%0 Journal Article
%T The 2-dimension of a Tree
%J Communications in Combinatorics and Optimization
%I Azarbaijan Shahid Madani University
%Z 2538-2128
%A Hedetniemi, Jason
%A Hedetniemi, Stephen
%A Renu C. Laskar, Renu C.
%A Mulder, Henry Martyn
%D 2020
%\ 06/01/2020
%V 5
%N 1
%P 69-81
%! The 2-dimension of a Tree
%K resolvability
%K location number
%K 2-dimension
%K tree
%K 2-locating set
%R 10.22049/cco.2019.26495.1119
%X 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.
%U http://comb-opt.azaruniv.ac.ir/article_13979_67e6ec33d043a864ea37af1094c77ac3.pdf