Identifying codes in graphs of given maximum degree: Characterizing trees

TL;DR

This work proves the identifying code conjecture for all trees by precisely characterizing the extremal trees that force a positive constant $c$ in the bound $gamma^{ID}(G) ext{≤ }((Δ-1)/Δ)n + c$, showing that for $Δ≥3$ one may take $c=1/Δ$ and that only the $Δ$-star (and, for $Δ=3$, a finite exceptional set of twelve trees) require $c>0$. The authors introduce a tight upper bound relating the identification number to the domination number on trees, and develop a decomposition framework via appending stars to smaller identifiable graphs to preserve the bound. They classify appended-star configurations, proving that all such trees satisfy the bound with $c=0$ except the listed exceptional cases, and they provide explicit constructions demonstrating tightness and near-tightness of the bound in various regimes. The results lay groundwork for proving the conjecture for all triangle-free graphs in a companion paper, where the same exceptional trees determine the constant behavior. Overall, the paper advances the understanding of identifying codes in trees and offers precise extremal characterizations that underpin broader conjectures in locating structures within graphs.

Abstract

An identifying code of a closed-twin-free graph $G$ is a dominating set $S$ of vertices of $G$ such that any two vertices in $G$ have a distinct intersection between their closed neighborhoods and $S$. It was conjectured that there exists an absolute constant $c$ such that for every connected graph $G$ of order $n$ and maximum degree $Δ$, the graph $G$ admits an identifying code of size at most $( \frac{Δ-1}Δ )n +c$. We provide significant support for this conjecture by exactly characterizing every tree requiring a positive constant $c$ together with the exact value of the constant. Hence, proving the conjecture for trees. For $Δ=2$ (the graph is a path or a cycle), it is long known that $c=3/2$ suffices. For trees, for each $Δ\ge 3$, we show that $c=1/Δ\le 1/3$ suffices and that $c$ is required to have a positive value only for a finite number of trees. In particular, for $Δ= 3$, there are 12 trees with a positive constant $c$ and, for each $Δ\ge 4$, the only tree with positive constant $c$ is the $Δ$-star. Our proof is based on induction and utilizes recent results from [F. Foucaud, T. Lehtilä. Revisiting and improving upper bounds for identifying codes. SIAM Journal on Discrete Mathematics, 2022]. We remark that there are infinitely many trees for which the bound is tight when $Δ=3$; for every $Δ\ge 4$, we construct an infinite family of trees of order $n$ with identification number very close to the bound, namely $\left( \frac{Δ-1+\frac{1}{Δ-2}}{Δ+\frac{2}{Δ-2}} \right) n > (\frac{Δ-1}Δ ) n -\frac{n}{Δ^2}$. Furthermore, we also give a new tight upper bound for identification number on trees by showing that the sum of the domination and identification numbers of any tree $T$ is at most its number of vertices.

Figures (6)

• Figure 1: The family $\mathcal{T}_{3}$ of trees of maximum degree 3 requiring $c>0$ in Conjecture \ref{['conj_G Delta_UB']}. The set of black vertices in each figure constitutes an identifying code of the tree.
• Figure 2: Examples for the cases in the proof of Lemma \ref{['lem_(m,Delta)-broom']}. The black vertices in each graph $G$ constitute an identifying code.
• Figure 3: Figures illustrating the cases in Lemma \ref{['lem_T2 rhd S']}. The black vertices constitute an identifying code.
• Figure 4: Tree $T_7 \in \mathcal{T}_{3}$: ${\rm diam}(T_7) = 6$ and $\gamma^{{\rm ID}}(T_7) = 9$. The set of black vertices constitutes an identifying code of $T_7$.
• Figure 5: Graph $Z$: $\gamma^{{\rm ID}}(Z) = 8$. The set of black vertices constitutes an identifying code of $Z$.

Theorems & Definitions (48)

• Conjecture 1: foucaud2012size
• Theorem 2
• Remark 3
• Theorem 5: BCHL2004
• Lemma 6: FL22
• Lemma 7: FL22
• Theorem 8
• proof
• Lemma 9
• Lemma 10