Codiameters of 3-domination critical graphs with toughness more than one

Author name used in this publication: C. T. Ng2008-2009 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishe

Hamilton-connectivity of 3-domination critical graphs with α=δ+1≥5

Author name used in this publication: C. T. Ng2007-2008 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishe

Critical concepts in domination, independence and irredundance of graphs

The lower and upper independent, domination and irredundant numbers of the graph G = (V, E) are denoted by i ( G) , f3 ( G), 'Y ( G), r ( G), ir ( G) and IR ( G) respectively. These six numbers are called the domination parameters. For each of these parameters n:, we define six types of criticality. The graph G is n:-critical (n:+ -critical) if the removal of any vertex of G causes n: (G) to decrease (increase), G is n:-edge-critical (n:+-edge-critical) if the addition of any missing edge causes n: (G) to decrease (increase), and G is Ir-ER-critical (n:- -ER-critical) if the removal of any edge causes n: (G) to increase (decrease). For all the above-mentioned parameters n: there exist graphs which are n:-critical, n:-edge-critical and n:-ER-critical. However, there do not exist any n:+-critical graphs for n: E {ir,"f,i,/3,IR}, no n:+-edge-critical graphs for n: E {ir,"f,i,/3} and non:--ER-critical graphs for: E {'Y,/3,r,IR}. Graphs which are "I-critical, i-critical, "I-edge-critical and i-edge-critical are well studied in the literature. In this thesis we explore the remaining types of criticality. We commence with the determination of the domination parameters of some wellknown classes of graphs. Each class of graphs we consider will turn out to contain a subclass consisting of graphs that are critical according to one or more of the definitions above. We present characterisations of "I-critical, i-critical, "I-edge-critical and i-edge-critical graphs, as well as ofn:-ER-critical graphs for n: E {/3,r,IR}. These characterisations are useful in deciding which graphs in a specific class are critical. Our main results concern n:-critical and n:-edge-critical graphs for n: E {/3, r, IR}. We show that the only /3-critical graphs are the edgeless graphs and that a graph is IRcritical if and only if it is r-critical, and proceed to investigate the r-critical graphs which are not /3-critical. We characterise /3-edge-critical and r-edge-critical graphs and show that the classes of IR-edge-critical and r-edge-critical graphs coincide. We also exhibit classes of r+ -critical, r+ -edge-critical and i- -ER-critical graphs.Mathematical SciencesD. Phil. (Mathematics

Two conjectures on 3-domination critical graphs

For a graph G = (V (G), E (G)), a set S ~ V (G) dominates G if each vertex in V (G) \S is adjacent to a vertex in S. The domination number I (G) (independent domination number i (G)) of G is the minimum cardinality amongst its dominating sets (independent dominating sets). G is k-edge-domination-critical, abbreviated k-1- critical, if the domination number k decreases whenever an edge is added. Further, G is hamiltonian if it has a cycle that passes through each of its vertices. This dissertation assimilates research generated by two conjectures: Conjecture I. Every 3-1-critical graph with minimum degree at least two is hamiltonian. Conjecture 2. If G is k-1-critical, then I ( G) = i ( G). The recent proof of Conjecture I is consolidated and presented accessibly. Conjecture 2 remains open for k = 3 and has been disproved for k :::>: 4. The progress is detailed and proofs of new results are presented.Mathematical ScienceM. Sc. (Mathematics