Deciding Whether a Grid is a Topological Subgraph of a Planar Graph is NP-Complete

The Topological Subgraph Containment (TSC) Problem is to decide, for two given graphs G and H, whether H is a topological subgraph of G. It is known that the TSC Problem is NP-complete when H is part of the input, that it can be solved in polynomial time when H is fixed, and that it is fixed-parameter tractable by the order of H. Motivated by the great significance of grids in graph theory and algorithms due to the Grid-Minor Theorem by Robertson and Seymour, we investigate the computational complexity of the Grid TSC Problem in planar graphs. More precisely, we study the following decision problem: given a positive integer k and a planar graph G, is the k × k grid a topological subgraph of G? We prove that this problem is NP-complete, even when restricted to planar graphs of maximum degree six, via a novel reduction from the Planar Monotone 3-SAT Problem.

Subjects

grids

NP-complete

planar graph

subdivision

subgraph homeomorphism

topological subgraph

DDC Class

510: Mathematik

Lizenz

https://creativecommons.org/licenses/by-nc-nd/4.0/

Name

1-s2.0-S1571066119300994-main.pdf

Size

271.08 KB

Format

Adobe PDF

Options

Deciding Whether a Grid is a Topological Subgraph of a Planar Graph is NP-Complete