Minimal k-Connected Graphs with Minimal Number of Vertices of Degree k

Abstract

A graph is k-connected if it has at least k+1 vertices and remains connected after deleting any k−1 vertices. A k-connected graph is said to be minimal if any its subgraph obtained by deleting any edge is not k-connected. W. Mader proved that any minimal k-connected graph with n vertices has at least $$ \frac{\left(k-1\right)n+2k}{2k-1} $$ vertices of degree k. The main result of the present paper is that any minimal k-connected graph with minimal number of vertices of degree k is isomorphic to a graph G k,T , where T is a tree the maximal vertex degree of which is at most k + 1. The graph G k,T is constructed from k disjoint copies of the tree T in the following way. If a is a vertex of T of degree j and a 1 , . . . , a k are the corresponding vertices of the copies of T, then k + 1 − j new vertices of degree k, which are adjacent to {a 1 , . . . , a k }, are added. Bibliography: 10 titles.