Parameterized Complexity of Simultaneous Planarity

TL;DR

This paper investigates the parameterized complexity of the Simultaneous Embedding With Fixed Edges (SEFE) problem, showing fixed-parameter tractability for certain combined parameters and NP-completeness for others, revealing the problem's computational boundaries.

Contribution

It establishes fixed-parameter tractability results for SEFE with respect to specific combined parameters and proves NP-completeness for various parameters of the shared graph.

Findings

SEFE is FPT with respect to k plus vertex cover or feedback edge set number of the union graph.

SEFE remains NP-complete even if the shared graph is a degree-4 tree.

Certain parameter combinations are intractable, while others are fixed-parameter tractable.

Abstract

Given $k$ input graphs $G_1, \dots ,G_k$, where each pair $G_i$, $G_j$ with $i \neq j$ shares the same graph $G$, the problem Simultaneous Embedding With Fixed Edges (SEFE) asks whether there exists a planar drawing for each input graph such that all drawings coincide on $G$. While SEFE is still open for the case of two input graphs, the problem is NP-complete for $k \geq 3$ [Schaefer, JGAA 13]. In this work, we explore the parameterized complexity of SEFE. We show that SEFE is FPT with respect to $k$ plus the vertex cover number or the feedback edge set number of the the union graph $G^\cup = G_1 \cup \dots \cup G_k$. Regarding the shared graph $G$, we show that SEFE is NP-complete, even if $G$ is a tree with maximum degree 4. Together with a known NP-hardness reduction [Angelini et al., TCS 15], this allows us to conclude that several parameters of $G$, including the maximum degree, the maximum number of degree-1 neighbors, the vertex cover number, and the number of cutvertices are intractable. We also settle the tractability of all pairs of these parameters. We give FPT algorithms for the vertex cover number plus either of the first two parameters and for the number of cutvertices plus the maximum degree, whereas we prove all remaining combinations to be intractable.