Vertex Vulnerability Parameter of Gear Graphs

Abstract

For a vertex v of a graph G = (V,E), the independent domination number (also called the lower independence number) iv(G) of G relative to v is the minimum cardinality of a maximal independent set in G that contains v. The average lower independence number of G is [Formula: see text]. In this paper, this parameter is defined and examined, also the average lower independence number of gear graphs is considered. Then, an algorithm for the average lower independence number of any graph is offered.

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

“…The toughness [13], the integrity [8], the domination number [16], the bondage number [5,6], the edge eccentric connectivity number [28], etc., have been proposed for measuring the vulnerability of networks. Recently, some average vulnerability parameters like the average lower independence number [4,17], the average lower domination number [2,7,11,17,23], the average connectivity number [10], the average lower connectivity number [1] and the average lower bondage number [24] have been defined.…”

“…In 2004, Henning introduced the concept of average domination and average independence in [17]. Moreover, the average lower domination and average lower independence number are the theoretical vulnerability parameters for a network that modeled a graph [4,7]. The average lower domination number of a graph G, denoted by…”

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

“…The toughness [6], the integrity [7], the domination number [8], the bondage number [9,10], the edge eccentric connectivity number [11], etc., have been proposed for measuring the vulnerability of networks. Recently, some average vulnerability parameters like the average lower independence number [12,13], the average lower domination number [13][14][15][16][17], the average connectivity number [18], the average lower connectivity number [19] and the average lower bondage number [4] have been defined.…”

“…In 2004, Henning introduced the concept of average domination and average independence in [13]. Moreover, the average lower domination and average lower independence number are the theoretical vulnerability parameters for a network that modeled a graph [12,15]. The average lower domination number of a graph G, denoted by γ av pGq, is defined as follows:…”

Average lower independence number in splitting graphs

“…Computer or communication networks are so designed that they do not easily get disrupted under external attack and, furthermore, are easily reconstructible if they do get disrupted. If we think of a graph as modeling a network, several vulnerability measures have been used to describe the stability of networks, including connectivity, toughness, scattering number, binding number and integrity [2,6,8]. Each of these parameters have been used to measure the vulnerability of networks in the case the communication vertices are damaged.…”