On the Density of Self-identifying Codes in $K_m \times P_n$ and $K_m \times C_n$

TL;DR

The paper analyzes self-identifying codes in the direct product graphs $K_m \times P_n$ and $K_m \times C_n$, establishing tight lower and upper bounds on the minimum size $\gamma^{\mathrm{SID}}$ and proving that the asymptotic density $\gamma^{\mathrm{SID}}/|V(G)|$ tends to $\frac{1}{3}$ for both graph families. It combines structural necessary conditions, explicit column- and pattern-based constructions, and symmetry arguments to derive bounds that match in the asymptotic regime. Importantly, the results show that the stronger self-identifying constraint does not increase the asymptotic density relative to standard identifying codes in these products. The methods are applicable for $m \ge 3$ with appropriate ranges of $n$ and suggest exploring SID densities in other graph products.

Abstract

We study the asymptotic density of self-identifying codes in the direct product graphs $K_m \times P_n$ and $K_m \times C_n$ (the direct product of complete graphs with paths and cycles). A self-identifying code is a dominating set $S$ where each vertex $u$ (in $G$) is uniquely determined by the intersection $\bigcap_{c \in N[u] \cap S} N[c]$. Let $γ^{\rm SID}(G)$ denote its minimum size. For these product graphs, we establish new lower and upper bounds on $γ^{\rm SID}$. Crucially, from these bounds we prove that the asymptotic density $γ^{\rm SID}(G) / |V(G)|$ of a smallest self-identifying code converges to 1/3 for both families of graphs. This value matches the known asymptotic density of standard identifying codes in the same graphs, as established by Shinde and Waphare. Our result therefore shows that the stronger self-identifying constraint does not necessitate a higher density of vertices in this product setting.

Figures (9)

• Figure 1: Illustration of the direct product structure and a self-identifying code.
• Figure 2: Visualization of the constructed self-identifying code $S$ (black vertices) for $K_m \times P_n$ when $n$ is even.
• Figure 3: Visualization of the constructed self-identifying code $S$ (black vertices) for $K_m \times P_n$ when $n$ is odd.
• Figure 4: Schematic of the constructed self-identifying code $S$ (black vertices) for $K_m \times C_n$ when $k = \lfloor n/3 \rfloor$ is even.
• Figure 5: Schematic of the constructed self-identifying code $S$ (black vertices) for $K_m \times C_n$ when $k = \lfloor n/3 \rfloor$ is odd.

Theorems & Definitions (30)

• Theorem 1.1: Main Result
• Definition 2.1
• Definition 2.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.1
• Theorem 2.2
• Lemma 3.1