Irregular orbital domination in graphs

For a non-negative integer r, the r-orbit of a vertex v in a connected graph G of order n is the set of vertices at distance r from v. A sequence of positive integers with is called an irregular orbital dominating sequence of G if for every pair i, j of integers with and G contains distinct vertices such that We investigate graphs that possess and graphs do not possess an irregular orbital dominating sequence. It is shown that a non-trivial tree has an irregular orbital dominating sequence if and only if it is neither a star, a path of order 2, nor a path of order 6.