  Proposed as a general framework, Liu and Yu(Discrete Math. 231 (2001)
311-320) 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 $n+2k+d\leq |V(G)|-2$ and
$|V(G)|-n-d$ is even. If when deleting any $n$ vertices from $G$, the remaining
subgraph $H$ of $G$ contains a $k$-matching and each such $k$- matching can be
extended to a defect-$d$ matching in $H$, then $G$ is called an
$(n,k,d)$-graph. In \cite{Liu}, the recursive relations for distinct parameters
$n, k$ and $d$ were presented and the impact of adding or deleting an edge also
was discussed for the case $d = 0$. In this paper, we continue the study begun
in \cite{Liu} and obtain new recursive results for $(n,k,d)$-graphs in the
general case $d \geq0$.
