Abstract

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.