Computing the metric dimension of a graph from primary subgraphs

Abstract

Let G be a connected graph. Given an ordered set W = {w 1 , w 2 , . . . w k } ⊆ V (G) and a vertex u ∈ V (G), the representation of u with respect to W is the ordered k-tuple (d(u, w 1 ), d(u, w 2 ), . . . , d(u, w k )), where d(u, w i ) denotes the distance between u and w i . The set W is a metric generator for G if every two different vertices of G have distinct representations. A minimum cardinality metric generator is called a metric basis of G and its cardinality is called the metric dimension of G. It is well known that the problem of finding the metric dimension of a graph is NP-Hard. In this paper we obtain closed formulae for the metric dimension of graphs with cut vertices. The main results are applied to specific constructions including rooted product graphs, corona product graphs, block graphs and chains of graphs.

Metric dimension and diameter in bipartite graphs

“…If D ≡ 0 (mod 3), then A = D+1 3 , and so D − A + 1 = 2 D+1 3 , which is even. Combining equations (7), (8)) and (9) we now obtain the desired value of |V (G k,D )| in all three cases.…”

“…If G is a connected graph, then a set W of vertices of G is a resolving set if every vertex of G is uniquely identified by its distances to the vertices in W . The minimum cardinality of a resolving set of G is called the metric dimension of G. These notions were introduced in papers by Slater [11] and Harary and Melter [5] and studied extensively over the past decades (see [2,4,9,10,[12][13][14] for some recent results).…”

On the mutual visibility in Cartesian products and triangle-free graphs

“…The corona G • H of disjoint graphs G and H was introduced in [11] as the graph obtained from the disjoint union of G and n(G) copies of H by joining the i th vertex of G to every vertex in the i th copy of H. This graph operation has so far been explored from many aspects, see [8,12,17]. Here we add the following result which is particular yields a large class of graphs with µ = µ i .…”

Strong resolving partitions for strong product graphs and Cartesian product graphs

“…The parameters related to metric dimension have not escaped to this. For instance, the metric dimensions of Cartesian product graphs, strong product graphs, lexicographic product graphs, corona product graphs and rooted product graphs were studied in [1,9,21,24,29] and [13], respectively. The strong metric dimensions of Cartesian product graph, strong product graph, corona product graphs and rooted product graphs were studied in [14][15][16]23] and [17], respectively.…”