Simultaneous Orthogonal Planarity

TL;DR

This paper investigates the computational complexity of the OrthoSEFE-k problem, which involves creating orthogonal planar drawings of multiple graphs sharing the same vertices, and identifies cases where the problem is NP-complete or polynomial-time solvable.

Contribution

The paper establishes NP-completeness for k ≥ 3 under certain conditions and provides polynomial-time algorithms for specific cases, also showing bounds on edge bends in positive instances.

Findings

NP-complete for k ≥ 3 with shared Hamiltonian cycle

Polynomial-time solvable for k=2 with maximum degree five

Positive instances have at most three bends per edge

Abstract

We introduce and study the $\textit{OrthoSEFE}-k$ problem: Given $k$ planar graphs each with maximum degree 4 and the same vertex set, do they admit an OrthoSEFE, that is, is there an assignment of the vertices to grid points and of the edges to paths on the grid such that the same edges in distinct graphs are assigned the same path and such that the assignment induces a planar orthogonal drawing of each of the $k$ graphs? We show that the problem is NP-complete for $k \geq 3$ even if the shared graph is a Hamiltonian cycle and has sunflower intersection and for $k \geq 2$ even if the shared graph consists of a cycle and of isolated vertices. Whereas the problem is polynomial-time solvable for $k=2$ when the union graph has maximum degree five and the shared graph is biconnected. Further, when the shared graph is biconnected and has sunflower intersection, we show that every positive instance has an OrthoSEFE with at most three bends per edge.