On the domination number of permutation graphs and an application to strong fixed points

A permutation graphGπ is a simple graph with vertices corresponding to the elements of π and an edge between i and j when i and j are inverted in π. A set of vertices D is said to dominate a graph G when every vertex in G is either an element of D, or adjacent to an element of D. The domination numb...

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Bibliographic Details
Published in:Discrete Applied Mathematics 2021-01, Vol.288, p.20-34
Main Authors: Baren, Theresa, Cory, Michael, Friedberg, Mia, Gardner, Peter, Hammer, James, Harrington, Joshua, McGinnis, Daniel, Waechter, Riley, Wong, Tony W.H.
Format: Article
Language:English
Subjects:
Apexes
Graph domination
Graph theory
Graphs
Permutation graphs
Permutations
Strong fixed points
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Summary:A permutation graphGπ is a simple graph with vertices corresponding to the elements of π and an edge between i and j when i and j are inverted in π. A set of vertices D is said to dominate a graph G when every vertex in G is either an element of D, or adjacent to an element of D. The domination numberγ(G) is defined as the cardinality of a minimum dominating set of G. A strong fixed point of a permutation π of order n is an element k such that π−1(j)k. In this article, we count the number of connected permutation graphs on n vertices with domination number 1 and domination number n2. We further show that for a positive integer k≤n2, there exists a connected permutation graph on n vertices with domination number k. We find a closed expression for the number of permutation graphs dominated by a set with two elements, and we find a closed expression for the number of permutation graphs efficiently dominated by any set of vertices. We conclude by providing an application of these results to strong fixed points, proving some conjectures posed on the OEIS.
ISSN:0166-218X
1872-6771
DOI:10.1016/j.dam.2020.08.021