Minimum neighboring degree realization in graphs and trees

We study a graph realization problem that pertains to degrees in vertex neighborhoods. The classical problem of degree sequence realizability asks whether or not a given sequence of n positive integers is equal to the degree sequence of some n-vertex undirected simple graph. While the realizability problem of degree sequences has been well studied for different classes of graphs, there has been relatively little work concerning the realizability of other types of information profiles, such as the vertex neighborhood profiles. In this paper we introduce and explore the minimum degrees in vertex neighborhood profile as it is one of the most natural extensions of the classical degree profile to vertex neighboring degree profiles. Given a graph G = (V, E), the min-degree of a vertex v ∈ V , namely MinND(v), is given by min{deg(w) | w ∈ N[v]}. Our input is a sequence σ = (dⁿ<sub>`</sub><sup>`</sup>, · · ·, dⁿ₁¹ ), where di+1 > di and each ni is a positive integer. We provide some necessary and sufficient conditions for σ to be realizable. Furthermore, under the restriction that the realization is acyclic, i.e., a tree or a forest, we provide a full characterization of realizable sequences, along with a corresponding constructive algorithm. We believe our results are a crucial step towards understanding extremal neighborhood degree relations in graphs.