Recognizing Optimal 1-Planar Graphs in Linear Time

Abstract

A graph with n vertices is 1-planar if it can be drawn in the plane such that each edge is crossed at most once, and is optimal if it has the maximum of 4n − 8 edges. We show that optimal 1-planar graphs can be recognized in linear time. Our algorithm implements a graph reduction system with two rules, which can be used to reduce every optimal 1-planar graph to an irreducible extended wheel graph. The graph reduction system is non-deterministic, constraint, and non-confluent.

An annotated bibliography on 1-planarity

“…This is in contrast with maximal planar graphs, which have exactly 3n − 6 edges. Moreover, any n-vertex maximal 1-planar graph has at least 28 13 n − 10 3 edges [30]. Brandenburg et al [30] also studied the problem in the fixed rotation system setting.…”

A Survey on Graph Drawing Beyond Planarity

“…Recognizing 1-planar graphs remains NP-complete also when the input graph comes with a fixed rotation system, which must be preserved [28]. On the positive side, deciding whether a graph with n is optimal 1planar (i.e., it is 1-planar and has 4n − 8 edges) is O(n)-time solvable [59]; the testing algorithm exploits a structural characterization of optimal 1-planar graphs [192]. We refer the reader to the annotated bibliography by Kobourov et al for further references on recognizing meaningful subclasses of 1-planar graphs [159].…”

Packing Trees into 1-Planar Graphs

“…If n = 10, we have 7n 2 − 3 = 32 and 4n − 8 = 32, which means that any 1planar packing of three paths and a perfect matching with n = 10 vertices is an optimal 1-planar graph. It is known that every optimal 1-planar graph has at least eight vertices of degree exactly six [1]. On the other hand, in any 1-planar packing of three paths and a perfect matching all vertices, except the at most six end-vertices of the three paths, have degree seven, which implies that a 1-planar packing of three paths and a perfect matching does not exist.…”