Graph realizations: Maximum degree in vertex neighborhoods

This paper initiates the study of the maximum neighborhood degree realization problem. Given a sequence D=(d₁,…,d_n) of non-negative integers, the goal is to construct a simple graph with vertices v₁,…,v_n such that for every i∈[1,n], the maximum degree in the neighborhood of v_i is exactly d_i (or output null if no such graph exists). Depending upon whether or not the realizing graph is required to be connected, and whether or not the neighborhood of a vertex is closed (that is, the neighborhood includes the vertex itself), the problem has four natural settings. We provide complete realizability criteria for all four settings of the problem. Our conditions are verifiable in linear time and our realizations can be constructed in polynomial time. In addition, we prove tight/approximate bounds for the number of maximum neighboring degree profiles of length n that are realizable.