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

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 ∈ V correspond to the weights d_uv , for each {u,v}∈ 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.