Stratification and domination in graphs with minimum degree two

Abstract

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 . We investigate the F-domination number when F is a 2-stratified path P 3 on three vertices rooted at a blue vertex which is an end-vertex of the P 3 and is adjacent to a blue vertex with the remaining vertex colored red. We show that for a connected graph of order n with minimum degree at least two this parameter is bounded above by ( n - 1 ) / 2 with the exception of five graphs (one each of orders four, five and six and two of order eight). For n ⩾ 9 , we characterize those graphs that achieve the upper bound of ( n - 1 ) / 2 .