For an ordered set W = {w1, w2, . . . , wk} of vertices and a vertex v in a connected graph G, the ordered k-vector r(v|W) := (d(v, w1), d(v, w2), . . . , d(v, wk)) is called the metric representation of v with respect to W, where d(x, y) is the distance between vertices x and y. A set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W. The minimum cardinality of a resolving set for G is its metric dimension. In this paper, we characterize all graphs of order n with metric dimension n - 3.
Zero forcing number has recently become an interesting graph parameter studied in its own right since its introduction by the ''AIM Minimum Rank–Special Graphs Work Group'', whereas metric dimension is a well-known graph parameter. We investigate the metric dimension and the zero forcing number of some line graphs by first determining the metric dimension and the zero forcing number of the line graphs of wheel graphs and the bouquet of circles. We prove that Z(G) ≤ 2Z(L(G)) for a simple and connected graph G. Further, we show that Z(G) ≤ Z(L(G)) when G is a tree or when G contains a Hamiltonian path and has a certain number of edges. We compare the metric dimension with the zero forcing number of a line graph by demonstrating a couple of inequalities between the two parameters. We end by stating some open problems.