Generalization of matching extensions in graphs (II) ?
Proposed as a general framework, Liu and Yu [Generalization of matching extensions in graphs, Discrete Math.  231 (2001) 311320.] introduced (n,k,d) ( 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 n , k and d  be non-negative integers such that n+2k+d?|V(G)|-2 n + 2 k + d ? | V ( G ) | - 2 and |V(G)|-n-d | 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) ( n , k , d ) -graph . Liu and Yu's Papee's paper, the recursive relations for distinct parameters n,k n , k and d  were presented and the impact of adding or deleting an edge also was discussed for the case d=0 d = 0 . In this paper, we continue the study begun by Liu and Yu and obtain new recursive results for (n,k,d) ( n , k , d ) -graphs in the general case d?0 d ? 0 .
05c70