Stratification and domination in graphs II Original Research Article

A graph G is 2-stratified if its vertex set is partitioned into two classes (each of which is a stratum or a color class.) We color the vertices in one color class red and the other color class blue. Let F be a 2-stratified graph with one fixed blue vertex v specified. We say that F is rooted at v. The F-domination number of a graph G is the minimum number of red vertices of G in a red–blue coloring of the vertices of G such that every blue vertex v of G belongs to a copy of F rooted at v. In this paper we investigate the F-domination number when (i) F is a 2-stratified path P3 on three vertices rooted at a blue vertex which is a vertex of degree 1 in the P3 and is adjacent to a blue vertex and with the remaining vertex colored red, and (ii) F is a 2-stratified K3 rooted at a blue vertex and with exactly one red vertex.