Enumerating Minimal Connected Dominating Sets in Graphs of Bounded Chordality

Listing, generating or enumerating objects of specified type is one of the principal tasks in algorithmics. In graph algorithms one often enumerates vertex subsets satisfying a certain property. We study the enumeration of all minimal connected dominating sets of an input graph from various graph classes of bounded chordality. We establish enumeration algorithms as well as lower and upper bounds for the maximum number of minimal connected dominating sets in such graphs. In particular, we present algorithms to enumerate all minimal connected dominating sets of chordal graphs in time O(1.7159^n), of split graphs in time O(1.3803^n), and of AT-free, strongly chordal, and distance-hereditary graphs in time O^*(3^{n/3}), where n is the number of vertices of the input graph. Our algorithms imply corresponding upper bounds for the number of minimal connected dominating sets for these graph classes

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

Recognizing Graphs Close to Bipartite Graphs with an Application to Colouring Reconfiguration

We continue research into a well-studied family of problems that ask whether the vertices of a graph can be partitioned into sets $A$ and~$B$, where $A$ is an independent set and $B$ induces a graph from some specified graph class ${\cal G}$. We let ${\cal G}$ be the class of $k$-degenerate graphs. This problem is known to be polynomial-time solvable if $k=0$ (bipartite graphs) and NP-complete if $k=1$ (near-bipartite graphs) even for graphs of maximum degree $4$. Yang and Yuan [DM, 2006] showed that the $k=1$ case is polynomial-time solvable for graphs of maximum degree $3$. This also follows from a result of Catlin and Lai [DM, 1995]. We consider graphs of maximum degree $k+2$ on $n$ vertices. We show how to find $A$ and $B$ in $O(n)$ time for $k=1$, and in $O(n^2)$ time for $k\geq 2$. Together, these results provide an algorithmic version of a result of Catlin [JCTB, 1979] and also provide an algorithmic version of a generalization of Brook's Theorem, which was proven in a more general way by Borodin, Kostochka and Toft [DM, 2000] and Matamala [JGT, 2007]. Moreover, the two results enable us to complete the complexity classification of an open problem of Feghali et al. [JGT, 2016]: finding a path in the vertex colouring reconfiguration graph between two given $\ell$-colourings of a graph of maximum degree $k$

Boundary classes for graph problems involving non-local properties

We continue the study of boundary classes for NP-hard problems and focus on seven NP-hard graph problems involving non-local properties: HAMILTONIAN CYCLE, HAMILTONIAN CYCLE THROUGH SPECIFIED EDGE, HAMILTONIAN PATH, FEEDBACK VERTEX SET, CONNECTED VERTEX COVER, CONNECTED DOMINATING SET and GRAPH VCCON DIMENSION. Our main result is the determination of the first boundary class for FEEDBACK VERTEX SET. We also determine boundary classes for HAMILTONIAN CYCLE THROUGH SPECIFIED EDGE and HAMILTONIAN PATH and give some insights on the structure of some boundary classes for the remaining problems

Towards Constant-Factor Approximation for Chordal / Distance-Hereditary Vertex Deletion

For a family of graphs ?, Weighted ?-Deletion is the problem for which the input is a vertex weighted graph G = (V, E) and the goal is to delete S ? V with minimum weight such that G?S ? ?. Designing a constant-factor approximation algorithm for large subclasses of perfect graphs has been an interesting research direction. Block graphs, 3-leaf power graphs, and interval graphs are known to admit constant-factor approximation algorithms, but the question is open for chordal graphs and distance-hereditary graphs. In this paper, we add one more class to this list by presenting a constant-factor approximation algorithm when ? is the intersection of chordal graphs and distance-hereditary graphs. They are known as ptolemaic graphs and form a superset of both block graphs and 3-leaf power graphs above. Our proof presents new properties and algorithmic results on inter-clique digraphs as well as an approximation algorithm for a variant of Feedback Vertex Set that exploits this relationship (named Feedback Vertex Set with Precedence Constraints), each of which may be of independent interest