A note on Fibonacci and Lucas number of domination in path

Abstract

Let G = (V(G), E(G)) be a path of order n ≥ 1. Let f_m(G) be a path with m ≥ 0 independent dominating vertices which follows a Fibonacci string of binary numbers where 1 is the dominating vertex. A set F(G) contains all possible f_m(G), m ≥ 0, having the cardinality of the Fibonacci number F_{n + 2}. Let F_d(G) be a set of f_m(G) where m = i(G) and F_d^max(G) be a set of paths with maximum independent dominating vertices. Let l_m(G) be a path with m ≥ 0 independent dominating vertices which follows a Lucas string of binary numbers where 1 is the dominating vertex. A set L(G) contains all possible l_m(G), m ≥ 0, having the cardinality of the Lucas number L_n. Let L_d(G) be a set of l_m(G) where m = i(G) and L_d^max(G) be a set of paths with maximum independent dominating vertices. This paper determines the number of possible elements in the sets F_d(G), L_d(G), F_d^max(G) and L_d^max(G) by constructing a combinatorial formula. Furthermore, we examine some properties of F(G) and L(G) and give some important results.