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 rid and the king grid. We prove that seven out of eight of our constructions have optimal densities.

翻译：本文针对每个正整数$r$，引入了图论中两类新的覆盖码。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最优局部1-识别码的规模，得到了渐近紧致的上下界。这些界共同表明，将覆盖码转化为局部1-识别码的成本可忽略不计。对于部分较小的$n$，我们给出了最优构造。此外，上界是通过线性码构造获得的。同时，我们研究了无穷正方形网格、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。证明表明，我们的八种构造中有七种达到了最优密度。