Complexity of the Minimum Single Dominating Cycle Problem for Graph Classes

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<p>In 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 N(V(C)) of neighbor vertices of C, V(G)=V(C)∪N(V(C)) and |V(C)| is minimum over all dominating cycles in G [6], [17], [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 NP-hard to approximate even when restricted to planar, bipartite, chordal, or r-regular (r≥3). Then, we show the (lnn+1)-approximability and the (1-ε)lnn-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.</p>

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