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, the graph G admits an identifying code of size at most (-1) 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 =2 (the graph is a path or a cycle), it is long known that c=3/2 suffices. For trees, for each 3, we show that c=1/ 1/3 suffices and that c is required to have a positive value only for a finite number of trees. In particular, for = 3, there are 12 trees with a positive constant c and, for each 4, the only tree with positive constant c is the -star. Our proof is based on induction and utilizes recent results from F. Foucaud, T. Lehtil\"a. 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 =3; for every 4, we construct an infinite family of trees of order n with identification number very close to the bound, namely (-1+1{-2}+2{-2}) n > (-1) n -n². 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.