Weighted graphs with distances in given ranges

Abstract

Let ${\cal G}=(G,w)$ be a weighted simple finite connected graph, that is, let $G$ be a simple finite connected graph endowed with a function $w$ from the set of the edges of $G$ to the set of real numbers. For any subgraph $G'$ of $G$, we define $w(G')$ to be the sum of the weights of the edges of $G'$. For any $i,j $ vertices of $G$, we define $D_{\{i,j\}} ({\cal G})$ to be the minimum of the weights of the simple paths of $G$ joining $i$ and $j$. The $D_{\{i,j\}} ({\cal G})$ are called $2$-weights of ${\cal G}$. Let $\{m_I\}_{I \in {\{1,...,n\} \choose 2}}$ and $\{M_I\}_{I \in {\{1,...,n\} \choose 2}}$ be two families of positive real numbers parametrized by the $2$-subsets of $ \{1,..., n\}$ with $m_I \leq M_I$ for any $I$; we study when there exist a positive-weighted graph ${\cal G}$ and an $n$-subset $\{1,..., n\}$ of the set of its vertices such that $D_I ({\cal G}) \in [m_I, M_I] $ for any $I \in {\{1,...,n\} \choose 2}$. Then we study the analogous problem for trees, both in the case of positive weights and in the case of general weights.