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.

Given a connected graph G = (V, E) on n vertices, the MAXIMUM r-REGULAR INDUCED CONNECTED SUBGRAPH (r-MaxRICS) problem asks for a maximum sized subset of vertices S ⊆ V such that the induced subgraph G[S] on S is connected and r-regular. It is known that 2-MaxRICS and 3-MaxRICS are NP-hard. Moreover, 2-MaxRICS cannot be approximated within a factor of n1-ε in polynomial time for any ε > 0 unless P= NP. In this paper, we show that r-MaxRICS are NP-hard for any fixed integer r ≥ 4. Furthermore, we show that for any fixed integer r ≥ 3, r-MaxRICS cannot be approximated within a factor of n1/6-ε in polynomial time for any ε > 0 unless P= NP.

This paper studies generalized variants of the MAXIMUM INDEPENDENT SET problem, called the MAXIMUM DISTANCE-d INDEPENDENT SET problem (MaxDdIS for short). For an integer d≥2, a distance-d independent set of an unweighted graph G=(V, E) is a subset S⊆V of vertices such that for any pair of vertices u, v∈S, the number of edges in any path between u and v is at least d in G. Given an unweighted graph G, the goal of MaxDdIS is to find a maximum-cardinality distance-d independent set of G. In this paper, we analyze the (in)approximability of the problem on r-regular graphs (r≥3) and planar graphs, as follows: (1) For every fixed integers d≥3 and r≥3, MaxDdIS on r-regular graphs is APX-hard. (2) We design polynomial-time O(rd-1)-approximation and O(rd-2/d)-approximation algorithms for MaxDdIS on r-regular graphs. (3) We sharpen the above O(rd-2/d)-approximation algorithms when restricted to d=r=3, and give a polynomial-time 2-approximation algorithm for MaxD3IS on cubic graphs. (4) Finally, we show that MaxDdIS admits a polynomial-time approximation scheme (PTAS) for planar graphs.