The problem of determining the maximum number of maximal independent sets in certain graph classes dates back to a paper of Miller and Muller and a question of Erdős and Moser from the 1960s. The minimum was always considered to be less interesting due to simple examples such as stars. In this paper we show that the problem becomes interesting when restricted to twin-free graphs, where no two vertices have the same open neighbourhood. We consider the question for arbitrary graphs, bipartite graphs and trees. The minimum number of maximal independent sets turns out to be logarithmic in the number of vertices for arbitrary graphs, linear for bipartite graphs and exponential for trees. In the latter case, the minimum and the extremal graphs have been determined earlier by Taletskii and Malyshev, but we present a shorter proof.
@article{10_37236_12789,
author = {Stijn Cambie and Stephan Wagner},
title = {The minimum number of maximal independent sets in twin-free graphs},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {4},
doi = {10.37236/12789},
zbl = {1557.05070},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12789/}
}
TY - JOUR
AU - Stijn Cambie
AU - Stephan Wagner
TI - The minimum number of maximal independent sets in twin-free graphs
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/12789/
DO - 10.37236/12789
ID - 10_37236_12789
ER -
%0 Journal Article
%A Stijn Cambie
%A Stephan Wagner
%T The minimum number of maximal independent sets in twin-free graphs
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/12789/
%R 10.37236/12789
%F 10_37236_12789
Stijn Cambie; Stephan Wagner. The minimum number of maximal independent sets in twin-free graphs. The electronic journal of combinatorics, Tome 31 (2024) no. 4. doi: 10.37236/12789
