Realizability of the Total Domination Criticality Index
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.
Haynes, T. W.; Mynhardt, C. M.; and Van Der Merwe, L. C.. 2005. Realizability of the Total Domination Criticality Index. Utilitas Mathematica. Vol.67 3-8. ISSN: 0315-3681