Generalization of matching extensions in graphs

Abstract Let G be a graph with vertex set V(G) . Let n, k and d be non-negative integers such that n+2k+d⩽|V(G)|−2 and |V(G)|−n−d is even. A matching which covers exactly |V(G)|−d vertices of G is called a defect - d matching of G . If when deleting any n vertices of G the remaining subgraph contains a matching of k edges and every k -matching can be extended to a defect- d matching, then G is called a (n,k,d) - graph . In this paper a characterization of (n,k,d) -graphs is given and several properties (such as connectivity, minimum degree, hierarchy, etc.) of (n,k,d) -graphs are investigated.