The Simultaneous Fractional Dimension of Graph Families

TL;DR

This paper introduces and studies the concept of simultaneous fractional dimension in graph families, extending the resolving set idea to a fractional setting and analyzing bounds and specific cases like vertex transitive graphs and graph pairs.

Contribution

It defines the new concept of simultaneous fractional dimension for graph families and characterizes bounds, providing exact values for certain classes like trees and unicyclic graphs.

Findings

Established lower and upper bounds for ${ m Sd}_f(\

Determined ${ m Sd}_f(G,ar{G})$ for trees and unicyclic graphs.

Analyzed ${ m Sd}_f(\

Abstract

A subset $S$ of the vertices $V$ of a connected graph $G$ resolves $G$ if no two vertices of $V$ share the same list of distances (shortest-path metric) with respect to the vertices of $S$ listed in a given order. The choice of such an $S$ in $V$ amounts to selecting a binary valued function $g$, said to be a resolving function, on $V$. The notion of a fractional resolving function is obtained by relaxing the codomain of $g$ to be the unit interval. Let $|g|=\sum_{v\in V}g(v)$. Given a finite collection $\mathcal{G}$ of connected graphs on a common vertex set $V$, the simultaneous metric dimension of $\mathcal{G}$ is the minimum cardinality of $|S|$ over all $S$ which resolve each member graph of $\mathcal{G}$. In this paper, we initiate the study of simultaneous fractional dimension ${\rm Sd}_f(\mathcal{G})$ of a graph family $\mathcal{G}$, defined to be the minimum $|g|$ over all functions $g$ each resolving all members of $\mathcal{G}$. We characterize the lower bound and examine the upper bound satisfied by ${\rm Sd}_f(\mathcal{G})$. We examine ${\rm Sd}_f(\mathcal{G})$ for families of vertex transitive graphs and for pairs $\{G,\overline{G}\}$ of complementary graphs, determining ${\rm Sd}_f(G,\overline{G})$ when $G$ is a tree or a unicyclic graph.