Optimal‐Time Algorithm for the k‐Node‐Connectivity Augmentation Problem for Ternary Trees

Abstract

AbstractGiven a graph and a positive integer k, in the k‐node‐connectivity augmentation problem (k‐NCAP), a set of edges is determined to be added to convert the graph into a k‐node‐connected graph with the minimum sum of the added edge‐weights. It is known that 1‐NCAP, for the directed acyclic graph where the weight is restricted to 1 and 2, is NP‐complete. It is also known that when the weight of the edges are all equal, 1‐NCAP for any directed graph can be solved in O([V] + [E]) time, and when the graph is restricted to the directed binary tree, k‐NCAP (k ≥ 2) is solved in O(k |y|) time. In the preceding, |V and |E are the number of nodes and the number of edges of the given graph, respectively. This paper discusses k‐NCAP (k ≥ 2) for the directed graph, where all edge weights are equal. A solution for k‐NCAP is shown for the family of graphs properly containing the directed ternary trees. It is shown that the problem can be solved in (k ≥ 3) time O(k |V) for the directed ternary tree, which is the optimal‐time algorithm within a constant factor. In this paper, a new family of regular and k‐node‐connected graphs called a k‐bouquet is defined. The solution for k‐NCAP is obtained by constructing a k‐bouquet by adding edges to the given graph.