For a graph G = (V, E), a set S ⊆ V is a total dominating set if every vertex in V is adjacent to some vertex in S. The smallest cardinality of any total dominating set is the total domination number γt(G). For an arbitrary edge e εE(Ḡ), γt(G) - 2 ≤ γt(G + e) ≤ γt(G); if the latter inequality is strict for each e ε E(Ḡ) ≠ φ, then G is said to be γt-critical. The criticality index of an edge e ε E(Ḡ) is γt(e) = γt(G) - γt(G + e). Let E(Ḡ) = [e1...,em} and S = ∑j=1m̄ci(ej). The criticality index of G is ci(G) = S/m̄. For any rational number k, 0 ≤ k ≤ 2, we construct a graph G with ci(G) = k. For 1 ≤ k ≤ 2, we construct graphs with this property that are γt-critical as well as graphs that are not γt-critical.
| Identifer | oai:union.ndltd.org:ETSU/oai:dc.etsu.edu:etsu-works-19674 |
| Date | 01 May 2005 |
| Creators | Haynes, T. W., Mynhardt, C. M., Van Der Merwe, L. C. |
| Publisher | Digital Commons @ East Tennessee State University |
| Source Sets | East Tennessee State University |
| Detected Language | English |
| Type | text |
| Source | ETSU Faculty Works |
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