On Coalition Graphs and Coalition Count of Graphs
Swathi Shetty, Sayinath Udupa N. V., B. R. Rakshith

TL;DR
This paper explores the concept of coalition graphs and counts in graphs, constructing smaller graphs with specific coalition properties and characterizing graphs with a coalition count of one.
Contribution
It constructs smaller graphs with given coalition graphs and characterizes all graphs with a coalition count of one, advancing understanding of coalition structures in graphs.
Findings
Constructed smaller graphs with specified coalition graphs.
Characterized all graphs with coalition count equal to one.
Studied properties of coalition count under certain conditions.
Abstract
Let be graph with vertex set and order . A coalition in a graph consists of two disjoint sets of vertices and , neither of which is a dominating set but whose union is a dominating set. A coalition partition, abbreviated -partition, in a graph is a vertex partition such that every set of is either a singleton dominating set, or is not a dominating set but forms a coalition with another set in . The sets and are coalition partners in . The coalition number equals the maximum order of a -partition of . For any graph with a -partition , the coalition graph of is a graph with vertex set , corresponding one-to-one with the set , and two vertices and …
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Taxonomy
TopicsAdvanced Graph Theory Research · Game Theory and Voting Systems · Complexity and Algorithms in Graphs
