K -vertex-connectivity minimum augmentation for undirected unweighted graphs*

Abstract

Abstract For an undirected unweighted graph G 0 = (V 0 E 0) and a positive integer K, the K-vertex-connectivity minimum augmentation problem (K-VCMAP) is to find a minimum set of edges E min such that the graph H 0 = (V 0 E 0 ⋓ E min) is K-vertex-connected. Results in the literature have given polynomial time algorithms for K-VCMAP in several special cases such as where k⩽3,> or G 0 is a tree. However, it still remains open whether or not there Exist polynomial time algorithms for K-VCMAP for any graph G 0 and any integer K. In this paper, we settle the problem by describing an efficient algorithm (KUCA) with time-complexity of O(K | V(G 0)|5) for the K-VCMAP for any G 0 and any positive integer K.