The Number of Maximal Independent Sets in Quasi-Tree Graphs and Quasi-Forest Graphs

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.


Introduction and Preliminary
Let ( ) be a simple undirected graph. An independent set is a subset S of V such that no two vertices in S are adjacent. A maximal independent set is an independent set that is not a proper subset of any other independent set. The set of all maximal independent sets of a graph G is denoted by ( ) MI G and its cardinality by ( ) mi G . The problem of determining the largest value of ( ) mi G in a general graph of order n and those graphs achieving the largest number was proposed by Erdös and Moser, and solved by Moon and Moser [1]. It was then studied for various families of graphs, including trees, forests, (connected) graphs with at most one cycle, (connected) triangle-free graphs, (k-)connected graphs, bipartite graphs; for a survey see [2]. Jin and Li [3] investigated the second largest number of ( ) mi G among all trees of order n. 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 ( ) such that G x − is a tree (respectively, forest). The concept of quasi-tree graphs was mentioned by Liu and Lu in [6]. Recently, the problem of determining the largest and the second largest numbers of ( ) mi G among all quasi-tree graphs and quasi-forest graphs of order n was solved by Lin [7] [8].
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.
, the deletion of A from G is the graph G A − obtained from G by removing all vertices in A and their incident edges. Two nG is the short notation for the union of n copies of disjoint graphs isomorphic to G. Denote by n C a cycle with n vertices and n P a path with n vertices.
Throughout this paper, for simplicity, let 2 r = .

Survey on the Large Numbers of Maximal Independent Sets
In this section, we survey on the large numbers of maximal independent sets among all trees, forests, quasi-trees and quasi-forests. The results of the largest numbers of maximal independent sets among all trees and forests are described in B i j is the set of batons, which are the graphs obtained from the basic path P of 1 i ≥ vertices by attaching 0 j ≥ paths of length two to the endpoints of P in all possible ways (see Figure 1).
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.
, if is even, 2 9 9 , if is odd. 2 Q n is shown in Figure 6.
, if is even, The results of the second largest numbers of maximal independent sets among all quasi-tree graphs and quasi-forest graphs are described in Theorems 2.9 and 2.10, respectively.
Q n is shown in Figure 7 and Figure 8.
, if is even, 5 , if is odd.
where W is a bow, that is, two triangles 3 C having one common vertex.
A graph is said to be unicyclic if it contains exactly one cycle. The result of the second largest number of maximal independent sets among all connected unicyclic graphs are described in Theorems 2.11.   Figure 9.

Main Results
In this section, we determine the third largest values of ( ) mi G among all quasi-tree graphs and quasi-forest graphs of order 7 n ≥ , respectively. Moreover, the extremal graphs achieving these values are also determined.
which is a contradiction. We obtain that Q is a connected unicyclic graph, thus the result follows from Theorem 2.11.
Theorem 3.2 If Q is a quasi-tree graph of even order ( ) This is a contradiction. So we assume that ( ) • deg 2 x = . There are 6 possibilities for graph Q. See Figure 11. Note that ( )   Hence we obtain that This is a contradiction. So we assume that ( ) The equalities holding imply that 10 n = , that is, Hence we obtain that ( ) − . There are 7 possibilities for graph Q. See Figure 13.
Note that ( ) In the following, we will investigate the same problem for quasi-forest graphs.  Figure 15.       which is a contradiction. Hence we obtain that s is even and