Generalization of matching extensions in graphs (III)

Abstract

Proposing them as a general framework, Liu and Yu (2001) [6] introduced $(n,k,d)$-graphs to unify the concepts of deficiency of matchings, $n$-factor-criticality and $k$-extendability. Let $G$ be a graph and let $n,k$ and $d$ be non-negative integers such that $n+2k+d+2⩽\left|V\left(G\right)\right|$ and $\left|V\left(G\right)\right|-n-d$ is even. If on deleting any $n$ vertices from $G$ the remaining subgraph $H$ of $G$ contains a $k$-matching and each $k$-matching can be extended to a defect-$d$ matching in $H$, then $G$ is called an $(n,k,d)$-graph. In this paper, we obtain more properties of $(n,k,d)$-graphs, in particular the recursive relations of $(n,k,d)$-graphs for distinct parameters $n,k$ and $d$. Moreover, we provide a characterization for maximal non-$(n,k,d)$-graphs.