The adjacency dimension of graphs

Abstract

It is known that the problem of computing the adjacency dimension of a graph is NPhard. This suggests finding the adjacency dimension for special classes of graphs or obtaining good bounds on this invariant. In this work we obtain general bounds on the adjacency dimension of a graph G in terms of known parameters of G. We discuss the tightness of these bounds and, for some particular classes of graphs, we obtain closed formulae. In particular, we show the close relationships that exist between the adjacency dimension and other parameters, like the domination number, the location-domination number, the 2-domination number, the independent 2-domination number, the vertex cover number, the independence number and the super domination number.

The Geodetic Number for the Unit Graphs Associated with Rings of Order P and P2

“…Furthermore, Chartrand et al [11] showed that, for any connected graph G, g(G) ≤ n − diam(G) + 1, where dim(G) = max{d(u, v) : u, v ∈ V}. Several researchers have studied this topic and presented some results [12][13][14][15][16][17][18].…”