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.

This paper deals with a problem to decide whether a given graph structure appears as a pattern in the structure of a given graph. A graph pattern is a triple p=(V,E,H), where (V,E) is a graph and H is a set of variables, which are ordered lists of vertices in V. A variable can be replaced with an arbitrary connected graph by a kind of hyperedge replacements. A substitution is a collection of such replacements. The graph pattern matching problem (GPMP) is the computational problem to decide whether or not a given graph G is obtained from a given graph pattern p by a substitution. In this paper, we show that GPMP for a graph pattern p and a graph G is solvable in polynomial time if the length of every variable in p is 2, p is of bounded treewidth, and G is connected.

Suppose that each arc in a digraph D = (V,A) has two costs of non-negative integers, called a spine cost and a leaf cost. A caterpillar is a directed tree consisting of a single directed path (of spine arcs) and leaf vertices each of which is incident to the directed path by exactly one incoming arc (leaf arc). For a given terminal set K ⊆ V, we study the problem of finding a caterpillar in D such that it contains all terminals in K and its total cost is minimized, where the cost of each arc in the caterpillar depends on whether it is used as a spine arc or a leaf arc. In this paper, we first show that the problem is NP-hard for any fixed constant number of terminals with |K| ≥ 3, while it is solvable in polynomial time for at most two terminals. We also give an inapproximability result for any fixed constant number of terminals with |K| ≥ 3. Finally, we give a linear-time algorithm to solve the problem for digraphs with bounded treewidth, where the treewidth for a digraph D is defined as the one for the underlying graph of D. Our algorithm runs in linear time even if |K| = O(|V|), and the hidden constant factor of the running time is just a single exponential of the treewidth.