On domination-type invariants of Fibonacci cubes and hypercubes

Abstract

The Fibonacci cube Γ n is the subgraph of the n-dimensional cube Q n induced by the vertices that contain no two consecutive 1s. Using integer linear programming, exact values are obtained for γ t (Γ n), n ≤ 12. Consequently, γ t (Γ n) ≤ 2F n−10 + 21F n−8 holds for n ≥ 11, where F n are the Fibonacci numbers. It is proved that if n ≥ 9, then γ t (Γ n) ≥ (F n+2 − 11)/(n − 3) − 1. Using integer linear programming exact values for the 2packing number, connected domination number, paired domination number, and signed domination number of small Fibonacci cubes and hypercubes are obtained. A conjecture on the total domination number of hypercubes asserting that γ t (Q n) = 2 n−2 holds for n ≥ 6 is also disproved in several ways.

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

“…We start by presenting some known values of these numbers in Table 1, which was given in [1,14]. Table 1.…”

“…Note that, in [1] by using integer programming it is shown that 54 ≤ γ(Γ 12 ) ≤ 61 and 97 ≤ γ t (Γ 13 ) ≤ 101.…”

“…By using the fundamental decomposition (2) of Γ n , the following upper bound on γ t (Γ n ) is obtained in [1].…”

“…In [13], upper and lower bounds for the domination number of Γ n are obtained and a comparison with the domination number of Lucas cubes is given. In [14], an integer programming method is used to compute the exact values of the domination number of Γ n for n ≤ 10 and then this technique is used in [1] to obtain the exact value of domination for n = 11. Also in [1] total domination number of a graph is defined and upper and lower bounds are obtained.…”

“…In [14], an integer programming method is used to compute the exact values of the domination number of Γ n for n ≤ 10 and then this technique is used in [1] to obtain the exact value of domination for n = 11. Also in [1] total domination number of a graph is defined and upper and lower bounds are obtained. In addition, using integer programming method exact values for total domination number is obtained for n ≤ 12.…”