We study a graph parameter related to resolving sets and metric dimension,
namely the resolving number, introduced by Chartrand, Poisson and Zhang. First,
we establish an important difference between the two parameters: while
computing the metric dimension of an arbitrary graph is known to be NP-hard, we
show that the resolving number can be computed in polynomial time. We then
relate the resolving number to classical graph parameters: diameter, girth,
clique number, order and maximum degree. With these relations in hand, we
characterize the graphs with resolving number 3 extending other studies that
provide characterizations for smaller resolving number.Comment: 13 pages, 3 figure
