Perfect matchings and $K_{1, p}$-restricted graphs

A graph is called $K_{1, p}$-restricted ($p \ge 3$) if for every vertex of the graph there are at least $p - 2$ edges between any $p$ of its neighbours. We establish sufficient conditions for the existence of a perfect matching in $K_{1, p}$-restricted graphs in terms of their connectivity and vertex degrees. These conditions imply, in particular, the classical Petersen's result: any $2$-edge-connected $3$-regular graph contains a perfect matching.