Realizable Triples for Stratified Domination in Graphs

A graph is 2-stratified if its vertex set is partitioned into two classes, where the vertices in one class are colored red and those in the other class are colored blue. Let F be a 2-stratified graph rooted at some blue vertex v. An F -coloring of a graph G is a red-blue coloring of the vertices of G in which every blue vertex v belongs to a copy of F rooted at v. The F -domination number γF (G) is the minimum number of red vertices in an F -coloring of G. In this paper, we study F -domination where F is a red-blue-blue path of order 3 rooted at a blue end-vertex. It is shown that a triple (A,B, C) of positive integers with A 6 B 6 2A and B > 2 is realizable as the domination number, open domination number, and F -domination number, respectively, for some connected graph if and only if (A,B, C) 6= (k, k, C) for any integers k and C with C > k > 2.