Open packing, total domination, and theP3-Radon number

A Taxonomy of Perfect Domination

“…For all graphs G, r o (G) ≤ g t (G), where g t (G) denotes the total domination number [33]. Rall proved that r o (T) = g t (T) for all trees with at least two vertices [49].…”

Total and Double Total Domination Number on Hexagonal Grid

“…Graph dominations are widely applied in different problems such as dominating queens, computer network, school bus routing, and social network problems. Specifically, graph dominations have huge applications in chemistry [1][2][3][4][5]. Chemical structures can be represented by graphs, where vertices and edges represent atoms and chemical bonds, respectively.…”

“…Because of such a correspondence, many chemical and physical properties of molecules are in correlation with graph theoretical invariants. One very important such invariant is the total (double) domination number [2,[6][7][8][9][10].…”

Domination, Total Domination and Open Packing of the Corcor Domain of Graphene

“…Open packing is the natural dual object of total dominating sets, and so ρ 0 (G) ≤ γ t (G) holds for every graph G, [5]. Moreover, if there exists a total dominating set D of G and D is also an open packing, then ρ 0 (G) = γ t (G) = |D|.…”

“…For computing γ t (G), it is enough to calculate |T |.|T | = 2|{v 1,3 , v 1,4 , v 2,1 , v 2,2 , v 2,7 , v 2,8 , v 2,9 , v 3,1 , v 3,2 , v 3,7 , v3,8 , v 3,13 , v 3,14 , v 4,2 , v 4,5 , v 4,6 , v 4,11 , v 4,12 , v 4,15 , v 5,2 , v 5,3 , v 5,8 , v 5,9 , v 5,14 , v 5,15 , v 6,4 }|.Then|T | ≤ |V (G[n])| 6n − 2 = 14n 2 − 2n. Since ρ 0 (G) ≤ γ t (G) holds for every graph G[5]. So that the inequality is hold.…”