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 Sciences / D. Phil. (Mathematics)
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:unisa/oai:uir.unisa.ac.za:10500/16885 |
| Date | 11 1900 |
| Creators | Grobler, Petrus Jochemus Paulus |
| Contributors | Mynhardt, C. M. |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | 1 online resource (iii, 83 leaves) |
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