Complexity of the Maximum k-Path Vertex Cover Problem

Eiji MIYANO  Toshiki SAITOH  Ryuhei UEHARA  Tsuyoshi YAGITA  Tom C. van der ZANDEN  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E103-A   No.10   pp.1193-1201
Publication Date: 2020/10/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.2019DMP0014
Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)
Category: complexity theory
Maximum k-path vertex cover,  NP-hardness,  polynomial time algorithm,  split graphs,  bounded treewidth,  

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This paper introduces the maximization version of the k-path vertex cover problem, called the MAXIMUM K-PATH VERTEX COVER problem (MaxPkVC for short): A path consisting of k vertices, i.e., a path of length k-1 is called a k-path. If a k-path Pk includes a vertex v in a vertex set S, then we say that v or S covers Pk. Given a graph G=(V, E) and an integer s, the goal of MaxPkVC is to find a vertex subset SV of at most s vertices such that the number of k-paths covered by S is maximized. The problem MaxPkVC is generally NP-hard. In this paper we consider the tractability/intractability of MaxPkVC on subclasses of graphs. We prove that MaxP3VC remains NP-hard even for split graphs. Furthermore, if the input graph is restricted to graphs with constant bounded treewidth, then MaxP3VC can be solved in polynomial time.