An Extension of the Win Theorem: Counting the Number of Maximum Independent Sets

Win proved a well-known result that the graph $G$ of connectivity$kappa(G)$ with $alpha(G) leq kappa(G) + k - 1$ $(kgeq2)$ has aspanning $k$-ended tree, i.e., a spanning tree with at most $k$leaves. In this paper, the authors extended the Win theorem in casewhen $kappa(G)=1$ to the following: Let $G$ be a simple connectedgraph of order large enough such that $alpha(G)leq k+1$ $(kgeq3)$and such that the number of maximum independent sets of cardinality$k+1$ is at most $n-2k-2$. Then $G$ has a spanning $k$-ended tree.