Local Construction of Connected and Plane Spanning Subgraphs under Acyclic Redundancy

Plane graphs play a major role for local routing and some other local network protocols in wireless communication. With such local algorithms each node requires information about its neighborhood only. It is assumed that nodes are deployed on the plane and each node knows its position in a given coordinate system. An arbitrary graph drawn on the plane can be transformed into a plane spanning subgraph by deleting edges. However, to assure connectivity at the same time some additional structural graph properties are required. Current graph classes that assure the existence of connected plane spanning subgraphs require assumptions, that are not very likely to hold for wireless network structures. In this work we develop the acyclic redundancy condition. This is a novel graph class with only one property that assures the existence of a connected plane spanning subgraph. Furthermore, we describe local algorithms that construct a connected plane spanning subgraph for graphs satisfying the acyclic redundancy condition. With numerical studies we confirm that the acyclic redundancy condition is a more realistic condition than existing graph classes that were required so far to construct connected plane spanning subgraphs.