The minimum number of maximal independent sets in graphs with given order and independence number

Abstract

Let $MIS(G)$ be the set of all maximal independent sets in a graph $G$, and let $mis(G)=|MIS(G)|$. In this paper, we show that for any tree $T$ with $n$ vertices and independence number $α$, \[mis(T)\geq f(n-α),\] and for any unicyclic graph $G$ with $n$ vertices and independence number $α$, \begin{align*} mis(G)\geq \begin{cases} 2, & \text{if} \ n=4\ \text{and}\ α=2, 3, & \text{if} \; α=n-2 \; \text{and} \; n\neq4, 2f(n-α), & \text{if} \; n\geq 5\; \text{and}\; \lceil \frac{n}{2} \rceil \leq α< n-2, f(n-α+2)-f(n-α-3), &\text{if} \; n\geq 5, \;\text{and}\ n \; \text{is odd}, \; \text{and} \; α= \lfloor \frac{n}{2} \rfloor, \end{cases} \end{align*} where $f(n)$ represent the $n$th Fibonacci number. Moreover, we also show that the above inequalities are sharp.

Figures (9)

• Figure 1: The structure of $T$.
• Figure 2: A tree $T$ of order $n$ and independence number $\alpha$ satisfying $mis(T)=g(n-\alpha)$.
• Figure 3: Subcase 1.2. $d=1$
• Figure 4: Subcase 1.3. $d=2$
• Figure 5: Subcase 1.4. $d \geq 3$

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Corollary 1
• Lemma 5
• Lemma 6
• Lemma 7