The k-Metric Dimension of a Unicyclic Graph

Abstract

Given a connected graph G=(V(G),E(G)), a set S⊆V(G) is said to be a k-metric generator for G if any pair of different vertices in V(G) is distinguished by at least k elements of S. A metric generator of minimum cardinality among all k-metric generators is called a k-metric basis and its cardinality is the k-metric dimension of G. We initially present a linear programming problem that describes the problem of finding the k-metric dimension and a k-metric basis of a graph G. Then we conducted a study on the k-metric dimension of a unicyclic graph.

The distance-k dimension of graphs

“…then u and w are called twin vertices of G. We begin with the following observations from [16,17,1,9] which we use in our proofs.…”

“…A subset S ⊆ V (G) is a distance-k resolving set of G if |S ∩ R k {x, y}| ≥ 1 for any pair of distinct vertices x and y in G, and the distance-k dimension of G, denoted by dim k (G), is the minimum cardinality over all distance-k resolving sets of G. The distance-k dimension of graphs was studied in [1], where it was also investigated more generally for metric spaces. The complexity of the problem was studied in [11] and [10], where it was shown that computing dim k (G) is an NP-hard problem for any positive integer k. The graphs G with dim k (G) = 1 were characterized in [9], which also investigated the problem in a more general setting.…”

A note on $k$-metric dimensional graphs

“…Let us recall the following definitions from [7]. A vertex of degree at least three in a graph G will be called a major vertex of G. Any end-vertex (a vertex of degree one) u of G is said to be a terminal vertex of a major vertex v of G if d G (u, v) < d G (u, w) for every other major vertex w of G. The terminal degree ter(v) of a major vertex v is the number of terminal vertices of v. Let M(G) be the set of exterior major vertices of G having terminal degree greater than one.…”

“…There are also several natural extensions of the definition of metric dimension in the literature, some of them, combining it with the idea of domination. See, for example [13,19,20] Another natural extension of metric dimension appears in [7]. See also [8,9,10,25].…”

“…In [7], the authors provide several bounds on Dim(G) and give some precise results in the case of trees. Herein, we provide some new bounds for this invariant and generalize some of their results.…”