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

Abstract

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 gridwhen $r\geq 1$ and $l=2$. The infinite king grid is the graph with vertex set $\mathbb Z^2$ and edge set$\{\{(x_1,y_1),(x_2,y_2)\}||x_1-x_2|\leq 1, |y_1-y_2|\leq 1, (x_1,y_1)\ne(x_2,y_2)\}.$