An Extremal Graph Problem on a Grid and an Isoperimetric Problem for Polyominoes

Abstract

Let $G$ denote the infinite grid graph with vertex set $\{(a,b)\ : \, a,b \in \mathbb{Z}\}$ and edge set $\big \{ \{u,v\} : |u-v|=1 \;\text{or}\; |u-v| = \sqrt{2} \big \}.$ A question in landscape ecology, restated in graph theoretic terms, asks the following. What is the maximum number of edges in an induced subgraph of $G$ of order $n$? It was conjectured by Taliceo and Fleron  that the maximum is $4n - \big \lceil \sqrt{28n-12} \, \big \rceil$. We prove the conjecture by formulating and solving a discrete version of the classical isoperimeteric problem.