On Directed Tree Realizations of Degree Sets

Abstract

Given a degree set D = {a ₁ < a ₂ < … < a _n } of non-negative integers, the minimum number of vertices in any tree realizing the set D is known [11]. 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. 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 μ _A (D) denotes the minimum number of vertices needed to realize any degree set, we show that a ₁ + a _n  + 1 ≤ μ _A (D) ≤ a _n − 1 + a _n  + 1. We also derive sufficiency conditions on a _i ’s under which the lower bound is achieved.

We study the following related algorithmic questions. (1) Given a degree set D and a non-negative integer r (as 1^r), test whether the set D can be realized by a tree of exactly μ _T (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 study the complexity of the problem in the case of directed trees as well.