Inverse total domination in graphs

Partitioning a graph into a dominating set, a total dominating set, and something else

“…Recently, several authors studied the cardinalities of pairs of disjoint dominating sets in graph [2,7,8,11,12]. The context of this research motivates the question for which graphs Theorem 1 is best-possible in the sense that the union D ∪ T of the two sets necessarily contains all vertices of the graph G. Our following main result gives a partial answer to this question.…”

The Mouse Cortical Interareal Network Reveals Well Defined Connectivity Profiles and an Ultra Dense Cortical Graph

“…28, 2017; A dominating set analysis on G19x47 provides further evidence that the FIN is indeed dense ( Figure S4A). In graph theory, a subset D of nodes of a graph G with node set V is said to be dominating G, if all elements of V have a link to at least one node in D (Kulli and Sigarkanti, 1991). Here we modify this definition slightly by saying that D dominates x% of the nodes of G, if an x% of all nodes in V are linked to one or more nodes in D. The x%=100% corresponds to "full" domination.…”

Equalityof Secure Domination and Inverse Secure Domination Numbers

“…A set D of vertices in a graph G is a dominating set if every vertex in V -D is adjacent to somevertexin D. The domination number (G) of G is the minimum cardinality of a dominating set of G. Recently severaldomination parameters are given in the books by Kulli in 2,3,4 . Let D be a minimum dominating set of G. If V -D contains a dominating set D' of G, then D' is called an inverse dominating set of G with respect to D. The inverse domination number  -1 (G) of G is the minimum cardinality of an inverse dominating set of G. This concept was introduced by Kulliand Sigarkanti in 5 . Many other inverse domination parameters were studied, for example, in 6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,24 .…”