Map Graphs

TL;DR

This paper introduces a new concept of map graphs based on a relaxed adjacency definition, provides an NP characterization, and offers an efficient recognition algorithm for certain restricted cases.

Contribution

It defines map graphs with boundary-sharing adjacencies, characterizes them NP-comprehensively, and develops a cubic time recognition algorithm for maps without holes and limited boundary intersections.

Findings

NP characterization of map graphs

Cubic time recognition algorithm for specific map configurations

Efficient recognition for maps with no holes and limited boundary points

Abstract

We consider a modified notion of planarity, in which two nations of a map are considered adjacent when they share any point of their boundaries (not necessarily an edge, as planarity requires). Such adjacencies define a map graph. We give an NP characterization for such graphs, and a cubic time recognition algorithm for a restricted version: given a graph, decide whether it is realized by adjacencies in a map without holes, in which at most four nations meet at any point.