Optimal Deployment Patterns for Full Coverage and $k$ -Connectivity $(k \leq 6)$ Wireless Sensor

In this paper, we study deployment patterns to achieve full coverage and k -connectivity (k ≤ 6) under different ratios of the sensor communication range (denoted by R_c) to the sensing range (denoted by R_s) for homogeneous wireless sensor networks (WSNs). In particular, we propose new patterns for 3- and 5-connectivity. We also discover that there exists a hexagon-based universally elemental pattern that can generate all known optimal patterns. The previously proposed Voronoi-based approach cannot be applied to prove the optimality of the new patterns due to their special features. We propose a new deployment-polygon-based methodology. We prove the optimality of deployment patterns to achieve 3-connectivity, 4-connectivity, and 5-connectivity for certain ranges of R_c/R_s, respectively, and prove the optimality of deployment patterns to achieve 6-connectivity under all ranges of R_c/R_s.