Recognizing Hole-Free 4-Map Graphs in Cubic Time

Abstract

We present a cubic-time algorithm for the following problem: Given a simple graph, decide whether it is realized by adjacencies of countries in a map without holes, in which at most four countries meet at any point.

Recognizing Optimal 1-Planar Graphs in Linear Time

“…10a by exchanging edges ( p, s), (c, h) and ( p, h), (c, s). Consider candidate c in B 17 . Then H (c) = (2, 3, 4, 4, 4, 5, 6) violates optimal 1-planarity.…”

“…In his introductory paper on 1-planar graphs, Ringel studied the coloring problem and observed that a pair of crossing edges can be completed to K 4 by adding planar edges. 1-Planar graphs generalize 4-map graphs, which are the graphs of adjacencies of nations of a map [16,17]. Two nations are adjacent if they share a The first study of structural properties of 1-planar graphs is by Bodendiek, Schumacher, and Wagner [7,8].…”

“…A 1-planar graph G is maximally dense or maximum [36] if there is no 1-planar graph of the same size with more edges. It is maximal 1-planar if the addition of any edge destroys 1-planarity and planar-maximal or triangulated [17] if no further edge can be added without introducing a crossing. Clearly, optimal 1-planar graphs are maximally dense, and maximally dense 1-planar graphs are maximal, but not conversely.…”

“…It can be extended to maximally dense 1-planar graphs and specialized to 5-connected optimal 1-planar graphs. Our algorithm improves upon the cubic running time algorithm of Chen et al [17], which solves a more general problem and searches 4-cycles and other types of separators. Combinatorial properties of the reductions are explored in [10].…”

How to Visualize the K-root Name Server

“…In [31] it is shown that testing if a graph is a map graph can be done in polynomial time. However, in [9] it is argued that the exponent of the polynomial bounding its running time from above is about 120. This makes it impractical to use the algorithm in [31].…”