Improved Upper Bounds on the Domination Number of Graphs With Minimum Degree at Least Five

Bujtás, Csilla; Klavžar, Sandi

doi:10.1007/s00373-015-1585-7

Cited by 15 publications

(5 citation statements)

References 22 publications

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“…First, (4.1) also follows from a more general result on transversals in hypergraphs due to Alon [1]. Second, the state of the art on the upper bounds on the domination number in terms of the minimum degree and the order of a given graph is given in [8].…”

Section: On Total Domination In Hypercubesmentioning

confidence: 97%

On domination-type invariants of Fibonacci cubes and hypercubes

Azarija¹,

Klavžar

Rho³

et al. 2017

Ars Math. Contemp.

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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.

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Section: On Total Domination In Hypercubesmentioning

confidence: 97%

On domination-type invariants of Fibonacci cubes and hypercubes

Azarija¹,

Klavžar

Rho³

et al. 2017

Ars Math. Contemp.

Self Cite

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“…To defend our approach, we recall that game invariants have already proved useful to give a new insight on the classical invariant they are related. We cite in particular [5], where a greedy like strategy used in the study of the domination game is used to improve several upper bounds on the domination number. See also [12] for an application of the coloring game to the graph packing problem.…”

Section: Background and Definition Of The Gamementioning

confidence: 99%

A New Game Invariant of Graphs: the Game Distinguishing Number

Gravier,

Meslem,

Schmidt

et al. 2014

Preprint

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The distinguishing number of a graph G is a symmetry related graph invariant whose study started two decades ago. The distinguishing number D(G) is the least integer d such that G has a distinguishing d-coloring. A distinguishing d-coloring is a coloring c : V (G) → {1, • • • , d} invariant only under the trivial automorphism. In this paper, we introduce a game variant of the distinguishing number. The distinguishing game is a game with two players, the Gentle and the Rascal, with antagonist goals. This game is played on a graph G with a set of d ∈ N * colors. Alternately, the two players choose a vertex of G and color it with one of the d colors. The game ends when all the vertices have been colored. Then the Gentle wins if the coloring is distinguishing and the Rascal wins otherwise. This game leads to define two new invariants for a graph G, which are the minimum numbers of colors needed to ensure that the Gentle has a winning strategy, depending on who starts. These invariants could be infinite, thus we start by giving sufficient conditions to have infinite game distinguishing numbers. We also show that for graphs with cyclic automorphism group of prime odd order, both game invariants are finite. After that, we define a class of graphs, the involutive graphs, for which the game distinguishing number can be quadratically bounded above by the classical distinguishing number. The definition of this class is closely related to imprimitive actions whose blocks have size 2. Then, we apply results on involutive graphs to compute the exact value of these invariants for hypercubes and even cycles. Finally, we study odd cycles, for which we are able to compute the exact value when their order is not prime. In the prime order case, we give an upper bound of 3.

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“…A similar proof technique was introduced in [2], later it was used in [3,4,18] for obtaining upper bounds on the game domination number (see [1] for the definition) and in [15,16] for proving bounds on the game total domination number [14]. Based on this approach we also obtained improvements for the upper bounds on the domination number [6], and in the conference paper [5] we presented a preliminary version of this algorithm to estimate the 2-domination number of graphs of minimum degree 8.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Bounds on the Domination Number

Haynes¹,

Hedetniemi²,

Slater³

2013

Fundamentals of Domination in Graphs

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In a graph G, a set D ⊆ V (G) is called 2-dominating set if each vertex not in D has at least two neighbors in D. The 2-domination number γ 2 (G) is the minimum cardinality of such a set D. We give a method for the construction of 2-dominating sets, which also yields upper bounds on the 2-domination number in terms of the number of vertices, if the minimum degree δ(G) is fixed. These improve the best earlier bounds for any 6 ≤ δ(G) ≤ 21. In particular, we prove that γ 2 (G) is strictly smaller than n/2, if δ(G) ≥ 6. Our proof technique uses a weight-assignment to the vertices where the weights are changed during the procedure.

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Improved Upper Bounds on the Domination Number of Graphs With Minimum Degree at Least Five

Cited by 15 publications

References 22 publications

On domination-type invariants of Fibonacci cubes and hypercubes

On domination-type invariants of Fibonacci cubes and hypercubes

A New Game Invariant of Graphs: the Game Distinguishing Number

Bounds on the Domination Number

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