Fibonacci Polynomials and Parity Domination in Grid Graphs

Goldwasser, John L.; Klostermeyer, William F.; Ware, Henry Oliver T.

doi:10.1007/s003730200020

Cited by 22 publications

(15 citation statements)

References 6 publications

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“…Much of the attention on even and odd dominating sets has focused on grid graphs [2,6,9,10,11]. We show in this section that only a handful of grids have CODS.…”

Section: Grid Graphsmentioning

confidence: 73%

“…A detailed analysis of length of the periodicity of the related recurrence for even dominating sets of grids can be found in [9,11].…”

Section: Grid Graphsmentioning

confidence: 99%

“…An odd dominating set of a simple, undirected graph G = (V, E) is a set of vertices D ⊆ V such that |N [v] ∩ D| ≡ 1 mod 2 for all vertices v ∈ V , where N [v] denotes the closed neighborhood of v. Odd dominating sets and the analogously defined even dominating sets have received considerable attention in the literature, see [1,3,2,4,5,6,7,8,9,10,11,12,14]. Sutner proved that every graph contains at least one odd dominating set [14] and other proofs of this can be found in [4,8].…”

Section: Introductionmentioning

confidence: 99%

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Connected odd dominating sets in graphs

Caro

Klostermeyer

Yuster

2005

Discuss. Math. Graph Theory

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An odd dominating set of a simple, undirected graph G = (V, E) is a set of vertices D ⊆ V such that |N [v] ∩ D| ≡ 1 mod 2 for all vertices v ∈ V . It is known that every graph has an odd dominating set. In this paper we consider the concept of connected odd dominating sets. We prove that the problem of deciding if a graph has a connected odd dominating set is NP-complete. We also determine the existence or non-existence of such sets in several classes of graphs. Among other results, we prove there are only 15 grid graphs that have a connected odd dominating set.

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“…Much of the attention on even and odd dominating sets has focused on grid graphs [2,6,9,10,11]. We show in this section that only a handful of grids have CODS.…”

Section: Grid Graphsmentioning

confidence: 73%

“…A detailed analysis of length of the periodicity of the related recurrence for even dominating sets of grids can be found in [9,11].…”

Section: Grid Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Connected odd dominating sets in graphs

Caro

Klostermeyer

Yuster

2005

Discuss. Math. Graph Theory

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“…If the Fibonacci polynomials are defined over F 2 , the Fibonacci index of any irreducible polynomial is either a divisor of 2 m − 1 or 2 m + 1, with m being the degree of p(x) 75 .…”

Section: Definition Ii5 (Fibonacci Index)mentioning

confidence: 99%

Structure of the sets of mutually unbiased bases with cyclic symmetry

Seyfarth¹,

Sánchez-Soto²,

Leuchs³

2014

J. Phys. A: Math. Theor.

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Mutually unbiased bases that can be cyclically generated by a single unitary operator are of special interest, since they can be readily implemented in practice. We show that, for a system of qubits, finding such a generator can be cast as the problem of finding a symmetric matrix over the field F 2 equipped with an irreducible characteristic polynomial of a given Fibonacci index. The entanglement structure of the resulting complete sets is determined by two additive matrices of the same size.

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“…The puzzle on 5 5 grids was studied by Anderson and Feil [1998], who used linear algebra over ‫ޚ‬ 2 to classify winnable configurations. The analysis on n m grids was done using Fibonacci polynomials in 2002]. Earlier, Sutner [1989] had shown that the winnability of configurations in Lights Out is also equivalent to a question on finite cellular automata.…”

Section: Introductionmentioning

confidence: 99%

Lights Out on finite graphs

Edwards

Elandt

James

et al. 2010

Involve

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Lights Out is a one-player game played on a finite graph. In the standard game the vertices can be either on or off; pressing a vertex toggles its state and that of all adjacent vertices. The goal of the game is to turn off all of the lights. We study an extension of the game in which the state of a vertex may be one of a finite number of colors. We determine which graphs in certain families (spider graphs and generalized theta graphs) are winnable for every initial coloring. We also provide a construction that gives every always-winnable tree for any prime power number of colors.

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Fibonacci Polynomials and Parity Domination in Grid Graphs

Cited by 22 publications

References 6 publications

Connected odd dominating sets in graphs

Connected odd dominating sets in graphs

Structure of the sets of mutually unbiased bases with cyclic symmetry

Lights Out on finite graphs

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