Complexity of the Maximum k-Path Vertex Cover Problem

Eiji Miyano*, Toshiki Saitoh, Ryuhei Uehara, Tsuyoshi Yagita, Tom C. van der Zanden

*Corresponding author for this work

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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 S⊆V 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.
Original languageEnglish
Pages (from-to)1193-1201
Number of pages9
JournalIeice Transactions on Fundamentals of Electronics Communications and Computer Sciences
Issue number10
Publication statusPublished - Oct 2020


  • Maximum k-path vertex cover
  • NP-hardness
  • polynomial time algorithm
  • split graphs
  • bounded treewidth

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