On minimum m -connected k -dominating set problem in unit disc graphs

Minimum m-connected k-dominating set problem is as follows: Given a graph G= (V ,E) and two natural numbers m and k, find a subset S ⊆ V of minimal size such that every vertex in V \ S is adjacent to at least k vertices in S and the induced graph of S is m-connected. In this paper we study this problem with unit disc graphs and small m, which is motivated by the design of fault-tolerant virtual backbone for wireless sensor networks. We propose two approximation algorithms with constant performance ratios for m≤ 2. We also discuss how to design approximation algorithms for the problem with arbitrarily large m.