Map graphs

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, derive some consequences regarding sparsity and coloring, and survey some algorithmic results.

Recognizing Hole-Free 4-Map Graphs in Cubic Time

“…For the details and a comprehensive survey of known results on map graphs, we refer the reader to [3]. Here we only describe a brief history of research on map graphs.…”

“…Here we only describe a brief history of research on map graphs. In [2] and [3], the authors gave a simple nondeterministic polynomial-time algorithm for recognizing map graphs and investigated the structure and the number of maximal cliques in a map graph. Subsequently, Thorup [7] presented a polynomial-time algorithm for recognizing map graphs.…”

“…In fact, it is still unknown if k-map graphs (respectively, hole-free k-map graphs) for k ≥ 5 can be recognized in polynomial time. We note in passing that for every k ≥ 4, neither the class of k-map graphs nor the class of hole-free k-map graphs can be characterized by forbidden subgraphs or minors (because there are a hole-free 4-map graph G and an edge e in G such that G − e is not a map graph [3]). …”

“…Theorem 3.1 in [3] shows that each clique in a map graph can be realized in only four different ways by a map. The basic idea behind our cubic-time algorithm is to figure out the correct way of realizing each maximal clique C of the input graph G in the target map.…”

“…It is known [3] that for every integer k ≥ 3, each k-map graph has no clique of size larger than 3k/2 . So, G has no 7-clique.…”