[Paper Review] The Domination Polynomials of Cubic graphs of order 10

Saieed Akbari, ‎Saeid Alikhani+1 more|ArXiv.org|May 20, 2009

Advanced Graph Theory Research3 citations

TL;DR

This paper investigates the domination polynomials of all 3-regular (cubic) graphs of order 10, computing their dominating sets and analyzing D-polynomial equivalence. Using vertex-transitivity and combinatorial counting, it proves the Petersen graph is uniquely determined by its domination polynomial, establishing it as D-unique, while identifying several other D-unique graphs and D-equivalence classes among the 19 cubic graphs of order 10.

ABSTRACT

Let G be a simple graph of order n. The domination polynomial of G is the polynomial D(G,x)=\sum_{i=γ(G)}^{n} d(G,i) x^{i}, where d(G,i) is the number of dominating sets of G of size i, and γ(G) is the domination number of G. In this paper we study the domination polynomials of cubic graphs of order 10. As a consequence, we show that the Petersen graph is determined uniquely by its domination polynomial.

Motivation & Objective

• To compute and analyze the domination polynomials of all cubic graphs of order 10.
• To determine which of these graphs are D-unique, i.e., uniquely identified by their domination polynomial.
• To classify the D-equivalence classes among cubic graphs of order 10, identifying graphs with identical domination polynomials.
• To investigate the structural and combinatorial properties of dominating sets in cubic graphs, especially in the Petersen graph.
• To establish that the Petersen graph is D-unique, meaning no other graph shares its domination polynomial.

Proposed method

• Computing the domination polynomial $ D(G,x) = \sum_{i=\gamma(G)}^{n} d(G,i)x^i $, where $ d(G,i) $ counts dominating sets of size $ i $.
• Applying Lemma 1: for vertex-transitive graphs, $ d(G,i) = \frac{n}{i} d_v(G,i) $, enabling efficient computation of $ d(G,i) $ via vertex-specific counts.
• Using Theorem 2 to infer minimum degree constraints from the point where $ d(G,j) = \binom{n}{j} $, indicating all $ j $-subsets are dominating sets.
• Enumerating $ \gamma $-sets (minimum dominating sets) for each cubic graph of order 10 to compare structural properties.
• Applying Theorem 6 on disconnected graphs: $ D(G,x) = D(G_1,x) \cdots D(G_m,x) $, to compute polynomials for graphs with multiple components.
• Comparing $ d(G,i) $ values across graphs to identify D-equivalence classes and D-uniqueness.

Experimental results

Research questions

• RQ1 Is the Petersen graph uniquely determined by its domination polynomial among all cubic graphs of order 10?
• RQ2 Which cubic graphs of order 10 are D-unique, i.e., have no non-isomorphic graph sharing their domination polynomial?
• RQ3 What are the D-equivalence classes among the 19 cubic graphs of order 10, and which graphs are equivalent under the D-equivalence relation?
• RQ4 How do the counts of dominating sets of various sizes differ across cubic graphs of order 10, and what structural features influence these counts?
• RQ5 Can the domination polynomial distinguish between non-isomorphic cubic graphs of order 10, particularly in the case of the Petersen graph?

Key findings

• The Petersen graph is D-unique: no other graph has the same domination polynomial, as shown by the fact that $ d(G_9,4) = 91 > d(P,4) $, proving $ [P] = \{P\} $.
• Graph $ G_9 $ is also D-unique, as it has a distinct domination polynomial with $ d(G_9,4) = 91 $, exceeding that of the Petersen graph.
• The graphs $ G_6 $ and $ G_{10} $ form a D-equivalence class with the same domination polynomial: $ x^{10} + \binom{10}{9}x^9 + \binom{10}{8}x^8 + \binom{10}{7}x^7 + (\binom{10}{6}-10)x^6 + (\binom{10}{5}-60)x^5 + 85x^4 + 10x^3 $.
• The graphs $ G_7 $ and $ G_8 $ are D-equivalent, both having $ d(G_i,4) = 80 $, computed via Lemma 1 and vertex-specific dominating set enumeration.
• The graphs $ G_{12}, G_{13}, G_{14}, G_{16}, G_{19} $ are D-unique, as their $ \gamma $-set counts ($ |\Gamma(G)| $) differ from all others: 15, 7, 22, 13, and 13 respectively.
• The graphs $ G_{20} $ and $ G_{21} $ are D-equivalent, both having the domination polynomial $ x^{10} + 10x^9 + 45x^8 + 120x^7 + 203x^6 + 216x^5 + 134x^4 + 36x^3 $, derived from the product of the polynomials of their components.