Sensor redundancy check without geometric information

In this paper we study sufficient conditions for sensor redundancy that can be used without any coordinate or geometric information. We assume boolean sensing model: each sensor observes a disc of radius r_s around his position, and can communicate with other sensors up to a distance r_c. We prove that if r_s ≥ r_c, any sensor with five neighbors that are pairwise not neighbors is redundant, and this is best possible. With a slightly stronger model that sensors can distinguish close and distant neighbors, we generalizes to the case where r_s ≤ r_c in several ways: any sensor is redundant if it has m neighbors within distance α_mr_s, being pairwise farther than r_s, where m = 3, 4, 5 and α₃ ≈ 0.62, α₄ ≈ 0.81, α₅ = 1; or if it has g(α) neighbors within distance r_s that are pairwise farther than αr_s, where α ∈]0, 1], g(1) = 5 and 15 ≤ g((√6 -- √2)/2) ≤ 16.