Optimal (r,?3) ( r , ? 3 ) -locatingdominating codes in the infinite king grid
Assume that G=(V,E) G = ( V , E ) is an undirected graph with vertex set V V and edge set E E . The ball Br(v) B r ( v ) denotes the vertices within graphical distance r r from v v . A subset C?V C ? V is called an (r,?l) ( r , ? l ) -locatingdominating code of type B if the sets Ir(F)=?v?F(Br(v)?C) I r ( F ) = ? v ? F ( B r ( v ) ? C ) are distinct for all subsets F?V?C F ? V ? C with at most l l vertices. We give examples of optimal (r,?3) ( r , ? 3 ) -locatingdominating codes of type B B in the infinite king grid for all r?N+ r ? N + and prove optimality. The infinite king grid is the graph with vertex set Z2 Z 2 and edge set {{(x1,y1),(x2,y2)}?|x1?x2|?1,|y1?y2|?1} { { ( x 1 , y 1 ) , ( x 2 , y 2 ) } ? | x 1 ? x 2 | ? 1 , | y 1 ? y 2 | ? 1 } .
king grid graph locatingdominating code density