Abstract:

It is said that a graph G is independent-set-deletable factor-critical (in short, ID-factor-critical), if, for every independent set I which has the same parity as |V(G)|, G-I has a perfect matching. A graph G is strongly IM-extendable, if for every spanning supergraph H of G, every induced matching of H is included in a perfect matching of H. The k-th power of G, denoted by G^k, is the graph with vertex set V(G) in which two vertices are adjacent if and only if they have distance at most k in G. ID-factor-criticality and IM-extendability of power graphs are discussed
in this article. The author shows that, if G is a connected graph, then G³ and T(G) (the total graph of G) are ID-factor-critical, and G⁴ (when |V(G)| is even) is strongly IM-extendable; if G is 2-connected, then D² is ID-factor-critical.

Key words: Independent set, perfect matching, induced matching, ID-factor-critical, IM-extendable, power of a graph