Generalization of Matching Extensions in Graphs (IV): Closures.

Let G be a graph and n, k and d be non-negative integers such that $$|V(G)|\geqslant n+2k+d+2$$ and $$|V(G)|-n-d \equiv 0 \pmod {2}$$ . A graph is called an ( n, k, d)- graph if deleting any n vertices from G the remaining subgraph of G contains k-matchings and each k-matching in the subgraph can be extended to a defect- d matching. We study the relationships between ( n, k, d)-graphs and various closure operations, which are usually considered in the theory of hamiltonian graphs. In particular, we obtain some necessary and sufficient conditions for the existence of ( n, k, d)-graphs in terms of these closures. [ABSTRACT FROM AUTHOR]