[Paper Review] Computing the alliance polynomial of a graph

Walter Carballosa, José M. Rodrı́guez+2 more|arXiv (Cornell University)|Oct 11, 2014

Graph theory and applications15 references3 citations

TL;DR

This paper introduces the alliance polynomial of a graph as a novel graph invariant that encodes the number of connected exact defensive $k$-alliances for all $k$. It presents an algorithm to compute the polynomial and proves that path, cycle, complete, and star graphs are uniquely characterized by their alliance polynomials—offering a stronger distinguishing power than traditional graph polynomials like chromatic or matching polynomials.

ABSTRACT

The alliance polynomial of a graph $Γ$ with order $n$ and maximum degree $δ_1$ is the polynomial $A(Γ; x) = \sum_{k=-δ_1}^{δ_1} A_{k}(Γ) \, x^{n+k}$, where $A_{k}(Γ)$ is the number of exact defensive $k$-alliances in $Γ$. We provide an algorithm for computing the alliance polynomial. Furthermore, we obtain some properties of $A(Γ; x)$ and its coefficients. In particular, we prove that the path, cycle, complete and star graphs are characterized by their alliance polynomials. We also show that the alliance polynomial characterizes many graphs that are not distinguished by other usual polynomials of graphs.

Motivation & Objective

• To define and formalize the alliance polynomial as a new graph invariant based on defensive $k$-alliances.
• To develop an algorithm for computing the alliance polynomial of a graph.
• To investigate whether the alliance polynomial can uniquely characterize graphs that other polynomials fail to distinguish.
• To explore the structural and algebraic properties of the alliance polynomial, including unimodality of coefficients.
• To compare the distinguishing power of the alliance polynomial with established graph polynomials such as chromatic, matching, and Tutte polynomials.

Proposed method

• Define a defensive $k$-alliance as a nonempty, connected subset $S \subseteq V$ where each vertex $v \in S$ satisfies $\delta_S(v) \geq \delta_{\bar{S}}(v) + k$.
• Introduce the exact index of alliance $k_S = \min_{v \in S} \{ \delta_S(v) - \delta_{\bar{S}}(v) \}$ to classify $S$ as an exact defensive $k_S$-alliance.
• Construct the alliance polynomial as $A(\Gamma; x) = \sum_{k \in \mathcal{K}} A_k(\Gamma) \, x^{n+k}$, where $A_k(\Gamma)$ counts connected exact defensive $k$-alliances.
• Implement an algorithm to compute $A(\Gamma; x)$ by enumerating all connected subsets and determining their exact $k$-indices.
• Prove that the alliance polynomial is symmetric for certain graph families, such as $P_4$ and $P_2 \boxtimes P_3$, using structural properties.
• Use known results on graph polynomials (e.g., from [6], [26], [27]) to compare distinguishing power and identify non-isomorphic graphs with identical classical polynomials but distinct alliance polynomials.

Experimental results

Research questions

• RQ1 Can the alliance polynomial uniquely characterize fundamental graph families such as paths, cycles, complete graphs, and stars?
• RQ2 How does the alliance polynomial compare in distinguishing power to classical graph polynomials like the chromatic, matching, and Tutte polynomials?
• RQ3 Are there graphs that are indistinguishable under standard graph polynomials but uniquely identified by their alliance polynomial?
• RQ4 What structural properties, such as unimodality, do the coefficients of the alliance polynomial exhibit?
• RQ5 Is there a finite set of graphs for which the alliance polynomial fails to distinguish non-isomorphic graphs, or is it potentially a universal distinguishing invariant?

Key findings

• The alliance polynomial uniquely characterizes path, cycle, complete, and star graphs, meaning no non-isomorphic graph shares the same polynomial.
• The alliance polynomial distinguishes graphs that share the same chromatic, matching, independence, or Tutte polynomials, as demonstrated by counterexamples in Figure 2 and Figure 3.
• For the complete bipartite graph $K_{3,3}$, the alliance polynomial is computed explicitly, showing $A_k(K_{3,3}) = 0$ for $k \in \{-2, 0, 2\}$ due to its cubic structure.
• The alliance polynomial of $P_4$ and $K_{1,3}$ differs despite having identical Tutte polynomials, confirming the polynomial's superior distinguishing power.
• The alliance polynomial is symmetric for some graphs (e.g., $P_4$), while others (e.g., $P_2 \boxtimes P_3$) are not, indicating structural sensitivity in the polynomial's form.
• The paper provides evidence that the alliance polynomial may distinguish all graphs, though a general proof remains open, suggesting it could be a universal graph invariant.