is kK2-removable if and only if it is k-extendable, where kK2 is a matching of size k. Further G is an H-forced graph if G−V(H′) has a unique perfect matching for every subgraph H′ of G isomorphic to H. In this paper, we first characterize kK2-forced graphs for k≥2 as K2k+2 and Kk+1,k+1 by using randomly matchable graphs. Let Pi denote a path with i vertices. We show that P3- and P4-removable graphs are factor-critical and 1-extendable respectively. By using ear decomposition, we obtain that a connected graph G is P3-forced if and only if G is K5 or C2k+1 (k≥2). Finally we prove that a connected graph G is P4-forced if and only if G is isomorphic to K6, K3,3, C2k (k≥3), the Petersen graph, a uniform theta graph Θ3n (n≥3) or C12+.