Beyond Bidimensionality: Parameterized Subexponential Algorithms on Directed Graphs

We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approaches with two well studied problems. For the first problem, {\sc $k$-Leaf Out-Branching}, which is to find an oriented spanning tree with at least $k$ leaves, we obtain an algorithm solving the problem in time $2^{O(\sqrt{k} \log k)} n+ n^{O(1)}$ on directed graphs whose underlying undirected graph excludes some fixed graph $H$ as a minor. For the special case when the input directed graph is planar, the running time can be improved to $2^{O(\sqrt{k})}n + n^{O(1)}$. The second example is a generalization of the {\sc Directed Hamiltonian Path} problem, namely {\sc $k$-Internal Out-Branching}, which is to find an oriented spanning tree with at least $k$ internal vertices. We obtain an algorithm solving the problem in time $2^{O(\sqrt{k} \log k)} + n^{O(1)}$ on directed graphs whose underlying undirected graph excludes some fixed apex graph $H$ as a minor. Finally, we observe that for any $\epsilon>0$, the {\sc $k$-Directed Path} problem is solvable in time $O((1+\epsilon)^k n^{f(\epsilon)})$, where $f$ is some function of \ve. Our methods are based on non-trivial combinations of obstruction theorems for undirected graphs, kernelization, problem specific combinatorial structures and a layering technique similar to the one employed by Baker to obtain PTAS for planar graphs

Oriented trees and paths in digraphs

Which conditions ensure that a digraph contains all oriented paths of some given length, or even a all oriented trees of some given size, as a subgraph? One possible condition could be that the host digraph is a tournament of a certain order. In arbitrary digraphs and oriented graphs, conditions on the chromatic number, on the edge density, on the minimum outdegree and on the minimum semidegree have been proposed. In this survey, we review the known results, and highlight some open questions in the area

Quick Trips: On the Oriented Diameter of Graphs

In this dissertation, I will discuss two results on the oriented diameter of graphs with certain properties. In the first problem, I studied the oriented diameter of a graph G. Erdos et al. in 1989 showed that for any graph with |V | = n and δ(G) = δ the maximum the diameter could possibly be was 3 n/ δ+1. I considered whether there exists an orientation on a given graph with |G| = n and δ(G) = δ that has a small diameter. Bau and Dankelmann (2015) showed that there is an orientation of diameter 11 n/ δ+1 + O(1), and showed that there is a graph which the best orientation admitted is 3 n/ δ+1 + O(1). It was left as an open question whether the factor of 11 in the first result could be reduced to 3. The result above was improved to 7 n / δ+1 +O(1) by Surmacs (2017) and I will present a proof of a further improvement of this bound to 5 n/δ−1 + O(1). It remains open whether 3 is the best answer. In the second problem, I studied the oriented diameter of the complete graph Kn with some edges removed. We will show that given Kn with n \u3e= 5 and any collection of edges Ev, with |Ev| = n − 5, that there is an orientation of this graph with diameter 2. It remains a question how many edges we can remove to guarantee larger diameters

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application