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：我们为每个正整数$r$引入了图覆盖码的两个新类别。这些新码分别称为局部$r$识别码与局部$r$定位支配码，它们分别由$r$识别码与$r$定位支配码衍生而来。我们研究了二进制超立方中最优局部1-识别码的规模。我们获得了渐近紧致的上下界。这些界限共同表明，将覆盖码转换为局部1-识别码的代价是可忽略的。对于某些较小的$n$，我们得到了最优构造。此外，上界是通过线性码构造获得的。我们还研究了无限方格、六角格、三角格及王格中最优局部1-识别码与局部1-定位支配码的密度。我们证明了八种构造中有七种具有最优密度。