Weighted graphs with distances in given ranges

TL;DR

This paper investigates conditions under which weighted graphs and trees can be constructed so that the shortest path distances between specified vertex pairs fall within given ranges, extending classical distance realization problems.

Contribution

It provides new criteria for the existence of weighted graphs and trees with prescribed distance intervals between vertex pairs, covering both positive and general weights.

Findings

Derived necessary and sufficient conditions for graphs with distance ranges.

Extended results to trees with positive and arbitrary weights.

Established frameworks for constructing such graphs and trees.

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.