The Metric Dimension of Regular Bipartite Graphs

Details The Metric Dimension of Regular Bipartite Graphs The Metric Dimension of Regular Bipartite Graphs

2011-01-19

arxiv.org/abs/1101.3624v1

Bull. Math. Soc. Sci. Math. Roumanie Tome 54(102) No. 1, 2011, 15-28

Mathematics

Combinatorics

To appear in "Bull. Math. Soc. Sci. Math. Roumanie"

Scientific paper

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 $k=n-2$.

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