The Threshold Strong Dimension of a Graph

TL;DR

This paper introduces the concept of threshold strong dimension of a graph, providing a geometric characterization, and explores its properties, including bounds and exact values for specific graph classes.

Contribution

It defines the threshold strong dimension, distinguishes it from related concepts, and characterizes it geometrically using path embeddings and strong products.

Findings

Threshold strong dimension differs from the previously studied threshold dimension.

Graphs with strong dimension 1 are paths; those with dimension 2 are characterized explicitly.

Sharp bounds and exact values are obtained for certain subclasses of trees.

Abstract

Let $G$ be a connected graph and $u,v$ and $w$ vertices of $G$. Then $w$ is said to {\em strongly resolve} $u$ and $v$, if there is either a shortest $u$-$w$ path that contains $v$ or a shortest $v$-$w$ path that contains $u$. A set $W$ of vertices of $G$ is a {\em strong resolving set} if every pair of vertices of $G$ is strongly resolved by some vertex of $W$. A smallest strong resolving set of a graph is called a {\em strong basis} and its cardinality, denoted $\beta_s(G)$, the {\em strong dimension} of $G$. The {\em threshold strong dimension} of a graph $G$, denoted $\tau_s(G)$, is the smallest strong dimension among all graphs having $G$ as spanning subgraph. A graph whose strong dimension equals its threshold strong dimension is called $\beta_s$-{\em irreducible}. In this paper we establish a geometric characterization for the threshold strong dimension of a graph $G$ that is expressed in terms of the smallest number of paths (each of sufficiently large order) whose strong product admits a certain type of embedding of $G$. We demonstrate that the threshold strong dimension of a graph is not equal to the previously studied threshold dimension of a graph. Graphs with strong dimension $1$ and $2$ are necessarily $\beta_s$-irreducible. It is well-known that the only graphs with strong dimension $1$ are the paths. We completely describe graphs with strong dimension $2$ in terms of the strong resolving graphs introduced by Oellermann and Peters-Fransen. We obtain sharp upper bounds for the threshold strong dimension of general graphs and determine exact values for this invariant for certain subclasses of trees.