Author/Authors :
Hedetniemi, Jason T Department of Mathematics - Wingate University, Wingate, NC 28174 U.S.A. , Hedetniemi, Stephen T Department of Mathematics - Wingate University, Wingate, NC 28174 U.S.A. , Emeritus, Professor School of Computing - Clemson University, Clemson, SC 29634 U.S.A. , Laskar, Renu C School of Computing - Clemson University, Clemson, SC 29634 U.S.A. , Emerita, Professor Department of Mathematical Sciences - Clemson University, Clemson, SC 29634 U.S.A.
Abstract :
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.
Keywords :
2-locating set , tree , 2-dimension , location number , resolvability
