An Upper Bound on F -domination Number in Grids †

A graph G is 2-stratified if its vertex set is colored into two nonempty classes, where one class of vertices colored red and the other color class blue. Let F be a 2-stratified graph rooted at one fixed blue vertex v. The F -domination number of G is the minimum number of red vertices of G in a red-blue coloring of the vertices of G such that for every blue vertex v of G, there is a copy of F in G rooted at v. In this paper, we explore an upper bound on the F -domination number for a specified 2-stratified graphs F in grids, where F is a path P3 rooted at a blue vertex that is adjacent to a blue vertex and with the remaining vertex colored red. As far as we know, no such an F -domination number was known for grids.