[Paper Review] Dominating sets and Domination polynomials of Cycles

‎Saeid Alikhani, Yee-Hock Peng|ArXiv.org|May 20, 2009

Advanced Graph Theory Research5 citations

TL;DR

This paper develops a recursive formula for counting dominating sets of cycles, introducing the domination polynomial of cycles and deriving its structural properties. The key contribution is a recurrence relation that enables efficient computation of the number of dominating sets of each size, leading to closed-form expressions and combinatorial identities for the domination polynomial coefficients.

ABSTRACT

Let G=(V,E) be a simple graph. A set S\subset V is a dominating set of G, if every vertex in V§is adjacent to at least one vertex in S. Let {\mathcal C}_n^i be the family of dominating sets of a cycle C_n with cardinality i, and let d(C_n,i) = |{\mathcal C}_n^i. In this paper, we construct {\mathcal C}_n^i, and obtain a recursive formula for d(C_n, i). Using this recursive formula, we consider the polynomial D(C_n, x) = \sum_{i=1}^n d(C_n, i)x^i, which we call domination polynomial of cycles and obtain some properties of this polynomial.

Motivation & Objective

• To characterize and enumerate all dominating sets of a cycle graph $C_n$ by size.
• To define and study the domination polynomial $D(C_n, x) = \sum_{i=\lceil n/3 \rceil}^n d(C_n, i) x^i$, where $d(C_n, i)$ counts dominating sets of size $i$.
• To derive a recursive formula for $d(C_n, i)$ based on smaller cycles $C_{n-1}, C_{n-2}, C_{n-3}$.
• To establish combinatorial identities and recurrence relations for the coefficients of the domination polynomial.
• To analyze the behavior of $d(C_n, i)$ across different values of $n$ and $i$, including extremal cases and symmetry properties.

Proposed method

• Construct the family $\mathcal{C}_n^i$ of dominating sets of size $i$ in $C_n$ using recursive construction from $\mathcal{C}_{n-1}^{i-1}, \mathcal{C}_{n-2}^{i-1}, \mathcal{C}_{n-3}^{i-1}$.
• Establish a recurrence: $d(C_n, i) = d(C_{n-1}, i-1) + d(C_{n-2}, i-1) + d(C_{n-3}, i-1)$, valid under specific conditions on $n$ and $i$.
• Use lemmas to eliminate impossible configurations and reduce the number of cases to five valid recursive combinations.
• Prove that the domination polynomial $D(C_n, x)$ satisfies the recurrence $S_n = S_{n-1} + S_{n-2} + S_{n-3}$, where $S_n = \sum_{j=\lceil n/3 \rceil}^n d(C_n, j)$.
• Derive closed-form expressions and identities for $d(C_n, i)$ in special cases, such as $i = n$, $i = n-1$, $i = n-2$, and $i = \lceil n/3 \rceil$.
• Use induction and case analysis to validate identities for $d(C_n, i)$ across different $n$ and $i$, including the identity $\sum_{i=j}^{3j} d(C_i, j) = 3 \sum_{i=j-1}^{3j-3} d(C_i, j-1)$.

Experimental results

Research questions

• RQ1 How can the number of dominating sets of size $i$ in a cycle $C_n$ be recursively computed using smaller cycles?
• RQ2 What are the structural and combinatorial properties of the domination polynomial $D(C_n, x)$ for cycles?
• RQ3 What are the extremal values of $d(C_n, i)$, such as $d(C_n, n)$, $d(C_n, n-1)$, and $d(C_n, \lceil n/3 \rceil)$?
• RQ4 How does the sequence $d(C_n, i)$ behave as $n$ increases for fixed $i$, particularly in terms of unimodal growth?
• RQ5 What recurrence or identity governs the total number of dominating sets of all sizes in $C_n$?

Key findings

• The domination polynomial of a cycle satisfies $S_n = S_{n-1} + S_{n-2} + S_{n-3}$ with initial values $S_1 = 1$, $S_2 = 3$, $S_3 = 7$, where $S_n$ is the total number of dominating sets in $C_n$.
• $d(C_n, n) = 1$ and $d(C_n, n-1) = n$, corresponding to the unique full set and all sets missing one vertex.
• $d(C_n, n-2) = \frac{n(n-1)}{2}$, derived from the recurrence and verified via induction.
• $d(C_n, n-3) = \frac{(n-4)n(n+1)}{6}$, established through inductive proof using the recurrence.
• For $n = 3k$, $d(C_{3k}, k) = 3$, corresponding to the three perfect dominating sets of size $k$.
• The sequence $d(C_n, i)$ for fixed $i$ increases up to $n = 2i$, then decreases, with $d(C_{3i}, i) = 3$, showing a unimodal pattern.