on location-domination in graphs

a subset s of the vertex set v (g) of a graph g is a dominating set if each vertex vєv(g)-s is adjacent to at least one vertex of s. the minimum dominating set of graph g is called domination number of g and donoted by γ(g). a dominating set s is defined to be locating if for any two distinct vertices v and u in v(g)-s, n_g(v)∩s≠n_g(u)∩s, where n_g(v) (respectively n_g(u)) is the set of vertices adjacent to v (respectively u) in g. the location-domination number of g, denoted by γi(g) is the minimum cardinality of a locating-dominating set of g. the bandage number, b (g) of a graph g is the cardinality of the smallest set e' of edges for which γl(g-e')>γl(g). the subdivision number, sd_γ(g) of a graph g is the minimum number of edges that must be subdivided (where each edge in g can be subdivided at most once)in order to increase the domination number of g. the location- dimination bondage number, b₁(g) of a fraph g is the cardinality of the smallest set e' of edges for which γ_l(g-e')> γ_l(g). the location- dimination subdivision number, sd_γi(g) of a graph g is the minimum number of edges that must be subdiviled (where each edge in g can be subdivided at most once) in order to increase that location- domination number of g. in this paper we establish the locating bondage number and location- domination domination subdivision number of paths and cycles.