Irreversible k-threshold conversion processes on graphs
| dc.contributor.author | Wodlinger, Jane | |
| dc.contributor.supervisor | Mynhardt, C.M. | |
| dc.date.accessioned | 2018-04-30T16:37:58Z | |
| dc.date.available | 2018-04-30T16:37:58Z | |
| dc.date.copyright | 2018 | en_US |
| dc.date.issued | 2018-04-30 | |
| dc.degree.department | Department of Mathematics and Statistics | en_US |
| dc.degree.level | Doctor of Philosophy Ph.D. | en_US |
| dc.description.abstract | Given a graph G and an initial colouring of its vertices with two colours, say black and white, an irreversible k-threshold conversion process on G is an iterative process in which a white vertex becomes permanently coloured black at time t if at least k of its neighbours are coloured black at time t-1. A set S of vertices is an irreversible k-threshold conversion set (k-conversion set) of G if the initial colouring in which the vertices of S are black and the others are white results in the whole vertex set becoming black eventually. In the case where G is (k+1)-regular, it can be shown that the k-conversion sets coincide with the so-called feedback vertex sets, or decycling sets. In this dissertation we study the size and structure of minimum k-conversion sets in several classes of graphs. We examine conditions that lead to equality and inequality in existing bounds on the minimum size of a k-conversion set of G, for k- and (k+1)-regular graphs G. Furthermore, we derive new sharp lower bounds on this number for regular graphs of degree ranging from k+1 to 2k-1 and for graphs of maximum degree k+1. We determine exact values of the minimum size of a k-conversion set for certain classes of trees. We show that every (k+1)-regular graph has a minimum k-conversion set that avoids certain structures in its induced subgraph. These results lead to new proofs of several known results on colourings and forest partitions of (k+1)-regular graphs and graphs of maximum degree k+1. | en_US |
| dc.description.scholarlevel | Graduate | en_US |
| dc.identifier.uri | http://hdl.handle.net/1828/9282 | |
| dc.language | English | eng |
| dc.language.iso | en | en_US |
| dc.rights | Available to the World Wide Web | en_US |
| dc.subject | graph theory | en_US |
| dc.subject | conversion process | en_US |
| dc.subject | irreversible conversion process | en_US |
| dc.subject | k-conversion | en_US |
| dc.subject | decycling | en_US |
| dc.subject | feedback vertex | en_US |
| dc.subject | k-threshold | en_US |
| dc.subject | vertex arboricity | en_US |
| dc.subject | conversion set | en_US |
| dc.subject | forest partition | en_US |
| dc.title | Irreversible k-threshold conversion processes on graphs | en_US |
| dc.type | Thesis | en_US |