Irreversible k-threshold conversion processes on graphs

dc.contributor.authorWodlinger, Jane
dc.contributor.supervisorMynhardt, C.M.
dc.date.accessioned2018-04-30T16:37:58Z
dc.date.available2018-04-30T16:37:58Z
dc.date.copyright2018en_US
dc.date.issued2018-04-30
dc.degree.departmentDepartment of Mathematics and Statisticsen_US
dc.degree.levelDoctor of Philosophy Ph.D.en_US
dc.description.abstractGiven 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.scholarlevelGraduateen_US
dc.identifier.urihttp://hdl.handle.net/1828/9282
dc.languageEnglisheng
dc.language.isoenen_US
dc.rightsAvailable to the World Wide Weben_US
dc.subjectgraph theoryen_US
dc.subjectconversion processen_US
dc.subjectirreversible conversion processen_US
dc.subjectk-conversionen_US
dc.subjectdecyclingen_US
dc.subjectfeedback vertexen_US
dc.subjectk-thresholden_US
dc.subjectvertex arboricityen_US
dc.subjectconversion seten_US
dc.subjectforest partitionen_US
dc.titleIrreversible k-threshold conversion processes on graphsen_US
dc.typeThesisen_US

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