The number of maximal independent sets in a connected graph

@article{Griggs1988TheNO,
  title={The number of maximal independent sets in a connected graph},
  author={J. R. Griggs and Charles M. Grinstead and D. Guichard},
  journal={Discret. Math.},
  year={1988},
  volume={68},
  pages={211-220}
}
Abstract We determine the maximum number of maximal independent sets which a connected graph on n vertices can have, and we completely characterize the extremal graphs, thereby answering a question of Wilf. 
The maximum number of maximal independent sets in unicyclic connected graphs
TLDR
The maximum number of maximal independent sets in a unicyclic connected graph is determined and a class of graphs achieving this maximum value is found. Expand
Maximal independent sets in graphs with at most r cycles
We find the maximum number of maximal independent sets in two families of graphs: all graphs with $n$ vertices and at most $r$ cycles, and all such graphs that are also connected. In addition, weExpand
The Maximum Number of Maximal Independent Sets in Forests 1
Let G be a simple and undirected graph. By mi(G) we denote the number of maximal independent sets in G. In this note, we determine the maximum number of maximal independent sets among the set of aExpand
Maximal Independent Sets in Graphs with at Most One Cycle
In this paper, we determine the largest number of maximal independent sets among all connected graphs of order n, which contain at most one cycle. We also characterize those extremal graphs achievingExpand
THE NUMBER OF MAXIMUM INDEPENDENT SETS IN GRAPHS
In this paper, we study the problem of determining the largest number of maximum independent sets of a graph of order n. Solutions to this problem are given for various classes of graphs, includingExpand
Coverings, matchings and the number of maximal independent sets of graphs
TLDR
The maximum number of maximal independent sets of arbitrary graphs in terms of their covering numbers is determined and the extremal graphs are completely characterize. Expand
Maximal independent sets in graphs with at most r cycles
We find the maximum number of maximal independent sets in two families of graphs. The first family consists of all graphs with n vertices and at most r cycles. The second family is all graphs of theExpand
Maximal independent sets in bipartite graphs
  • Jiuqiang Liu
  • Mathematics, Computer Science
  • J. Graph Theory
  • 1993
TLDR
The maximum number of maximal independent sets among all bipartite graphs of order n and the extremal graphs as well as the corresponding results for connected bipartites graphs are determined. Expand
Trees without twin-leaves with smallest number of maximal independent sets
Abstract For any n, in the set of n-vertex trees such that any two leaves have no common adjacent vertex, we describe the trees with the smallest number of maximal independent sets.
Integers for the Number of Maximal Independent Sets in Graphs
Let G be a simple undirected graph. Denote by mi(G) (respectively, xi(G)) the number of maximal (respectively, maximum) independent sets in G. In this paper we determine the third and fouth largestExpand
...
1
2
3
4
5
...

References

SHOWING 1-4 OF 4 REFERENCES
The number of maximal independent sets in connected graphs
  • Z. Füredi
  • Mathematics, Computer Science
  • J. Graph Theory
  • 1987
TLDR
A theorem of Moon and Moser is generalized to determine the maximum number of maximal independent sets in a connected graph on n vertices for n sufficiently large, e.g., n > 50. Expand
The number of maximal independent sets in a tree
We find the largest number of maximal independent sets of vertices that any tree of n vertices can have.
On cliques in graphs
A clique is a maximal complete subgraph of a graph. The maximum number of cliques possible in a graph withn nodes is determined. Also, bounds are obtained for the number of different sizes of cliquesExpand
Graph theory