Discrete isoperimetric inequalities on the strong products of paths

TL;DR

This paper investigates the minimal vertex boundary sizes for subsets within the strong product of two paths, providing specific cases where these boundaries are minimized, thus advancing understanding of isoperimetric inequalities in graph products.

Contribution

It characterizes the conditions under which the vertex boundary is minimized in the strong product of two paths, a novel contribution to graph isoperimetric problems.

Findings

Identifies cases with minimal vertex boundary in strong product of paths

Provides exact characterizations of boundary-minimizing subsets

Advances understanding of isoperimetric inequalities in graph products

Abstract

For a graph $G=(V,\ E)$ and a nonempty set $S\subseteq V$, the \emph{vertex boundary} of $S$, denoted by $\partial_G(S)$, is defined to be the set of vertices that are not in $S$ but have at least one neighbor in $S$. In this paper, for $G$ being a strong product of two paths, we determine the cases in which $|\partial_G(S)|$ is minimized.