NP-Hardness and Fixed-Parameter Tractability of Realizing Degree Sequences with Directed Acyclic Graphs

TL;DR

This paper proves that realizing degree sequences with directed acyclic graphs is NP-complete, but fixed-parameter tractable when considering the maximum degree, resolving an open problem in graph theory.

Contribution

It establishes NP-completeness for the directed acyclic graph realization problem and shows fixed-parameter tractability based on maximum degree.

Findings

NP-completeness of the problem

Fixed-parameter tractability with respect to maximum degree

Answers an open question from prior research

Abstract

In graph realization problems one is given a degree sequence and the task is to decide whether there is a graph whose vertex degrees match to the given sequence. This realization problem is known to be polynomial-time solvable when the graph is directed or undirected. In contrary, we show NP-completeness for the problem of realizing a given sequence of pairs of positive integers (representing indegrees and outdegrees) with a directed acyclic graph, answering an open question of Berger and M\"uller-Hannemann [FCT 2011]. Furthermore, we classify the problem as fixed-parameter tractable with respect to the parameter "maximum degree".