The K-discretization and K-incident graphs for discretizable Distance Geometry

Abstract

The Distance Geometry Problem (DGP) is the problem of determining whether a realization for a simple weighted undirected graph $$G=(V,E,d)$$ in a given Euclidean space exists so that the distances between pairs of realized vertices u, $$v \in V$$ correspond to the weights $$d_{uv}$$, for each $$\{u,v\} \in E$$. We focus on a special class of DGP instances, referred to as the Discretizable DGP (DDGP), and we introduce the K-discretization and the K-incident graphs for the DDGP class. The K-discretization graph is independent on the vertex order that can be assigned to V, and can be useful for discovering whether one of such orders actually exists so that the DDGP assumptions are satisfied. The use of a given vertex order allows the definition of another important graph, the K-incident graph, which is potentially useful for performing pre-processing analysis on the solution set of DDGP instances.