Testing Simultaneous Planarity when the Common Graph is 2-Connected

TL;DR

This paper presents a linear-time algorithm for testing simultaneous planarity of two graphs sharing a 2-connected subgraph, extending to multiple graphs with shared or exclusive vertices and edges.

Contribution

It introduces an efficient algorithm for simultaneous planarity testing when the common subgraph is 2-connected, advancing the understanding of graph drawing problems.

Findings

Linear-time algorithm for two graphs with 2-connected shared subgraph

Extension to multiple graphs with shared or exclusive vertices/edges

Efficient planarity testing in complex graph sharing scenarios

Abstract

Two planar graphs G1 and G2 sharing some vertices and edges are `simultaneously planar' if they have planar drawings such that a shared vertex [edge] is represented by the same point [curve] in both drawings. It is an open problem whether simultaneous planarity can be tested efficiently. We give a linear-time algorithm to test simultaneous planarity when the two graphs share a 2-connected subgraph. Our algorithm extends to the case of k planar graphs where each vertex [edge] is either common to all graphs or belongs to exactly one of them.