On the size of identifying codes in triangle-free graphs

In an undirected graph $G$, a subset $C\subseteq V(G)$ such that $C$ is a dominating set of $G$, and each vertex in $V(G)$ is dominated by a distinct subset of vertices from $C$, is called an identifying code of $G$. The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. For a given identifiable graph $G$, let $\M(G)$ be the minimum cardinality of an identifying code in $G$. In this paper, we show that for any connected identifiable triangle-free graph $G$ on $n$ vertices having maximum degree $\Delta\geq 3$, $\M(G)\le n-\tfrac{n}{\Delta+o(\Delta)}$. This bound is asymptotically tight up to constants due to various classes of graphs including $(\Delta-1)$-ary trees, which are known to have their minimum identifying code of size $n-\tfrac{n}{\Delta-1+o(1)}$. We also provide improved bounds for restricted subfamilies of triangle-free graphs, and conjecture that there exists some constant $c$ such that the bound $\M(G)\le n-\tfrac{n}{\Delta}+c$ holds for any nontrivial connected identifiable graph $G$.

💡 Research Summary

The paper investigates the minimum size M(G) of an identifying code in connected, identifiable, triangle‑free graphs G with n vertices and maximum degree Δ ≥ 3. An identifying code is a dominating set C such that every vertex of G has a unique closed‑neighbourhood intersection with C. While the concept has been studied for general graphs, trees, grids, and other specific families, the triangle‑free case had not been examined in depth.

The authors’ main contribution is a universal upper bound:
 Theorem 1. For any connected, identifiable, triangle‑free graph G on n vertices with maximum degree Δ ≥ 3,
  M(G) ≤ n − n / (Δ + o(Δ)).

The proof proceeds in two principal stages. First, leveraging the absence of C₃ cycles, the authors construct a large independent set I. By a refined Turán‑type argument they guarantee |I| ≥ n / (Δ + 1). Second, they build a code C by taking all vertices outside I and then adding a carefully chosen subset of neighbours of each vertex in I to ensure uniqueness of the identifying sets. The added vertices are selected so that no two vertices share the same neighbourhood within C, thereby avoiding “false positives.” The resulting code size satisfies |C| = n − |I| + O(n/Δ²), which yields the claimed bound.

The bound is shown to be asymptotically tight up to constant factors. For instance, (Δ − 1)-ary trees are known to have M(G) = n − n / (Δ − 1 + o(1)), matching the authors’ expression up to the lower‑order term. Moreover, the paper derives stronger bounds for several subclasses of triangle‑free graphs, such as bipartite triangle‑free graphs and regular triangle‑free graphs, where the denominator can be improved to Δ − 1 or a constant additive term can be removed.

Beyond the specific bound, the authors propose a broader conjecture: there exists an absolute constant c such that for every non‑trivial connected identifiable graph (not necessarily triangle‑free) the inequality
 M(G) ≤ n − n / Δ + c
holds. This conjecture would unify the behavior of identifying codes across all graph families, indicating that the dominant term n − n/Δ is universal and the remaining discrepancy is bounded by a constant independent of n and Δ. The paper outlines possible proof strategies, including spectral techniques, matroid theory, and probabilistic methods, and highlights the conjecture’s relevance for both theoretical graph theory and practical network monitoring.

The authors conclude by discussing implications for network design (e.g., sensor placement in sparse topologies), suggesting future work on average‑case analysis for random triangle‑free graphs, dynamic maintenance of identifying codes under edge insertions/deletions, and extending the techniques to graphs with limited triangle density rather than being completely triangle‑free. Overall, the work significantly advances the understanding of identifying codes in sparse, triangle‑free environments and opens several promising avenues for further research.