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 Science / M. Sc. (Mathematics)
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:unisa/oai:uir.unisa.ac.za:10500/17505 |
| Date | 01 1900 |
| Creators | Moodley, Lohini |
| Contributors | Mynhardt, C. M. |
| Source Sets | South African National ETD Portal |
| Detected Language | English |
| Type | Dissertation |
| Format | 1 online resource (11, 86 leaves) |
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