The 2-Dimension of A Tree

Let x and y be two distinct vertices in a connected graph g. the x,y-location of a vertex w is the ordered pair of distances from w to x and y, that is, the ordered pair (d(x,w),d(y,w)). a set of vertices w in g is x,y-located if any two vertices in w have distinct x,y-locations. a set w of vertices in g is 2-located if it is x,y-located, for some distinct vertices x and y. the 2-dimension of g is the order of a largest set that is 2-located in g. note that this notion is related to the metric dimension of a graph, but not identical to it. we study in depth the trees t that have a 2-locating set, that is, have 2-dimension equal to the order of t. using these results, we have a nice characterization of the 2-dimension of arbitrary trees.