Irregular orbital domination in graphs. Issue 1 (2nd January 2022) - Broe, Peter; Chartrand, Gary; Zhang, Ping
Abstract : For a non-negative integer r, the r -orbit O r ( v ) of a vertex v in a connected graph G of order n is the set of vertices at distance r from v . A sequence s : r 1, r 2, …, r k of positive integers with 1 ≤ k ≤ n is called an irregular orbital dominating sequence of G if r i ≠ r j for every pair i, j of integers with 1 ≤ i < j ≤ k and G contains distinct vertices v 1, v 2, …, v k such that ⋃ i = 1 k O r i ( v i ) = V ( G ) . 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.