Abstract Given a graph G, the minimum Connected-k-Subgraph Cover problem (MinCkSC) is to find a minimum vertex subset C of G such that every connected subgraph of G on k vertices has at least one vertex in C. If furthermore the subgraph of G induced by C is connected, then the problem is denoted as MinCkSC $_{con}$ . In this paper, we first present a PTAS for MinCkSC on an H-minor-free graph, where H is a graph with a constant number of vertices. Then, we design an $O((\omega+1)(2(k-1)(\omega+2))^{3\omega+3})|V|$ -time FPT algorithm for MinCkSC $_{con}$ on a graph with treewidth $\omega$ , based on which we further design an $O(2^{O(\sqrt{t}\log t)}|V|^{O(1)})$ time subexponential FPT algorithm for MinCkSC $_{con}$ on an H-minor-free graph, where t is an upper bound of solution size.

Finding a connected dominating set (CDS) in a given graph is a fundamental problem and has been studied intensively for a long time because of its application in computer science and operations research, e.g., connected facility location and wireless networks. In some cases, fault-tolerance is desirable. Taking wireless networks as an example, since wireless nodes may fail due to accidental damage or energy depletion, it is desirable that the virtual backbone has some fault-tolerance. Such a problem can be modeled as finding a minimum k-connected m-fold dominating set ((k, m)-CDS) of a graph G = (V, E), which is a node set D such that every node outside of D has at least m neighbors in D and the subgraph of G induced by D is k-connected. In this paper, we study the minimum weight (1, m)-CDS problem ((1, m)-MWCDS), and present an (H(δ + m) + 2H(δ − 1))-approximation algorithm, where δ is the maximum degree of the graph and H(·) is the Harmonic number. Notice that the state-of-the-art algorithm achieves O(l...

To save energy and alleviate interference, connected dominating set (CDS) was proposed to serve as a virtual backbone of wireless sensor networks (WSNs). Because sensor nodes may fail due to accidental damages or energy depletion, it is desirable to construct a fault tolerant virtual backbone with high redundancy in both coverage and connectivity. This can be modeled as a k-connected m-fold dominating set (abbreviated as (k, m)-CDS) problem. A node set C ⊆ V (G) is a (k, m)-CDS of graph G if every node in V(G)\C is adjacent with at least m nodes in C and the subgraph of G induced by C is k-connected. Constant approximation algorithm is known for (3, m)-CDS in unit disk graph, which models homogeneous WSNs. In this paper, we present the first performance guaranteed approximation algorithm for (3, m)-CDS in a heterogeneous WSN. In fact, our performance ratio is valid for any topology. The performance ratio is at most γ, where γ = α + 8 + 2 ln(2α - 6) for α ≥ 4 and γ = 3α +2 ln 2 for α <; 4, and α is the performance ratio for the minimum (2, m)-CDS problem. Using currently best known value of α, the performance ratio is ln δ +o(ln δ), where δ is the maximum degree of the graph, which is asymptotically best possible in view of the non-approximability of the problem. Applying our algorithm on a unit disk graph, the performance ratio is less than 27, improving previous ratio 62.3 by a large amount for the (3, m)-CDS problem on a unit disk graph.