Complexity of the Minimum Single Dominating Cycle Problem for Graph Classes

Abstract

SUMMARYIn this paper, we study a variant of the Minimum Dominating Set problem. Given an unweighted undirected graph G = (V, E) of n = |V| vertices, the goal of the Minimum Single Dominating Cycle problem (MinSDC) is to find a single shortest cycle which dominates all vertices, i.e., a cycle C such that for the set V(C) of vertices in C and the set [24]. In this paper we consider the (in)approximability of MinSDC if input graphs are restricted to some special classes of graphs. We first show that MinSDC is still NPhard to approximate even when restricted to planar, bipartite, chordal, or r-regular (r ≥ 3). Then, we show the (ln n + 1)-approximability and the (1 − ε) ln n-inapproximability of MinSDC on split graphs under P NP. Furthermore, we explicitly design a linear-time algorithm to solve MinSDC for graphs with bounded treewidth and estimate the hidden constant factor of its running time-bound.