The Forcing Domination Number of Hamiltonian Cubic Graphs

Abstract

A set of vertices S in a graph G is called to be a Smarandachely dominating k-set, if each vertex of G is dominated by at least k vertices of S. Particularly, if k = 1, such a set is called a dominating set of G. The Smarandachely domination number k(G) of G is the minimum cardinality of a Smarandachely dominating set of G. For abbreviation, we denote 1(G) by (G). In 1996, Reed proved that the domination number (G) of every n-vertex graph G with minimum degree at least 3 is at most 3n/8. Also, he conjectured that (H) ≥ ⌈n/3⌉ for every connected 3-regular n-vertex graph H. In (?), the authors presented a sequence of Hamiltonian cubic graphs whose domination numbers are sharp and in this paper we study forcing domination number for those graphs.