The Nonnegative Node Weight j-Restricted k-Matching Problems

Abstract

Given a simple and undirected graph, nonnegative node weights, a nonnegative integer j, and a positive integer k, a k-matching in the graph is a subgraph with no isolated nodes and with maximum degree no more than k, a j-restricted k-matching is a k-matching with each connected component having at least j + 1 edges, and the total node weight of a j-restricted k-matching is the total weight of the nodes covered by the edges in the j-restricted k-matching. When j = 1 and k = 2, Kaneko [Kaneko A (2003) A necessary and sufficient condition for the existence of a path factor every component of which is a path of length at least two. J. Combin. Theory, B 88:195–218] and Kano et al. [Kano M, Katona G, Király Z (2005) Packing paths of length at least two. Discrete Math. 283:129–135] studied the problem of maximizing the number of nodes covered by the edges in a 1-restricted 2-matching. In this paper, we consider the problem of finding a j-restricted k-matching with the maximum total node weight. We present a polynomial-time algorithm for the problem as well as a min-max theorem in the case of j < k. We also prove that, when j ≥ k ≥ 2, the problem of maximizing the number of nodes covered by the edges in a j-restricted k-matching is NP-hard.