Upper Bounds on the Domination and Total Domination Number of Fibonacci Cubes

Abstract

Abstract:One of the basic model for interconnection networks is the n-dimensional hypercube graph Q n and the vertices of Q n are represented by all binary strings of length n. The Fibonacci cube Γ n of dimension n is a subgraph of Q n , where the vertices correspond to those without two consecutive 1s in their string representation. In this paper, we deal with the domination number and the total domination number of Fibonacci cubes. First we obtain upper bounds on the domination number of Γ n for n ≥ 13. Then using these result we obtain upper bounds on the total domination number of Γ n for n ≥ 14 and we see that these upper bounds improve the bounds given in [1].

On the chromatic polynomial and the domination number of k-Fibonacci cubes

“…Let of a total dominating set of G , denoted by γ t (G). Bounds on the domination number and total domination number of Γ n are obtained in [1,2,14,15] by using the degree information and the decomposition of Γ n . Some improvements appear in [16].…”

On the domination number and the total domination number of Fibonacci cubes

“…These two upper bounds further improved in [15] by using the values of γ(Γ n ) and the fundamental decomposition (1.1) of Γ n more than once.…”

“…The exact values of γ(Γ n ) and γ t (Γ n ) are also considered by using integer programming in [1]. The upper bound on γ t (Γ n ) given in [1] is improved in [15]. We summarize these results in Section 2.…”

“…We summarize these results in Section 2. The aim of this work is to improve some of the results given in [1] and [15].…”

“…In recent years results on disjoint hypercubes in Γ n are presented in [5,13,16] and the cube enumerator polynomial of Γ n is considered in [10,11,17] and many combinatorial results are given. The domination-type invariants of Γ n are considered in [1,2,8,14,15] and some numerical results and bounds are presented.…”