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 domination number is an important subject that it has become one of the most widely studied topics in graph theory, and also is the most often studied property of vulnerability of communication networks. The vulnerability value of a communication network shows the resistance of the network after the disruption of some centers or connection lines until a communication breakdown. Let G = (V (G), E(G)) be a simple graph. The bondage number b(G) of a nonempty graph G is the smallest number of edges whose removal from G result in a graph with domination number greater than that of G. If we think a graph as a modeling of network, the average lower bondage number of a graph is a new measure of the graph vulnerability and it is defined by bav, where the lower bondage number, denoted by be(G), of the graph G relative to e is the minimum cardinality of bondage set in G that contains the edge e. In this paper, the above mentioned new parameter has been defined and examined. Then upper bounds, lower bounds and exact formulas have been obtained for any graph G. Finally, the exact values have been determined for some well-known graph families.
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.
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