Revisiting and Improving Upper Bounds for Identifying Codes

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