Characterization of n-Vertex Graphs of Metric Dimension n − 3 by Metric Matrix

Abstract

Let G = ( V ( G ) , E ( G ) ) be a connected graph. An ordered set W ⊂ V ( G ) is a resolving set for G if every vertex of G is uniquely determined by its vector of distances to the vertices in W. The metric dimension of G is the minimum cardinality of a resolving set. In this paper, we characterize the graphs of metric dimension n − 3 by constructing a special distance matrix, called metric matrix. The metric matrix makes it so a class of graph and its twin graph are bijective and the class of graph is obtained from its twin graph, so it provides a basis for the extension of graphs with respect to metric dimension. Further, the metric matrix gives a new idea of the characterization of extremal graphs based on metric dimension.

Automorphism Group and Other Properties of Zero Component Graph over a Vector Space

“…In [15], using various algebraic, combinatorial, and geometric approaches, Bailey et al studied the constructions of resolving sets of Kneser and Johnson graphs and provided bounds on their metric dimension. For more works on the fixing number and metric dimension, the reader is referred to [9,[16][17][18][19][20][21] and the references therein.…”

Removing Twins in Graphs to Break Symmetries

“…These sets are a generalization of resolving sets, independently introduced by Slater [5] and Harary and Melter [6], motivated by the problem of identifying the location of an intruder in a network, by means of distances. Resolving sets and some related sets were recently studied in [7][8][9][10][11][12]. Determining sets and resolving sets were jointly studied (see [13,14]).…”