Total Domination in Graphs

WITHDRAWN: On graphs with disjoint dominating and 2-dominating sets

“…Various graph theoretic properties and parameters of graphs having disjoint dominating sets are studied in [1,8,9,10,14,20,21]. Characterizations of graphs with disjoint dominating and total dominating sets are given in [11,12,13,16,17,19,25], while in [2,4,5,6,18] graphs which have the property that their vertex set can be partitioned into two disjoint total dominating sets are studied. Conditions which guarantee the existence of a dominating set whose complement contains a 2-dominating set, a paired-dominating set or an independent dominating set are presented in [7,12,15,19,20,22,26].…”

Total Domination Versus Domination in Cubic Graphs

“…In this case G ∼ = C n and n ≥ 3. It is well known (see, for example, [9,11,15]) that γ t (C n ) = n/2 + n/4 − n/4 and γ (C n ) = n/3 . As an immediate consequence of these results, we note that…”

“…For ≥ 1, let G be the graph constructed in [8] as follows. (This construction is also described in Chapter 5, p. 44, of [15].) Consider two copies of the path P 2 with respective vertex sequences Let H even be the subfamily of H consisting of all graphs H where is even; that is, H even = {H | ≥ 2 is even}.…”

“…For a recent book on total domination in graphs we refer the reader to [15]. A survey of total domination in graphs can also be found in [14].…”

Disjoint Paired-Dominating sets in Cubic Graphs

“…The notions of graph domination, total domination, and paired-domination have been investigated extensively in the literature (see e.g., [4, 5, 7–11] and the references therein). We recall the main definitions.…”