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 max d (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 max d (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 max d (G) and L max d (G) by constructing a combinatorial formula. Furthermore, we examine some properties of F (G) and L(G) and give some important results.

Odd and even repetition sequences of independent domination number

“…Corollary 2.13. If ≡ 0( 2) and ≡ ( ) = ≡ 1 ( 2), then = 3 − 1 for some ∈ ℕ. Corollary 2.14. If ≡ 0( 2) and ≡ ( ) = ≡ 1 ( 2), then (3 −1)/2 = ( ) for some ∈ ℕ.…”

“…Therefore, if we combine the two cases, then this completes the proof. The next Corollaries are quick consequences of Theorem 2.Corollary If If ≡ 1( 2) and ≡ ( ) = ≡ 1( 2), then (3 −1)/2 = ( ) or (3 +1)/2 = ( ) for some ∈ ℕ.…”

“…A path is a walk that does not repeat edges and does not end where it starts, i.e., 1 → 2 → … → , 1 ≠ . A path of order and length − 1 is denoted by [1,2,10]. Then, neighborhood of is the set ( ) = ( ) = { ∈ ( ): ∈ ( )} of ( ) is a dominating set of if for every ∈ ( )\ , there exists ∈ such that is an edge of , i.e., [ ] = ( ).…”

“…Figure 2 shows the independent dominating vertices in that follow a unique configuration of odd and even repetition sequences when ≡ 1 ( 2) and ≡ 0( 2), respectively. A Fibonacci number can be obtained by the following equation, = −2 + −1 , for ∈ ℕ\{1, 2}, where 1 = 2 = 1 [2,12]. The Fibonacci sequence has been widely studied by many researchers and covers a wide range of interests in combinatorial mathematics as it appears intriguing in the works of Vajda [14].…”

On Triangular Secure Domination Number

“…The smallest cardinality of a dominating set is called dominating number of graph denoted by ( ). Domination concepts in graphs and some of its variants can be found in [4] , where is the triangular number in graph [13] [2] [14]. The minimum cardinality of a secure dominating set of , denoted by ( ) is called a secure domination number of .…”