A note on efficient domination in a superclass of P5-free graphs

In a graph G=(V,E), an efficient dominating set is a subset D@?V such that D is an independent set such that each vertex outside D has exactly one neighbor in D. E stands for the graph with vertices a,b,c,d,e,f and edges ab,bc,cd,de and cf, while xNet stands for the graph with vertices a,b,c,d,e,f,g and edges ad,be,cf,de,ef,fd and cg. This note shows that, given an (E,xNet)-free graph, a minimum efficient dominating set (if any) can be found in polynomial time, extending two polynomially solvable cases for efficient dominating sets recently found by Milanic (2013) [4], and Brandstadt et al. (2013) [2].