On k-leaf-connected graphs
U. S. R. Murty gave a generalization of the concept of Hamilton- connectivity of simple graphs calling a graph G k-leaf-connected if \(| V(G)| >k\) and for each subset S of V(G) with \(k=| S|\) there exists a spanning tree T with precisely S as set of endvertices. Clearly G is Hamilton-connected iff G is 2-leaf-connected. The authors prove that the property that a graph is k-leaf-connected is \((| V(G)| +k-1)\)-stable, which means that if u, v are non- adjacent vertices of G with \(d(u)+d(v)\geq | V(G)| +k-1\) and \(G+uv\) is k-leaf-connected, then also G is k-leaf-connected. With other results they show that for all naturals n, k, \(2\leq k<n-2\) there are k- leaf-connected graphs with \(\lceil \frac{(k+1)n}{2}\rceil\) edges, the minimum number of edges that a k-leaf-connected graph on n vertices can have.
scientific article; zbMATH DE number 3966118