Title:

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 4Colouring problem which, as a graph problem, is NPcomplete. However, when a map is converted into a graph, we observe that this graph has structural properties, namely being (K_5, K_{3,3})minorfree which can be exploited and as such there exist algorithms which can find 4colourings of maps in polynomial time. This thesis looks at problems which are NPcomplete 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 NPcomplete. This thesis looks at four problems over four chapters, the first being Parallel KnockOut. This chapter will show that Parallel KnockOut can be solved in O(n+m) time on P_4free graphs, also known as cographs, however, remains hard on split graphs, a subclass of P_5free 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 3connected planar graphs but NPhard for 2connected 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 HColouring. This chapter shows that Surjective HColouring is NPcomplete, for any fixed, nonloop 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 HColouring on all fixed graphs H of size at most 4.
