Families of $2$-weights of some particular graphs

Let $\mathcal{G}=(G,w)$ 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 $D_{i,j}(\mathcal{G})$ 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 ${D_I}_{I \in { {1,…, n} \choose 2}}$,thereexistsapositive-weightedgraph${\cal G} =(G,w) $intheclasswehavefixed,withvertexsetequalto${1,…,n}$andsuchthat$D_I ({\cal G}) =D_I$forany$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.