Degree-constrained editing of small-degree graphs

This thesis deals with degree-constrained graph modification problems. In particular, we investigate the computational complexity of DAG Realization and Degree Anonymity. The DAG Realization problem is, given a multiset of positive integer pairs, to decide whether there is a realizing directed acyclic graph (DAG), that is, pairs are one-to-one assigned to vertices such that the indegree and the outdegree of every vertex coincides with the two integers of the assigned pair. The Degree Anonymity problem is, given an undirected graph G and two positive integers k and s, to decide whether at most s graph modification operations can be performed in G in order to obtain a k-anonymous graph, that is, a graph where for each vertex there are k − 1 other vertices with the same degree. We classify both problems as NP-complete, that is, there are presumably no polynomial-time algorithms that can solve every instance of these problems. Confronted with this worst-case intractability, we perform a parameterized complexity study in order to detect efficiently solvable special cases that are still practically relevant. The goal herein is to develop fixed-parameter algorithms where the seemingly unavoidable exponential dependency in the running time is confined to a parameter of the input. If the parameter is small, then the corresponding fixed-parameter algorithm is fast. The parameter thus measures some structure in the input whose exploitation makes the particular input tractable. Considering Degree Anonymity, two natural parameters provided with the input are anonymity level k and solution size s. However, we will show that Degree Anonymity is W[1]-hard with respect to the parameter s even if k = 2. This means that the existence of fixed-parameter algorithms for s and k is very unlikely. Thus, other parameters have to be considered. We will show that the parameter maximum vertex degree is very promising for both DAG Realization and Degree Anonymity. Herein, for Degree Anonymity, we consider the maximum degree of the input graph. Considering DAG Realization, we take the maximum degree in a realizing DAG. Due to the problem definition, we can easily determine the maximum degree by taking the maximum over all integers in the given multiset. We provide fixed-parameter