A vertex subset I of a graph G is called a k-path vertex cover if every path on k vertices in G contains at least one vertex from I. The K-PATH VERTEX COVER RECONFIGURATION (K-PVCR) problem asks if one can transform one k-path vertex cover into another via a sequence of k-path vertex covers where each intermediate member is obtained from its predecessor by applying a given reconfiguration rule exactly once. We investigate the computational complexity of K-PVCR from the viewpoint of graph classes under the well-known reconfiguration rules: TS, TJ, and TAR. The problem for k=2, known as the VERTEX COVER RECONFIGURATION (VCR) problem, has been well-studied in the literature. We show that certain known hardness results for VCR on different graph classes can be extended for K-PVCR. In particular, we prove a complexity dichotomy for K-PVCR on general graphs: on those whose maximum degree is three (and even planar), the problem is PSPACE-complete, while on those whose maximum degree is two (i.e., paths and cycles), the problem can be solved in polynomial time. Additionally, we also design polynomial-time algorithms for K-PVCR on trees under each of TJ and TAR. Moreover, on paths, cycles, and trees, we describe how one can construct a reconfiguration sequence between two given k-path vertex covers in a yes-instance. In particular, on paths, our constructed reconfiguration sequence is shortest.

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.

Suppose that we are given two vertex covers C0 and Ct of a graph G, together with an integer threshold k ≥ max{|C0|, |Ct|}. Then, the vertex cover reconfiguration problem is to determine whether there exists a sequence of vertex covers of G which transforms C0 into Ct such that each vertex cover in the sequence is of cardinality at most k and is obtained from the previous one by either adding or deleting exactly one vertex. This problem is PSPACE-complete even for planar graphs. In this paper, we first give a linear-time algorithm to solve the problem for even-hole-free graphs, which include several well-known graphs, such as trees, interval graphs and chordal graphs. We then give an upper bound on k for which any pair of vertex covers in a graph G has a desired sequence. Our upper bound is best possible in some sense.

Suppose one of the edges is attacked in a graph G, where some number of guards are placed on some of its vertices. If a guard is placed on one of the end-vertices of the attacked edge, she can defend such an attack to protect G by passing over the edge. For each of such attacks, every guard is allowed either to move to a neighboring vertex, or to stay at where she is. The eternal vertex cover number τ∞(G) is the minimum number of guards sufficient to protect G from any length of any sequence of edge attacks. This paper derives the eternal vertex cover number τ∞(G) of such graphs constructed by replacing each edge of a tree by an arbitrary elementary bipartite graph (or by an arbitrary clique), in terms of easily computable graph invariants only, thereby showing that τ∞(G) can be computed in polynomial time for such graphs G.

A vertex cover of a given graph G = (V,E) is a subset N of V such that N contains either u or v for any edge (u,v) of E. The minimum weight vertex cover problem (MWVC for short) is the problem of finding a vertex cover N of any given graph G = (V,E), with weight w(v) for each vertex v of V, such that the sum w(N) of w(v) over all v of N is minimum. In this paper, we consider MWVC with w(v) of any v of V being a positive integer. We propose simple procedures as postprocessing of algorithms for MWVC. Furthremore, five existing approximation algorithms with/without the proposed procedures incorporated are implemented, and they are evaluated through computing experiment.

Chen and Kanj considered the VERTEX COVER problem for graphs with perfect matchings (VC-PM). They showed that: (i) There is a reduction from general VERTEX COVER to VC-PM, which guarantees that if one can achieve an approximation factor of less than two for VC-PM, then one can do so for general VERTEX COVER as well. (ii) There is an algorithm for VC-PM whose approximation factor is given as 1.069+0.069 where is the average degree of the given graph. In this paper we improve (ii). Namely we give a new VC-PM algorithm which greatly outperforms the above one and its approximation factor is roughly . Our algorithm also works for graphs with "large" matchings, although its approximation factor is degenerated.

For a property π on graphs, the edge-contraction problem with respect to π is defined as a problem of finding a set of edges of minimum cardinality whose contraction results in a graph satisfying the property π. This paper gives a lower bound for the approximation ratio for the problem for any property π that is hereditary on contractions and determined by biconnected components.

Reconfiguration of memory arrays using spare lines is known to be an NP-complete problem. In this paper, we present an algorithm that reconfigures a memory array without any faults by using spare lines effectively even if they contain faulty elements. First, the reconfiguration problem is transformed to an equivalent covering problem in which faulty elements are covered by imaginary fault-free spare lines. Next, the covering problem is heuristically solved by using the Dulmange-Mendelsohn decomposition. The experiments for recently designed memory arrays show that the proposed algorithm is fast and practical.