This paper addresses the following relay sensor placement problem: given the set of duty sensors in the plane and the upper bound of the transmission range, compute the minimum number of relay sensors such that the induced topology by all sensors is globally connected. This problem is motivated by practically considering the tradeoff among performance, lifetime, and cost when designing sensor networks. In our study, this problem is modelled by a NP-hard network optimization problem named Steiner Minimum Tree with Minimum number of Steiner Points and bounded edge length (SMT-MSP).We propose two approximate algorithms, together with their detailed performance analysis. The first one has performance ratio 3 and the second one has performance ratio 2.5.

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A graph is subcubic if its maximum degree is at most 3. The bipartite density of a graph G is max{ε(H )/ε(G): H is a bipartite subgraph of G}, where ε(H ) and ε(G) denote the numbers of edges in H and G, respectively. It is an NP-hard problem to determine the bipartite density of any given trianglefree cubic graph. Bondy and Locke gave a polynomial time algorithm which, given a triangle-free subcubic graph G, finds a bipartite subgraph of G with at least 4 5 ε(G) edges; and showed that the Petersen graph and the dodecahedron are the only triangle-free cubic graphs with bipartite density 4 5 . Bondy and Locke further conjectured that there are precisely seven triangle-free subcubic graphs with bipartite density 4 5 . We prove this conjecture of Bondy and Locke. Our result will be used in a forthcoming paper to solve a problem of Bollobás and Scott related to judicious partitions.