Angle Covers: Algorithms and Complexity

TL;DR

This paper studies the angle cover problem in graphs with rotation systems, providing algorithms for certain degrees, proving NP-hardness for degree 5, and exploring extensions and applications in graph theory.

Contribution

It introduces the angle cover problem, establishes complexity results for various degrees, and connects the problem to graph isomorphism and thickness issues.

Findings

Graphs with max degree 4 always admit an angle cover.

Deciding angle cover existence is polynomial for graphs without degree-3 vertices.

NP-hardness is proven for graphs with max degree 5.

Abstract

Consider a graph with a rotation system, namely, for every vertex, a circular ordering of the incident edges. Given such a graph, an angle cover maps every vertex to a pair of consecutive edges in the ordering -- an angle -- such that each edge participates in at least one such pair. We show that any graph of maximum degree 4 admits an angle cover, give a poly-time algorithm for deciding if a graph with no degree-3 vertices has an angle-cover, and prove that, given a graph of maximum degree 5, it is NP-hard to decide whether it admits an angle cover. We also consider extensions of the angle cover problem where every vertex selects a fixed number $a>1$ of angles or where an angle consists of more than two consecutive edges. We show an application of angle covers to the problem of deciding if the 2-blowup of a planar graph has isomorphic thickness 2.