Realizability of graph specifications: Characterizations and algorithms

Abstract

The study of graphs and networks often involves studying various parameters of the graph vertices, capturing different aspects of the graph structure, such as the vertex degrees or the distances between the vertices. Given an n-vertex graph G and a parameter of interest f, one may associate with G a vector F(G) = 〈f 1 , …, f n 〉 giving the value of f for each vertex. This vector can be thought of as the f-profile of the graph. This paper concerns the dual problem, where given an n-entry f-specification vector F = 〈f 1 , …, f n 〉, we need to decide whether it is possible to find a graph G realizing this specification, namely, whose f-profile F(G) conforms to F. The paper introduces the notion of graph realiziations and illustrates a number of example problems related to finding graph realiziations for given specifications.