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An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. These codes have been widely studied for over two decades. We give an improvement over all the best known upper bounds, some of which have stood for over 20 years, for identifying codes in trees, proving the upper bound of $(n+\ell)/2$, where $n$ is the order and $\ell$ is the number of leaves (pendant vertices) of the graph. In addition to being an improvement in size, the new upper bound is also an improvement in generality, as it actually holds for bipartite graphs having no twins (pairs of vertices with the same closed or open neighbourhood) of degree 2 or greater. We also show that the bound is tight for an infinite class of graphs and that there are several structurally different families of trees attaining the bound. We then use our bound to derive a tight upper bound of $2n/3$ for twin-free bipartite graphs of order $n$, and characterize the extremal examples, as $2$-corona graphs of bipartite graphs. This is best possible, as there exist twin-free graphs, and trees with twins, that need $n-1$ vertices in any of their identifying codes. We also generalize the existing upper bound of $5n/7$ for graphs of order $n$ and girth at least 5 when there are no leaves, to the upper bound $\frac{5n+2\ell}{7}$ when leaves are allowed. This is tight for the $7$-cycle $C_7$ and for all stars., 13 pages, 3 figures

International audience; We introduce the Red-Blue Separation problem on graphs, where we are given a graph $G = (V, E)$ whose vertices are colored either red or blue, and we want to select a (small) subset $S\subseteq V$ , called red blue separating set , such that for every red-blue pair of vertices, there is a vertex $s\in S$ whose closed neighborhood contains exactly one of the two vertices of the pair.We study the computational complexity of Red-Blue Separation, in which one asks whether a given red-blue colored graph has a red-blue separating set of size at most a given integer. We prove that the problem is NP-complete even for restricted graph classes. We also show that it is always approximable in polynomial time within a factor of $2\ln n$, where $n$ is the input graph's order. In contrast, for triangle-free graphs and for graphs of bounded maximum degree, we show that Red-Blue Separation is solvable in polynomial time when the size of the smaller color class is bounded by a constant. However, on general graphs, we show that the problem is W[2] -hard even when parameterized by the solution size plus the size of the smaller color class.We also consider the problem Max Red-Blue Separation where the coloring is not part of the input. Here, given an input graph $G$, we want to determine the smallest integer $k$ such that, for every possible red-blue-coloring of $G$, there is a red-blue separating set of size at most $k$. We derive tight bounds on the cardinality of an optimal solution of Max Red-Blue Separation, showing that it can range from logarithmic in the graph order, up to the order minus one. We also give bounds with respect to related parameters. For trees however we prove an upper bound of two-thirds the order. We then show that Max Red-Blue Separation is NP-hard, even for graphs of bounded maximum degree, but can be approximated in polynomial time within a factor of $O(\ln^2 n)$.

In this article, we study locating-dominating codes in binary Hamming spaces $$\mathbb {F}^n$$ . Locating-dominating codes have been widely studied since their introduction in 1980s by Slater and Rall. They are dominating sets suitable for distinguishing vertices in graphs. Dominating sets as well as locating-dominating codes have been studied in Hamming spaces in multiple articles. Previously, Honkala et al. (Discret Math Theor Comput Sci 6(2):265, 2004) have presented a lower bound for locating-dominating codes in binary Hamming spaces. In this article, we improve the lower bound for all values $$n\ge 10$$ . In particular, when $$n=11$$ , we manage to improve the previous lower bound from 309 to 317. This value is very close to the current best known upper bound of 320.