Recognizing generalized transmission graphs of line segments and circular sectors

Abstract

Suppose we have an arrangement A of n geometric objects x1, …, xn⊆ ℝ2 in the plane, with a distinguished point pi in each object xi. The generalized transmission graph of A has vertex set {x1, …, xn} and a directed edge xixj if and only if pj∈ xi. Generalized transmission graphs provide a generalized model of the connectivity in networks of directional antennas. The complexity class ∃ ℝ contains all problems that can be reduced in polynomial time to an existential sentence of the form ∃ x1, …, xn: ϕ(x1, ⋯, xn), where x1, …, xn range over ℝ and ϕ is a propositional formula with signature (+, -, ·, 0, 1). The class ∃ℝ aims to capture the complexity of the existential theory of the reals. It lies between NP and PSPACE. Many geometric decision problems, such as recognition of disk graphs and of intersection graphs of lines, are complete for ∃ℝ. Continuing this line of research, we show that the recognition problem of generalized transmission graphs of line segments and of circular sectors is hard for ∃ℝ. As far as we know, this constitutes the first such result for a class of directed graphs.