We introduce a new measure of network reliability related to the order of the largest component. This new connectivity measure considers a network to be operational if there is a component or order at least some fixed proportion, r, of the original order. Thus, the network is in a failure state if all components are sufficiently small. In this paper, we consider the parameters with vertex deletions as well as edge deletions for particular graph classes. We also find the minimum values of the parameter for graphs with a fixed size and order. We end with a discussion and some conjectures for the maximum value of the parameter for graphs with a fixed size and order.
Hanson, Nicholas; Shank, Nathan; Armbruster, Ashley; and Di, Jiequi
"Proportional Component Order Network Connectivity,"
Communications on Number Theory and Combinatorial Theory: Vol. 3, Article 4.
Available at: https://research.library.kutztown.edu/contact/vol3/iss1/4