Hamiltonicity of Connected Domination Critical Graphs.

A graph G is said to be k-gamma(c)-critical if the connected domination number gamma(c)(G) of G is k and gamma(c)(G + uv) < k for every uv (sic) E(G). The problem of interest for a positive integer 1 >= 2 is to determine whether or not l-connected k-gamma(c)-critical graphs are Hamiltonian. In this paper, for 1 >= 2, we prove that if k = 1, 2 or 3, then every l-connected k-gamma(c)-critical graph is Hamiltonian. We further show that, for n >= (k - 1)l+ 3, the class of 1-connected k-gamma(c)-critical non-Hamiltonian graphs of order n is empty if and only if k = 1,2 or 3.