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Given a requested area, the Minimum Connected Sensor Cover problem is to find a minimum number of sensors such that their communication ranges induce a connected graph and their sensing ranges cover the requested area. Several polynomial-time approximation algorithms have been designed previously in the literature. Their best known performance ratio is O(r ln n) where r is the link radius of the sensor network and n is the number of sensors. In this paper, we will present two polynomial-time approximation algorithms. The first one is a random algorithm, with probability 1 − ε, producing an approximation solution with performance ratio O(log 3 n log log n), independent from r. The second one is a deterministic approximation with performance ratio O(r), independent from n.

Let G be a planar graph with maximum degree Δ ≥ 8 and without chordal 5-cycles. Then χ l (G) = Δ and χ l (G) = Δ + 1.

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