Abstract

Map graphs generalize planar graphs and were introduced by Chen et al. (STOC 1998, J. ACM, 2002). They showed that the problem of recognizing map graphs is in NP by proving the existence of a planar witness graph W. Shortly after, Thorup (FOCS 1998) published a polynomial-time recognition algorithm for map graphs. However, the run time of this algorithm is estimated to be Ω(n120) for n-vertex graphs, and a full description of its details remains unpublished.We give a new and purely combinatorial algorithm that decides whether a graph G is a map graph having an outerplanar witness W. This is a step towards a first combinatorial recognition algorithm for general map graphs. The algorithm runs in time and space O(n+m). In contrast to Thorup’s approach, it computes the witness graph W in the affirmative case.

Highlights

  • Consider the adjacency graph of the states of the USA, where two states are adjacent if their borders intersect

  • Another motivation for d-map graphs is the study of 1-planar graphs, which are the graphs that can be embedded in the plane such that each edge crosses at most one other edge

  • In the following we show that cliques that cannot be represented by an intersection point in an outerplanar witness induce a special structure

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Summary

Introduction

Consider the adjacency graph of the states of the USA, where two states are adjacent if their borders intersect. Our algorithm runs in time and space O(n + m) and is certifying This is the first non-trivial step towards a combinatorial and efficient recognition algorithm for general map graphs. Even efficiently recognizing general 4-map graphs in polynomial time remains an open problem; Thorup’s algorithm does not necessarily give an embedding minimizing the maximum degree of the intersection points, so it cannot be used to recognize 4-map graphs. Another motivation for d-map graphs is the study of 1-planar graphs, which are the graphs that can be embedded in the plane such that each edge crosses at most one other edge. Fomin et al [9] gave PTAS’s for optimization problems on map graphs; they later improved these to EPTAS’s [8]

Preliminaries
Reduction along Small Separators
Map Graphs with a Tree Witness
Map Graphs with an Outerplanar Witness
Structural Properties of Map Graphs with Outerplanar Witness
Recognition Algorithm
Discussion
Full Text
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