# American Institute of Mathematical Sciences

February  2012, 6(1): 27-38. doi: 10.3934/amc.2012.6.27

## On $(r,\leq 2)$-locating-dominating codes in the infinite king grid

 1 Turku Centre for Computer Science TUCS, Department of Mathematics, University of Turku, FI-20014 Turku, Finland

Received  August 2010 Revised  November 2011 Published  January 2012

Assume that $G=(V,E)$ is an undirected graph with vertex set $V$ and edge set $E$. The ball $B_r(v)$ denotes the vertices within graphical distance $r$ from $v$. Let $I_r(F)=\bigcup$$v\in F$$(B_r(v) \cap C)$ be a set of codewords in the neighbourhoods of vertices $v\in F$. A subset $C\subseteq V$ is called an $(r,\leq l)$-locating-dominating code of type A if sets $I_r(F_1)$ and $I_r(F_2)$ are distinct for all subsets $F_1,F_2\subseteq V$ where $F_1\ne F_2$, $F_1\cap C= F_2 \cap C$ and $|F_1|,|F_2| \leq l$. A subset $C\subseteq V$ is an $(r,\leq l)$-locating-dominating code of type B if the sets $I_r(F)$ are distinct for all subsets $F\subseteq V\setminus C$ with at most $l$ vertices. We study $(r,\leq l)$-locating-dominating codes in the infinite king grid when $r\geq 1$ and $l=2$. The infinite king grid is the graph with vertex set $\mathbb Z^2$ and edge set $\{\{(x_1,y_1),(x_2,y_2)\}||x_1-x_2|\leq 1, |y_1-y_2|\leq 1, (x_1,y_1)\ne(x_2,y_2)\}.$
Citation: Mikko Pelto. On $(r,\leq 2)$-locating-dominating codes in the infinite king grid. Advances in Mathematics of Communications, 2012, 6 (1) : 27-38. doi: 10.3934/amc.2012.6.27
##### References:
 [1] I. Charon, I. Honkala, O. Hudry and A. Lobstein, The minimum density of an identifying code in the king lattice,, Discrete Math., 276 (2004), 95.  doi: 10.1016/S0012-365X(03)00306-6.  Google Scholar [2] I. Charon, O. Hudry and A. Lobstein, Identifying codes with small radius in some infinite regular graphs,, Electronic J. Combin., 9 (2002).   Google Scholar [3] G. Cohen, I. Honkala, A. Lobstein and G. Zémor, On codes identifying vertices in the two-dimensional square lattice with diagonals,, IEEE Trans. Comput., 50 (2001), 174.  doi: 10.1109/12.908992.  Google Scholar [4] I. Honkala and T. Laihonen, Codes for identification in the king lattice,, Graphs Combin., 19 (2003), 505.  doi: 10.1007/s00373-003-0531-2.  Google Scholar [5] I. Honkala and T. Laihonen, On identifying codes in the triangular and square grids,, SIAM J. Comput., 33 (2004), 304.  doi: 10.1137/S0097539703433110.  Google Scholar [6] I. Honkala and T. Laihonen, On locating-dominating sets in infinite grids,, European J. Combin., 27 (2006), 218.  doi: 10.1016/j.ejc.2004.09.002.  Google Scholar [7] I. Honkala and T. Laihonen, On a new class of identifying codes in graphs,, Inform. Proc. Letters, 102 (2007), 92.  doi: 10.1016/j.ipl.2006.11.007.  Google Scholar [8] I. Honkala, T. Laihonen and S. Ranto, On locating-dominating codes in binary Hamming spaces,, Discrete Math. Theor. Comput. Sci., 6 (2004), 265.   Google Scholar [9] M. G. Karpovsky, K. Chakrabarty and L. B. Levitin, On a new class of codes for identifying vertices in graphs,, IEEE Trans. Informa. Theory, IT-44 (1998), 599.  doi: 10.1109/18.661507.  Google Scholar [10] M. Pelto, New bounds for $(r,\leq 2)$-identifying codes in the infinite king grid,, Crypt. Commun., 2 (2010), 41.  doi: 10.1007/s12095-009-0015-1.  Google Scholar [11] M. Pelto, On locating-dominating codes for locating large numbers of vertices in the infinite king grid,, Australas. J. Combin., 50 (2011), 127.   Google Scholar [12] M. Pelto, On locating-dominating codes in the infinite king grid,, Ars Combin., ().   Google Scholar [13] M. Pelto, Optimal $(r,\leq 3)$-locating-dominating codes in the infinite king grid,, submitted., ().   Google Scholar [14] S. Ranto, "Identifying and Locating-Dominating Codes in Binary Hamming Spaces,'', Ph.D thesis, (2007).   Google Scholar [15] P. J. Slater, Domination and location in acyclic graphs,, Networks, 17 (1987), 55.  doi: 10.1002/net.3230170105.  Google Scholar [16] P. J. Slater, Dominating and reference sets in a graph,, J. Math. Phys. Sci., 22 (1988), 445.   Google Scholar [17] , URL, \url{http://www.infres.enst.fr/~lobstein/bibLOCDOMetID.html}, ().   Google Scholar

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##### References:
 [1] I. Charon, I. Honkala, O. Hudry and A. Lobstein, The minimum density of an identifying code in the king lattice,, Discrete Math., 276 (2004), 95.  doi: 10.1016/S0012-365X(03)00306-6.  Google Scholar [2] I. Charon, O. Hudry and A. Lobstein, Identifying codes with small radius in some infinite regular graphs,, Electronic J. Combin., 9 (2002).   Google Scholar [3] G. Cohen, I. Honkala, A. Lobstein and G. Zémor, On codes identifying vertices in the two-dimensional square lattice with diagonals,, IEEE Trans. Comput., 50 (2001), 174.  doi: 10.1109/12.908992.  Google Scholar [4] I. Honkala and T. Laihonen, Codes for identification in the king lattice,, Graphs Combin., 19 (2003), 505.  doi: 10.1007/s00373-003-0531-2.  Google Scholar [5] I. Honkala and T. Laihonen, On identifying codes in the triangular and square grids,, SIAM J. Comput., 33 (2004), 304.  doi: 10.1137/S0097539703433110.  Google Scholar [6] I. Honkala and T. Laihonen, On locating-dominating sets in infinite grids,, European J. Combin., 27 (2006), 218.  doi: 10.1016/j.ejc.2004.09.002.  Google Scholar [7] I. Honkala and T. Laihonen, On a new class of identifying codes in graphs,, Inform. Proc. Letters, 102 (2007), 92.  doi: 10.1016/j.ipl.2006.11.007.  Google Scholar [8] I. Honkala, T. Laihonen and S. Ranto, On locating-dominating codes in binary Hamming spaces,, Discrete Math. Theor. Comput. Sci., 6 (2004), 265.   Google Scholar [9] M. G. Karpovsky, K. Chakrabarty and L. B. Levitin, On a new class of codes for identifying vertices in graphs,, IEEE Trans. Informa. Theory, IT-44 (1998), 599.  doi: 10.1109/18.661507.  Google Scholar [10] M. Pelto, New bounds for $(r,\leq 2)$-identifying codes in the infinite king grid,, Crypt. Commun., 2 (2010), 41.  doi: 10.1007/s12095-009-0015-1.  Google Scholar [11] M. Pelto, On locating-dominating codes for locating large numbers of vertices in the infinite king grid,, Australas. J. Combin., 50 (2011), 127.   Google Scholar [12] M. Pelto, On locating-dominating codes in the infinite king grid,, Ars Combin., ().   Google Scholar [13] M. Pelto, Optimal $(r,\leq 3)$-locating-dominating codes in the infinite king grid,, submitted., ().   Google Scholar [14] S. Ranto, "Identifying and Locating-Dominating Codes in Binary Hamming Spaces,'', Ph.D thesis, (2007).   Google Scholar [15] P. J. Slater, Domination and location in acyclic graphs,, Networks, 17 (1987), 55.  doi: 10.1002/net.3230170105.  Google Scholar [16] P. J. Slater, Dominating and reference sets in a graph,, J. Math. Phys. Sci., 22 (1988), 445.   Google Scholar [17] , URL, \url{http://www.infres.enst.fr/~lobstein/bibLOCDOMetID.html}, ().   Google Scholar
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