Composed Degree-Distance Realizations of Graphs

Network realization problems require, given a specification π for some network parameter (such as degrees, distances or connectivity), to construct a network G conforming to π, or to determine that no such network exists. In this paper we study composed profile realization, where the given instance consists of two or more profile specifications that need to be realized simultaneously. To gain some understanding of the problem, we focus on two classical profile types, namely, degrees and distances, which were (separately) studied extensively in the past. We investigate a wide spectrum of variants of the composed distance and degree realization problem. For each variant we either give a polynomial-time realization algorithm or establish NP hardness. In particular: (i)We consider both precise specifications and range specifications, which specify a range of permissible values for each entry of the profile.(ii)We consider realizations by both weighted and unweighted graphs.(iii)We also study settings where the realizing graph is restricted to specific graph classes, including trees and bipartite graphs.