Existence of Connected Intersection-Free Subgraphs in Graphs with Redundancy and Coexistence Property

Abstract

Constructing connected intersection-free graphs is a relevant building block for local algorithmic solutions for data communication, task coordination and network maintenance in wireless sensor networks, sensor-actuator networks and distributed robotics. One way to construct such graph is to remove edges from the given network graph. Though an intersection-free graph can always be constructed that way, assuring connectivity at the same time is not possible for arbitrary graphs. It requires the underlying graph to have a supporting structure. In search of algorithms for constructing intersection-free subgraphs in wireless networks redundancy and coexistence have been identified as such properties. Practical evidence shows that these properties may hold with high probability in many practical wireless network graphs. In this work we study graphs obeying redundancy and coexistence. We demonstrate that so far existing solutions cannot guarantee connectivity of the constructed intersection-free subgraphs. Thus, one fundamental question stood open so far, if graphs obeying redundancy and coexistence property always contain a connected intersection-free subgraph at all. The contribution of this work is in answering this question in the positive.