The Metric Dimension of Regular Bipartite Graphs

TL;DR

This paper determines the metric dimension of specific regular bipartite graphs, specifically when each vertex is connected to all but one or two vertices in the other partition, providing exact values for these cases.

Contribution

The paper explicitly calculates the metric dimension for k-regular bipartite graphs G(n,n) when k equals n-1 or n-2, filling a gap in the understanding of these graph classes.

Findings

Metric dimension for k=n-1 case is established.

Metric dimension for k=n-2 case is established.

Provides exact values for these specific regular bipartite graphs.

Abstract

A set of vertices $W$ resolves a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $W$. A metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. A bipartite graph G(n,n) is a graph whose vertex set $V$ can be partitioned into two subsets $V_1$ and $V_2,$ with $|V_1|=|V_2|=n,$ such that every edge of $G$ joins $V_1$ and $V_2$. The graph $G$ is called $k$-regular if every vertex of $G$ is adjacent to $k$ other vertices. In this paper, we determine the metric dimension of $k$-regular bipartite graphs G(n,n) where $k=n-1$ or $k=n-2$.