We introduce two new classes of covering codes in graphs for every positive integer $r$. These new codes are called local $r$-identifying and local $r$-locating-dominating codes and they are derived from $r$-identifying and $r$-locating-dominating codes, respectively. We study the sizes of optimal local 1-identifying codes in binary hypercubes. We obtain lower and upper bounds that are asymptotically tight. Together the bounds show that the cost of changing covering codes into local 1-identifying codes is negligible. For some small $n$ optimal constructions are obtained. Moreover, the upper bound is obtained by a linear code construction. Also, we study the densities of optimal local 1-identifying codes and local 1-locating-dominating codes in the infinite square grid, the hexagonal grid, the triangular grid, and the king grid. We prove that seven out of eight of our constructions have optimal densities.

翻译：我们针对每个正整数$r$，引入了图上的两类新覆盖码。这些新码分别称为局部$r$-识别码和局部$r$-定位控制码，它们分别源自$r$-识别码和$r$-定位控制码。我们研究了二进制超立方体中局部1-识别码的最优尺寸，得到了渐近紧致的上下界。这两个界共同表明：将覆盖码转换为局部1-识别码的成本可忽略不计。对于若干较小的$n$，我们给出了最优构造。此外，上界是通过一种线性码构造实现的。我们还研究了无限正方网格、六边形网格、三角形网格和国王网格中最优局部1-识别码与局部1-定位控制码的密度，并证明了八个构造中有七个具有最优密度。