On the doubly connected domination number of a graph

Abstract

For a given connected graph G = (V, E), a set D ⊆ V (G) is a doubly connected dominating set if it is dominating and both D and V (G) − D are connected. The cardinality of the minimum doubly connected dominating set in G is the doubly connected domination number. We investigate several properties of doubly connected dominating sets and give some bounds on the doubly connected domination number.

Computation of Various Domination Numbers of Rolf Nevanlinna (RNP) Collaboration Graph

“…The outer connected domination number of G, denoted (G) γ~c is the cardinality of a minimum outer connected dominating set of G [6]. Note that every graph G has an outer connected dominating set.…”

“…The maximum cardinality of a 2-packing set of G is called the 2-packing number of G and is denoted by  2 (G). 6 ), e 6 = (u 1 , u 7 ), e 7 = (u 1 , u 8 ), e 8 = (u 1 , u 9 ), e 9 = (u 1 , u 10 ), e 10 = (u 2 , u 8 ), e 11 = (u 2 , u 13 ), e 12 = (u 2 , u 14 ), e 13 = (u 2 , u 17 ), e 14 = (u 2 , u 18 ), e 15 = (u 3 , u 11 ), e 16 = (u 4 , u 12 ), e 17 = (u 5 , u 16 ), e 18 = (u 6 , u 9 ), e 19 = (u 6 , u 16 ), e 20 = (u 6 , u 17 ), e 21 = (u 7 , u 8 ), e 22 = (u 7 , u 9 ), e 23 = (u 7 , u 17 ), e 24 = (u 7 , u 18 ), e 25 = (u 8 , u 9 ), e 26 = (u 8 , u 13 ), e 27 = (u 8 , u 17 ), e 28 = (u 8 , u 18 ), e 29 = (u 9 , u 10 ), e 30 = (u 9 , u 17 ), e 31 = (u 9 , u 19 ), e 32 = (u 10 , u 13 ), e 33 = (u 11 , u 15 ), e 34 = (u 12 , u 15 ), e 35 = (u 13 , u 17 ), e 36 = (u 13 , u 18 ), e 37 = (u 13 , u 19 ), e 38 = (u 13 , u 20 ), e 39 = (u 16 , u 17 ), e 40 = (u 17 , u 19 ), e 41 = (u 19 , u 20 ), e 42 = (u 1 , u 21 ), e 43 = (u 7 , u 21 ), e 44 = (u 9 , u 21 ), e 45 = (u 17 , u 22 ), e 46 = (u 19 , u 21 ), e 47 = (u 21 , u 22 ), e 48 = (u 22 , u 23 ). None of the eight RNPW'S have Erdos number 1.…”

“…In social networks, One can consider the assumption that dominating sets with minimum order contain agents that strongly influence the rest of the members in the network. The deemed elites are those who influences stealthily on the ambitious network, are considered in the sociology literature [6]. In order to detect elites is through their high degree and through k-cores [6].…”

“…The deemed elites are those who influences stealthily on the ambitious network, are considered in the sociology literature [6]. In order to detect elites is through their high degree and through k-cores [6]. Another way is to find for them inside a minimum order dominating set.…”

On Graph Domination Numbers

“…The outer connected domination number of G, denoted γ ∼ c (G) is the cardinality of a minimum outer connected dominating set of G [1]. Note that every graph G has an outer connected dominating set.…”