Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 5 1 2020 06 01 The 2-dimension of a Tree 69 81 13979 10.22049/cco.2019.26495.1119 EN Jason Hedetniemi Department of Mathematics Wingate University Wingate NC USA Stephen Hedetniemi School of Computing Clemson University Clemson, SC U.S.A. Renu C. Renu C. Laskar Clemson University Henry Martyn Mulder Econometrisch Instituut Erasmus Universiteit Rotterdam Netherlands Journal Article 2019 05 23 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. https://comb-opt.azaruniv.ac.ir/article_13979_67e6ec33d043a864ea37af1094c77ac3.pdf