Augmenting a graph of minimum degree 2 to have two disjoint total dominating sets

WITHDRAWN: On graphs with disjoint dominating and 2-dominating sets

“…Hence renaming s 1 and s 2 if necessary, we may assume that H) in G, contradicting our choice of F . This completes the proof of the statement (6).…”

“…The statements (1)- (5) are immediate consequences of the definition of the 2-subdivision graph. To prove (6), let G ′ be a spanning subgraph of G that is a 2subdivision graph and let F be a component of G ′ . Since G ′ is a 2-subdivision graph, so too is the graph F , i.e., F = S 2 (H ′ ) for some connected graph H ′ (and some functions ξ ′ : E H ′ → X 2 and α ′ : L H ′ → N).…”

“…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].…”

Minimal graphs with disjoint dominating and total dominating sets

“…Various graph theoretic properties and parameters of graphs having disjoint dominating sets are studied in [1,7,8,9,13,21,22]. Characterizations of graphs with disjoint dominating and total dominating sets (or with disjoint total dominating sets) are given in [2,3,4,5,10,11,12,16,17,19,20,26]. 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 [6,11,15,20,21,23,27].…”

Disjoint dominating and 2-dominating sets in graphs

“…Various graph theoretic properties and parameters of graphs having disjoint dominating sets are studied in [1,12,13,14,18,26,27]. Characterizations of graphs with disjoint dominating and total dominating sets are given in [15,16,17,20,21,24,29], while in [2,4,5,8,22] 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 [11,16,19,24,26,30].…”