Maximum Number of Minimum Dominating and Minimum Total Dominating Sets
Anant Godbole, Jessie Jamieson, and William Jamieson
TL;DR
This paper investigates the maximum number of minimum dominating and total dominating sets in connected graphs with a given domination number, providing exact results for the case when the domination number is 2 and bounds for larger values.
Contribution
It offers an exact characterization for the maximum counts when the domination number is 2 and establishes lower bounds for higher domination numbers.
Findings
Exact maximum number for =2
Lower bounds for
Provides insights into the structure of graphs with many minimum dominating sets.
Abstract
Given a connected graph with domination (or total domination) number \gamma>=2, we ask for the maximum number m_\gamma and m_{\gamma,T} of dominating and total dominating sets of size \gamma. An exact answer is provided for \gamma=2and lower bounds are given for \gamma>=3.
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems · Complexity and Algorithms in Graphs