The Average Lower 2-Domination Number of Wheels Related Graphs and an Algorithm

Abstract

The problem of quantifying the vulnerability of graphs has received much attention nowadays, especially in the field of computer or communication networks. In a communication network, the vulnerability measures the resistance of the network to disruption of operation after the failure of certain stations or communication links. If we think of a graph as modeling a network, the average lower 2-domination number of a graph is a measure of the graph vulnerability and it is defined by γ2av(G)=1|V(G)|∑v∈V(G)γ2v(G), where the lower 2-domination number, denoted by γ2v(G), of the graph G relative to v is the minimum cardinality of 2-domination set in G that contains the vertex v. In this paper, the average lower 2-domination number of wheels and some related networks namely gear graph, friendship graph, helm graph and sun flower graph are calculated. Then, we offer an algorithm for computing the 2-domination number and the average lower 2-domination number of any graph G.

On Arithmetic-Geometric Index (Ga) and Edge Ga Index

“…Definition 3.1. [6,7,13] The wheel W n with n (n ≥ 3) spokes is a graph that contains an n-cycle and one additional central vertex v c that is adjacent to all vertices of the cycle. The central vertex v c of the wheel W n has a vertex degree of n. Wheel graph has (n + 1) vertices and 2n edges.…”

“…By summing up the Cases 1 and 2, it is clear that GA( [6,7,13] The gear graph is a wheel graph with a vertex added between each pair adjacent graph vertices of the outer cycle. Gear graph G n includes an even cycle C 2n .…”

The Average Lower 2-Domination Number of Wheels Related Graphs and an Algorithm

“…In 2015, a new graph theoretical parameter namely the average lower 2-domination number was defined in [23,24]. The average lower 2-domination number of a graph G, denoted by γ 2av pGq, is defined as follows:…”

“…where the lower 2-domination number, denoted by γ 2v pGq, is the minimum cardinality of a dominating set of the graph G that contains the vertex v [23,24]. If we think of a graph as modeling a network, then the average lower 2-domination number can be more sensitive for the vulnerability of graphs than the other known vulnerability measures of a graph [23]. We consider two connected simple graphs G and H in Figure 1, where |VpGq| " |VpHq| " 10 and |EpGq| " |EpHq| " 17.…”

“…When we compute γ 2av pHq and γ 2av pGq, we get γ 2av pHq " 51{10 " 5.1 and γ 2av pGq " 50{10 " 5. So, the average lower 2-domination number may be used for distinguish between these two graphs G and H. Since γ 2av pGq ă γ 2av pHq, we can say that the graph H is more vulnerable than the graph G. In other words, the graph G is tougher than the graph H [23,24]. The wheel graph has been used in different areas such as the wireless sensor networks, the vulnerability of networks, and so on.…”