Graph Colouring with Input Restrictions

In this thesis, we research the computational complexity of the graph colouring problem and its variants including precolouring extension and list colouring for graph classes that can be characterised by forbidding one or more induced subgraphs. We investigate the structural properties of such graph classes and prove a number of new properties. We then consider to what extent these properties can be used for efficiently solving the three types of colouring problems on these graph classes. In some cases we obtain polynomial-time algorithms, whereas other cases turn out to be NP-complete. We determine the computational complexity of k-COLOURING, k-PRECOLOURING EXTENSION and LIST k-COLOURING on $P_k$-free graphs. In particular, we prove that k-COLOURING on $P_8$-free graphs is NP-complete, 4-PRECOLOURING EXTENSION $P_7$-free graphs is NP-complete, and LIST 4-COLOURING on $P_6$-free graphs is NP-complete. In addition, we show the existence of an integer r such that k-COLOURING is NP-complete for $P_r$-free graphs with girth 4. In contrast, we determine for any fixed girth $g\geq 4$ a lower bound $r(g)$ such that every $P_{r(g)}$-free graph with girth at least $g$ is 3-colourable. We also prove that 3-LIST COLOURING is NP-complete for complete graphs minus a matching. We present a polynomial-time algorithm for solving 4-PRECOLOURING EXTENSION on $(P_2+P_3)$-free graphs, a polynomial-time algorithm for solving LIST 3-Colouring on $(P_2+P_4)$-free graphs, and a polynomial-time algorithm for solving LIST 3-COLOURING on $sP_3$-free graphs. We prove that LIST k-COLOURING for $(K_{s,t},P_r)$-free graphs is also polynomial-time solvable. We obtain several new dichotomies by combining the above results with some known results

Graph Algorithms and Complexity Aspects on Special Graph Classes

Graphs are a very flexible tool within mathematics, as such, numerous problems can be solved by formulating them as an instance of a graph. As a result, however, some of the structures found in real world problems may be lost in a more general graph. An example of this is the 4-Colouring problem which, as a graph problem, is NP-complete. However, when a map is converted into a graph, we observe that this graph has structural properties, namely being (K_5, K_{3,3})-minor-free which can be exploited and as such there exist algorithms which can find 4-colourings of maps in polynomial time. This thesis looks at problems which are NP-complete in general and determines the complexity of the problem when various restrictions are placed on the input, both for the purpose of finding tractable solutions for inputs which have certain structures, and to increase our understanding of the point at which a problem becomes NP-complete. This thesis looks at four problems over four chapters, the first being Parallel Knock-Out. This chapter will show that Parallel Knock-Out can be solved in O(n+m) time on P_4-free graphs, also known as cographs, however, remains hard on split graphs, a subclass of P_5-free graphs. From this a dichotomy is shown on $P_k$-free graphs for any fixed integer $k$. The second chapter looks at Minimal Disconnected Cut. Along with some smaller results, the main result in this chapter is another dichotomy theorem which states that Minimal Disconnected Cut is polynomial time solvable for 3-connected planar graphs but NP-hard for 2-connected planar graphs. The third chapter looks at Square Root. Whilst a number of results were found, the work in this thesis focuses on the Square Root problem when restricted to some classes of graphs with low clique number. The final chapter looks at Surjective H-Colouring. This chapter shows that Surjective H-Colouring is NP-complete, for any fixed, non-loop connected graph H with two reflexive vertices and for any fixed graph H’ which can be obtained from H by replacing vertices with true twins. This result enabled us to determine the complexity of Surjective H-Colouring on all fixed graphs H of size at most 4

Complexity Framework for Forbidden Subgraphs III: When Problems Are Tractable on Subcubic Graphs

For any finite set H = {H1,. .. , Hp} of graphs, a graph is H-subgraph-free if it does not contain any of H1,. .. , Hp as a subgraph. In recent work, meta-classifications have been studied: these show that if graph problems satisfy certain prescribed conditions, their complexity can be classified on classes of H-subgraph-free graphs. We continue this work and focus on problems that have polynomial-time solutions on classes that have bounded treewidth or maximum degree at most 3 and examine their complexity on H-subgraph-free graph classes where H is a connected graph. With this approach, we obtain comprehensive classifications for (Independent) Feedback Vertex Set, Connected Vertex Cover, Colouring and Matching Cut. This resolves a number of open problems. We highlight that, to establish that Independent Feedback Vertex Set belongs to this collection of problems, we first show that it can be solved in polynomial time on graphs of maximum degree 3. We demonstrate that, with the exception of the complete graph on four vertices, each graph in this class has a minimum size feedback vertex set that is also an independent set

Complexity Framework for Forbidden Subgraphs {III:}: When Problems Are Tractable on Subcubic Graphs

For any finite set H = {H1, . . ., Hp} of graphs, a graph is H-subgraph-free if it does not contain any of H1, . . ., Hp as a subgraph. In recent work, meta-classifications have been studied: these show that if graph problems satisfy certain prescribed conditions, their complexity can be classified on classes of H-subgraph-free graphs. We continue this work and focus on problems that have polynomial-time solutions on classes that have bounded treewidth or maximum degree at most 3 and examine their complexity on H-subgraph-free graph classes where H is a connected graph. With this approach, we obtain comprehensive classifications for (Independent) Feedback Vertex Set, Connected Vertex Cover, Colouring and Matching Cut. This resolves a number of open problems. We highlight that, to establish that Independent Feedback Vertex Set belongs to this collection of problems, we first show that it can be solved in polynomial time on graphs of maximum degree 3. We demonstrate that, with the exception of the complete graph on four vertices, each graph in this class has a minimum size feedback vertex set that is also an independent set

Complexity Framework for Forbidden Subgraphs {III:}: When Problems are Tractable on Subcubic Graphs

For any finite set H={H1,…,Hp} of graphs, a graph is H-subgraph-free if it does not contain any of H1,…,Hp as a subgraph. In recent work, meta-classifications have been studied: these show that if graph problems satisfy certain prescribed conditions, their complexity is determined on classes of H-subgraph-free graphs. We continue this work and focus on problems that have polynomial-time solutions on classes that have bounded treewidth or maximum degree at most~3 and examine their complexity on H-subgraph-free graph classes where H is a connected graph. With this approach, we obtain comprehensive classifications for (Independent) Feedback Vertex Set, Connected Vertex Cover, Colouring and Matching Cut. This resolves a number of open problems. We highlight that, to establish that Independent Feedback Vertex Set belongs to this collection of problems, we first show that it can be solved in polynomial time on graphs of maximum degree 3. We demonstrate that, with the exception of the complete graph on four vertices, each graph in this class has a minimum size feedback vertex set that is also an independent set