Secure coalitions in graphs
Swathi Shetty, Sayinath Udupa N. V., B. R. Rakshith

TL;DR
This paper introduces the concept of secure coalitions in graphs, explores their properties, and characterizes graphs based on their secure coalition number, including proofs that all graphs admit such partitions.
Contribution
It defines secure coalition partitions, introduces secure coalition graphs, and characterizes graphs with specific secure coalition numbers, including all trees and graphs without isolated vertices.
Findings
Every graph admits a secure coalition partition.
Characterization of graphs with SEC(G) in {1, 2, n}.
All trees have SEC(T) = n-1.
Abstract
A secure coalition in a graph consists of two disjoint vertex sets and , neither of which is a secure dominating set, but whose union forms a secure dominating set. A secure coalition partition (-partition) of is a vertex partition where each set is either a secure dominating set consisting of a single vertex of degree , or a set that is not a secure dominating set but forms a secure coalition with some other set . The maximum cardinality of a secure coalition partition of is called the secure coalition number of , denoted . For every -partition of a graph , we associate a graph called the secure coalition graph of with respect to , denoted , where the vertices of correspond to the sets of , and two…
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Taxonomy
TopicsAdvanced Graph Theory Research · Game Theory and Voting Systems · Security in Wireless Sensor Networks
