Tree-Breadth of Graphs with Variants and Application

The tree-breadth of a graph is a recently introduced variant of the well known idea of decomposing a graph into a tree of bags. It is a parameter which adds a metric constraint to tree-decompositions limiting the radius of each bag. In this dissertation, we further investigate the tree-breadth of graphs. We present approaches to compute a tree-decomposition with small breadth for a given graph including approximation algorithms for general graphs as well as optimal solutions for special graph classes. Additionally, we introduce a variant of tree-breadth called strong tree-breadth. Next, we present various algorithms to approach the (Connected) r-Domination problem for graphs with bounded tree-breadth. One variant, called path-breadth, requires the decomposition to be a path instead of a tree. We use graphs with bounded path-breadth to construct efficient constant-factor approximation algorithms for the bandwidth and line-distortion problems. Motivated by these results, we introduce and investigate the new Minimum Eccentricity Shortest Path problem. We analyse the hardness of the problem, show algorithms to compute an optimal solution, and present approximation algorithms differing in quality and runtime.