Hierarchical Partial Planarity

TL;DR

This paper introduces a new graph layout problem where edges have importance levels, aiming to minimize crossings for more important edges, and provides polynomial-time algorithms for specific cases.

Contribution

It formalizes the hierarchical partial planarity problem with multiple importance levels and offers polynomial-time testing algorithms under certain connectivity conditions.

Findings

Polynomial-time testing algorithm when the two most important edge sets induce a biconnected graph

Formalization of hierarchical partial planarity with three importance levels

Discussion of relationships with other constrained planarity problems

Abstract

In this paper we consider graphs whose edges are associated with a degree of {\em importance}, which may depend on the type of connections they represent or on how recently they appeared in the scene, in a streaming setting. The goal is to construct layouts of these graphs in which the readability of an edge is proportional to its importance, that is, more important edges have fewer crossings. We formalize this problem and study the case in which there exist three different degrees of importance. We give a polynomial-time testing algorithm when the graph induced by the two most important sets of edges is biconnected. We also discuss interesting relationships with other constrained-planarity problems.