ON (r,<= 2)-LOCATING-DOMINATING CODES IN THE INFINITE KING GRID

: Pelto M

Publisher: AMER INST MATHEMATICAL SCIENCES

: 2012

Advances in Mathematics of Communications

ADVANCES IN MATHEMATICS OF COMMUNICATIONS

: ADV MATH COMMUN

: 1

: 6

: 1

: 27

: 38

: 12

: 1930-5346

DOI: https://doi.org/10.3934/amc.2012.6.27

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 distancerfromv. Let I-r(F) = U-v is an element of F(Br(v) boolean AND C)be a set of code words in the neighbourhoods of vertices v is an element of F. A subset C subset of V is called an (r, <= 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 subset of V where F-1 not equal F-2, F-1 boolean AND C = F-2 boolean AND C and vertical bar F-1 vertical bar, vertical bar F-2 vertical bar <= l. A subset C subset of V is an (r, <= l)-locating-dominating code of type B if the sets I-r(F) are distinct for all subsets F subset of V\C with at most l vertices. We study (r, <= l)-locating-dominating codes in the infinite king grid when r >= 1 and l = 2. The infinite king grid is the graph with vertex set Z(2) and edge set {{(x(1),y(1)),(x(2),y(2))}parallel to x(1) - x(2)vertical bar <= 1,vertical bar y(1) - y(2)vertical bar <= 1,(x(1),y(1))not equal(x(2),y(2))}.

Mathematics