Dynamic monopolies and feedback vertex sets in hexagonal grids

Adams, Sarah Spence; Troxell, Denise Sakai; Zinnen, S. Luke

doi:10.1016/j.camwa.2011.09.047

Cited by 15 publications

(10 citation statements)

References 14 publications

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“…Obviously, minseed(G, θ > ) = min-seed(G, 2). Since G has mn vertices and 3mn 2 − m edges (see Lemma 4.1 of [2]), by the result min-seed(G, θ > ) ≥ |V (G)|θ min −|E(G)| θ min presented in Lemma 2, we see at once that min-seed(G, 2) ≥ ⌈(mn + 2m)/4⌉.…”

Section: Theoremmentioning

confidence: 67%

“…We now turn to determine the exact value of min-seed(G, θ) for certain families of graphs G under specific threshold functions θ. Related results can be found in [1,2,4,5,7,8,11,13,14,15,16,24,25,27,28,29,33], where min-seed(G, θ) has been investigated for different types of network structure G such as bounded treewidth graphs, hexagonal grids, trees, cycle permutation graphs, generalized Petersen graphs, block-cactus graphs, chordal graphs, Hamming graphs, chordal rings, tori, meshes, butterflies, Cartesian products of two cycles.…”

Section: Introductionmentioning

confidence: 99%

“…In marketing setting, the threshold of a vertex (customer) v represents his/her latent tendency of buying the new product when his/her neighbors (friends) do (see [31]). There are two types of important and well-studied thresholds on a graph G called majority threshold and strict majority threshold (see [2,6,20,21,23,27,28] and references therein), which will be denoted by θ ≥ and θ > respectively throughout this paper. In a majority threshold we have θ ≥ (v) = ⌈d(v)/2⌉ for every vertex v of G, while in a strict majority threshold we have θ > (v) = ⌈(d(v) + 1)/2⌉ for every vertex v of G.…”

Section: Introductionmentioning

confidence: 99%

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Target Set Selection Problem for Honeycomb Networks

Chiang

Huang

Yeh³

2013

SIAM J. Discrete Math.

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Let G be a graph with a threshold function θ :Suppose we are given a target set S ⊆ V (G). The paper considers the following repetitive process on G. At time step 0 the vertices of S are colored black and the other vertices are colored white. After that, at each time step t > 0, the colors of white vertices (if any) are updated according to the following rule. All white vertices v that have at least θ(v) black neighbors at the time step t − 1 are colored black, and the colors of the other vertices do not change. The process runs until no more white vertices can update colors from white to black. The following optimization problem is called TARGET SET SELECTION: Finding a target set S of smallest possible size such that all vertices in G are black at the end of the process. Such an S is called an optimal target set for G under the threshold function θ. We are interested in finding an optimal target set for the well-known class of honeycomb networks under an important threshold function called strict majority threshold, where θ(v) = ⌈(d G (v) + 1)/2⌉ for each vertex v in G. In a graph G, a feedback vertex set is a subset S ⊆ V (G) such that the subgraph induced by V (G) \ S is cycle-free. In this paper we give exact value on the size of the optimal target set for various kinds of honeycomb networks under strict majority threshold, and as a by-product we also provide a minimum feedback vertex set for different kinds regular graphs in the class of honeycomb networks.

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Section: Theoremmentioning

confidence: 67%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Target Set Selection Problem for Honeycomb Networks

Chiang

Huang

Yeh³

2013

SIAM J. Discrete Math.

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“…Exemplarily, here we consider one-way cellular automata with time-bounded nondeterminism, that is, the number of nondeterministic state transitions is a limited resource which depends on the length of the input (see [21] for further details, variants, and references). In order to distinguish between deterministic and nondeterministic steps a deterministic as well as a nondeterministic transition function is provided.…”

Section: Limited Nondeterminismmentioning

confidence: 99%

“…In fact, bounds have been derived for the minimum number of vertices one needs to activate in order to end up activating the whole graph. These sets of vertices are called irreversible dynamic monopolies or irreversible dynamos [20][21][22][23][24][25][26][27][28][29].…”

Section: Related and Previous Workmentioning

confidence: 99%