On the coalition number of trees
Davood Bakhshesh, Michael A. Henning, Dinabandhu Pradhan

TL;DR
This paper investigates the coalition number in graphs, characterizes specific graphs and trees based on their coalition partitions, and explores the structure of coalition graphs, solving several open problems in the field.
Contribution
It characterizes graphs with maximum coalition number and trees with near-maximum coalition number, and analyzes the structure of coalition graphs for paths, addressing open problems.
Findings
Graphs with minimum degree ≤ 1 and maximum coalition number equal to the number of vertices are characterized.
All trees with coalition number one less than the number of vertices are characterized.
The number of coalition graphs for a path is determined, and it is shown that no universal coalition path exists.
Abstract
Let be a graph with vertex set and of order , and let and be the minimum and maximum degree of , respectively. Two disjoint sets form a coalition in if none of them is a dominating set of but their union is. A vertex partition of is a coalition partition of if every set is either a dominating set of with the cardinality , or is not a dominating set but for some , and form a coalition. The maximum cardinality of a coalition partition of is the coalition number of . Given a coalition partition of , a coalition graph is associated on such that there is a one-to-one correspondence between its vertices and the members of , where two…
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Taxonomy
TopicsGame Theory and Voting Systems · Political Influence and Corporate Strategies
