Classifying Resolving Lists by Distances between Members
Paul Feit · Theory and Applications of Graphs · 2016
L Let $G$ be a connected graph and let $w_1,\cdots w_r$ be a list of vertices. We refer the choice of a triple $(r;G;w_1,\cdots w_r)$, as a {\em metric selection.} Let $\rho$ be the shortest path metric of $G$. We say that $w_1,\cdots w_r$ {\em resolves $G$ (metricly)\/} if the function $c:V(G)\mapsto\bbz^r$ given by \[ x\mapsto (\rho (w_1,x),\cdots ,\rho (w_r,x))\] is injective. We refer to this function the {\em code map,} and to its image as the {\em codes\/} of the triple $(r;G;w_1,\cdots ,w_r)$. %edit This paper proves basic results on the following questions: \begin{enumerate} \item What sets can be the image of a code map? \item Given the image of a graph's code map, what can we determine about the graph? \end{enumerate}