Families of $2$-weights of some particular graphs

Abstract

Let ${\cal G}=(G,w) $ be a positive-weighted graph, that is a graph $G$ endowed with a function $w$ from the edge set of $G$ to the set of positive real numbers; for any distinct vertices $i,j $, we define $D_{i,j}({\cal G})$ to be the weight of the path in $G$ joining $i$ and $j$ with minimum weight. In this paper we fix a particular class of graphs and we give a criterion to establish whether, given a family of positive real numbers $\{D_I\}_{I \in { \{1,...., n\} \choose 2}}$, there exists a positive-weighted graph ${\cal G} =(G,w) $ in the class we have fixed, with vertex set equal to $\{1,....,n\}$ and such that $D_I ({\cal G}) =D_I$ for any $I \in { \{1,...., n\} \choose 2}$. In particular, the classes of graphs we consider are the following: snakes, caterpillars, polygons, bipartite graphs, complete graphs, planar graphs.