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. / Graduate
| Identifer | oai:union.ndltd.org:uvic.ca/oai:dspace.library.uvic.ca:1828/9282 |
| Date | 30 April 2018 |
| Creators | Wodlinger, Jane |
| Contributors | Mynhardt, C.M. |
| Source Sets | University of Victoria |
| Language | English, English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
| Rights | Available to the World Wide Web |
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