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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.