4-cycle-free induced subgraphs of grid graphs

TL;DR

This paper characterizes maximal vertex sets in 2D grid graphs that induce subgraphs free of 4-cycles and provides bounds on the number of such subgraphs.

Contribution

It introduces a characterization of maximal C4-free induced subgraphs in 2D grid graphs and establishes bounds on their quantities.

Findings

Characterized maximal C4-free induced subgraphs in 2D grid graphs.

Provided upper and lower bounds on the number of near-maximum C4-free induced subgraphs.

Abstract

The avoidance of induced forests, or induced acyclic subgraphs, in $d$-dimensional grid graphs, or lattice graphs, has been studied in Alon et al. and later in Caragiannis et al., finding upper and lower bounds with respect to the number of vertices in a single dimension $n$ and the dimension $d$. In this work, we study the avoidance of induced $C_4$-free subgraphs, a superset of induced forests, of $2$-dimensional grid graphs $G$ and characterize the maximal sets $S \subseteq V$ such that the induced subgraph $G_S$ of $G$ with vertex set $S$ is $C_4$-free. Additionally, we will give upper and lower bounds on the number of $C_4$-free induced subgraphs with slightly fewer vertices than contained in the maximum.