  As a generalization of matching preclusion number of a graph, we provide the
(strong) integer $k$-matching preclusion number, abbreviated as $MP^{k}$ number
($SMP^{k}$ number), which is the minimum number of edges (vertices and edges)
whose deletion results in a graph that has neither perfect integer $k$-matching
nor almost perfect integer $k$-matching. In this paper, we show that when $k$
is even, the ($SMP^{k}$) $MP^{k}$ number is equal to the (strong) fractional
matching preclusion number. We obtain a necessary condition of graphs with an
almost-perfect integer $k$-matching and a relational expression between the
matching number and the integer $k$-matching number of bipartite graphs. Thus
the $MP^{k}$ number and the $SMP^{k}$ number of complete graphs, bipartite
graphs and arrangement graphs are obtained, respectively.

Combinatorics