Graph Transversals for Hereditary Graph Classes: a Complexity Perspective

Within the broad field of Discrete Mathematics and Theoretical Computer Science, the theory of graphs has been of fundamental importance in solving a large number of optimization problems and in modelling real-world situations. In this thesis, we study a topic that covers many aspects of Graph Theory: transversal sets. A transversal set in a graph G is a vertex set that intersects every subgraph of G that belongs to a certain class of graphs. The focus is on vertex cover, feedback vertex set and odd cycle transversal. The decision problems Vertex Cover, Feedback Vertex Set and Odd Cycle Transversal ask, for a given graph G and an integer k, whether there is a corresponding transversal of G of size at most k. These problems are NP-complete in general and our focus is to determine the complexity of the problems when various restrictions are placed on the input, both for the purpose of finding tractable cases and to increase our understanding of the point at which a problem becomes NP-complete. We consider graph classes that are closed under vertex deletion and in particular H-free graphs, i.e. graphs that do not contain a graph H as an induced subgraph. The first chapter is an introduction to the thesis. There we illustrate the motivation of our work and introduce most of the terminology we have used for our research. In the second chapter, we develop a number of structural results for some classes of H-free graphs. The third chapter looks at the Subset Transversal problems: there we prove that Feedback Vertex Set and Odd Cycle Transversal and their subset variants can be solved in polynomial time for both P_4-free and (sP_1+P_3)-free graphs, while for Subset Vertex Cover we show that it can be solved in polynomial time for (sP_1+P_4)-free graphs. The fourth chapter is entirely dedicated to the Connected Vertex Cover problem. The connectivity constraint requires additional proof techniques. We prove this problem can be solved in polynomial time for (sP_1+P_5)-free graphs, even when weights are given to the vertices of the graph. We continue the research on connected transversals in the fifth chapter: we show that Connected Feedback Vertex Set, Connected Odd Cycle Transversal and their extension variants can be solved in polynomial time for both P_4-free and (sP_1+P_3)-free graphs. In the sixth chapter we study the price of independence: can the size of a smallest independent transversal be bounded in terms of the minimum size of a transversal? We establish complete and almost-complete dichotomies which determine for which graph classes such a bound exists and for which cases such a bound is the identity

Covering Small Independent Sets and Separators with Applications to Parameterized Algorithms

We present two new combinatorial tools for the design of parameterized algorithms. The first is a simple linear time randomized algorithm that given as input a $d$-degenerate graph $G$ and an integer $k$, outputs an independent set $Y$, such that for every independent set $X$ in $G$ of size at most $k$, the probability that $X$ is a subset of $Y$ is at least $\left({(d+1)k \choose k} \cdot k(d+1)\right)^{-1}$.The second is a new (deterministic) polynomial time graph sparsification procedure that given a graph $G$, a set $T = \{\{s_1, t_1\}, \{s_2, t_2\}, \ldots, \{s_\ell, t_\ell\}\}$ of terminal pairs and an integer $k$, returns an induced subgraph $G^\star$ of $G$ that maintains all the inclusion minimal multicuts of $G$ of size at most $k$, and does not contain any $(k+2)$-vertex connected set of size $2^{{\cal O}(k)}$. In particular, $G^\star$ excludes a clique of size $2^{{\cal O}(k)}$ as a topological minor. Put together, our new tools yield new randomized fixed parameter tractable (FPT) algorithms for Stable $s$-$t$ Separator, Stable Odd Cycle Transversal and Stable Multicut on general graphs, and for Stable Directed Feedback Vertex Set on $d$-degenerate graphs, resolving two problems left open by Marx et al. [ACM Transactions on Algorithms, 2013]. All of our algorithms can be derandomized at the cost of a small overhead in the running time.Comment: 35 page

Bidimensionality and Geometric Graphs

In this paper we use several of the key ideas from Bidimensionality to give a new generic approach to design EPTASs and subexponential time parameterized algorithms for problems on classes of graphs which are not minor closed, but instead exhibit a geometric structure. In particular we present EPTASs and subexponential time parameterized algorithms for Feedback Vertex Set, Vertex Cover, Connected Vertex Cover, Diamond Hitting Set, on map graphs and unit disk graphs, and for Cycle Packing and Minimum-Vertex Feedback Edge Set on unit disk graphs. Our results are based on the recent decomposition theorems proved by Fomin et al [SODA 2011], and our algorithms work directly on the input graph. Thus it is not necessary to compute the geometric representations of the input graph. To the best of our knowledge, these results are previously unknown, with the exception of the EPTAS and a subexponential time parameterized algorithm on unit disk graphs for Vertex Cover, which were obtained by Marx [ESA 2005] and Alber and Fiala [J. Algorithms 2004], respectively. We proceed to show that our approach can not be extended in its full generality to more general classes of geometric graphs, such as intersection graphs of unit balls in R^d, d >= 3. Specifically we prove that Feedback Vertex Set on unit-ball graphs in R^3 neither admits PTASs unless P=NP, nor subexponential time algorithms unless the Exponential Time Hypothesis fails. Additionally, we show that the decomposition theorems which our approach is based on fail for disk graphs and that therefore any extension of our results to disk graphs would require new algorithmic ideas. On the other hand, we prove that our EPTASs and subexponential time algorithms for Vertex Cover and Connected Vertex Cover carry over both to disk graphs and to unit-ball graphs in R^d for every fixed d

Feedback Vertex Set Inspired Kernel for Chordal Vertex Deletion

Given a graph $G$ and a parameter $k$, the Chordal Vertex Deletion (CVD) problem asks whether there exists a subset $U\subseteq V(G)$ of size at most $k$ that hits all induced cycles of size at least 4. The existence of a polynomial kernel for CVD was a well-known open problem in the field of Parameterized Complexity. Recently, Jansen and Pilipczuk resolved this question affirmatively by designing a polynomial kernel for CVD of size $O(k^{161}\log^{58}k)$, and asked whether one can design a kernel of size $O(k^{10})$. While we do not completely resolve this question, we design a significantly smaller kernel of size $O(k^{12}\log^{10}k)$, inspired by the $O(k^2)$-size kernel for Feedback Vertex Set. Furthermore, we introduce the notion of the independence degree of a vertex, which is our main conceptual contribution