ON THE BOUNDARY AS AN x-GEODOMINATING SET IN GRAPHS

Given a graph G and a vertex x is an element of V(G), a vertex set S subset of V (G) is an x-geodominating set of G if each vertex v is an element of V (G) lies on an x y geodesic for some element y is an element of S. The minimum cardinality of an x-geodominating set of G is defined as the x-geodomination number of G, g(x)(G), and an x-geodominating set of cardinality g(x)(G) is called a g(x)-set. It is known that there is a unique g(x)-set for each vertex x. We prove that, in any graph G, the g(x)-set associated to a vertex x is the set of boundary vertices of x, that is partial derivative(x) = {v is an element of V (G) : for all w E N(v) : d(x,w) <= d(x, v)}. This characterization of g(x)-sets allows to deduce, in a easy way, different properties of these sets and also to compute both g(x)-sets and the x-geodomination number, in graphs obtained using different graphs products: Cartesian, strong and lexicographic.