1,962 research outputs found
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
Improved FPT algorithms for weighted independent set in bull-free graphs
Very recently, Thomass\'e, Trotignon and Vuskovic [WG 2014] have given an FPT
algorithm for Weighted Independent Set in bull-free graphs parameterized by the
weight of the solution, running in time . In this article
we improve this running time to . As a byproduct, we also
improve the previous Turing-kernel for this problem from to .
Furthermore, for the subclass of bull-free graphs without holes of length at
most for , we speed up the running time to . As grows, this running time is
asymptotically tight in terms of , since we prove that for each integer , Weighted Independent Set cannot be solved in time in the class of -free graphs unless the
ETH fails.Comment: 15 page
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 and~, where is
an independent set and induces a graph from some specified graph class
. We let be the class of -degenerate graphs. This
problem is known to be polynomial-time solvable if (bipartite graphs) and
NP-complete if (near-bipartite graphs) even for graphs of maximum degree
. Yang and Yuan [DM, 2006] showed that the case is polynomial-time
solvable for graphs of maximum degree . This also follows from a result of
Catlin and Lai [DM, 1995]. We consider graphs of maximum degree on
vertices. We show how to find and in time for , and in
time for . 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 -colourings of a graph
of maximum degree
On cycle transversals and their connected variants in the absence of a small linear forest.
A graph is H-free if it contains no induced subgraph isomorphic to H. We prove new complexity results for the two classical cycle transversal problems Feedback Vertex Set and Odd Cycle Transversal by showing that they can be solved in polynomial time for (sP1+P3) -free graphs for every integer s≥1 . We show the same result for the variants Connected Feedback Vertex Set and Connected Odd Cycle Transversal. For the latter two problems we also prove that they are polynomial-time solvable for cographs; this was known already for Feedback Vertex Set and Odd Cycle Transversal
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