### 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.

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PDFDOI: http://dx.doi.org/10.5614/ejgta.2018.6.2.11

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