Identifying codes in graphs of given maximum degree: Characterizing trees

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 $\left(\frac{\Delta -1}{\Delta }\right)n+c$. 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 $\Delta =2$ (the graph is a path or a cycle), it is long known that $c=3/2$ suffices. For trees, for each $\Delta \ge 3$, we show that $c=1/\Delta \le 1/3$ suffices and that c is required to have a positive value only for a finite number of trees. In particular, for $\Delta =3$, there are 12 trees with a positive constant c and, for each $\Delta \ge 4$, the only tree with positive constant c is the Δ-star. Our proof is based on induction and utilizes recent results from Foucaud and Lehtilä (2022) [17]. We remark that there are infinitely many trees for which the bound is tight when $\Delta =3$; for every $\Delta \ge 4$, we construct an infinite family of trees of order n with identification number very close to the bound, namely $\left(\frac{\Delta -1+\frac{1}{\Delta -2}}{\Delta +\frac{2}{\Delta -2}}\right)n>\left(\frac{\Delta -1}{\Delta }\right)n-\frac{n}{{\Delta }^2}$. 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.