On Directed Tree Realizations of Degree Sets

Given a degree set D = {a1 < a2 < . . . < an} of nonnegative integers, the minimum number of vertices in any tree realizing the set D is known[10]. In this paper, we study the number of vertices and multiplicity of distinct degrees as parameters of tree realizations of degree sets. We explore this in the context of both directed and undirected trees and asymmetric directed graphs (graphs which do not have a cycle of length two). We show the following results. – We show a tight lower bound on the maximum multiplicity needed for any tree realization of a degree set. – For the directed trees, we study two natural notions of realizability by directed graphs and show tight lower bounds on the number of vertices needed to realize any degree set. – For asymmetric graphs, if μA(D) denotes the minimum number of vertices needed to realize any degree set, we show that a1 +an +1 ≤ μA(D) ≤ an−1 +an +1. We also derive sufficiency conditions on ai’s under which the lower bound is achieved. – We study the following algorithmic questions related to our problem and study their complexity. (1) Given a degree set D and a nonnegative integer r (as 1), test whether the set D can be realized by a tree of exactly μT (D) + r number of vertices. We show that the problem is fixed parameter tractable under two natural parameterizations of |D| and r. We also study the variant of the problem : (2) Given a tree T , and a non-negative integer r (in unary), test whether there exists another tree T ′ such that T ′ has exactly r more vertices than T and has the same degree set as T . We show that this problem can be solved in log-space. – For directed trees, under the both notions of realizability, we show that if μ′(D) is the minimum number of vertices needed for any directed tree realization, then for any non-negative integer r, there is a directed tree with μ′(D) + r vertices realizing the same degree set.