Abstract
A maximal independent set is an independent set that is not a proper subset of any other independent set. A connected graph (respectively, graph) G with vertex set V(G) is called a quasi-tree graph (respectively, quasi-forest graph), if there exists a vertex x ∈V(G) such that G − x is a tree (respectively, forest). In this paper, we survey on the large numbers of maximal independent sets among all trees, forests, quasi-trees and quasi-forests. In addition, we further look into the problem of determining the third largest number of maximal independent sets among all quasi-trees and quasi-forests. Extremal graphs achieving these values are also given.
Highlights
Introduction and PreliminaryLet G = (V, E ) be a simple undirected graph
We survey on the large numbers of maximal independent sets among all trees, forests, quasi-trees and quasi-forests
We further look into the problem of determining the third largest number of maximal independent sets among all quasi-trees and quasi-forests
Summary
The results of the second largest numbers of maximal independent sets among all trees and forests are described in Theorems 2.3 and 2.4, respectively. The results of the third largest numbers of maximal independent sets among all trees and forests are described in Theorems 2.5 and 2.6, respectively. The results of the largest numbers of maximal independent sets among all quasi-tree graphs and quasi-forest graphs are described in Theorems 2.7 and 2.8, respectively.
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