The vertex isoperimetric problem for the powers of the diamond graph

Bezrukov, Sergei L.; Rius, Miquel; Serra, Oriol

doi:10.1016/j.disc.2007.04.060

Cited by 4 publications

(6 citation statements)

References 10 publications

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“…where m i is the total number of valid moves made at v i during all the process, and Note that the parameter B(G) may have some connection with the well-studied vertex isoperimetric number [2]. We refer to [15] for some known connection between the lit-only σ -game and isoperimetric problems.…”

Section: A Strategy To Play the Lit-only σ-Gamementioning

confidence: 98%

“…Let H be the matrix multiplicative semigroup generated by {P v,G : v ∈ V (G)}, which is rarely abelian. To study the lit-only σ -game on G is the same as to study the action of H on F V (G) 2 . Note that the existence of loops causes the non-invertibility of the lit-only σ -game.…”

Section: σ-Game and Lit-only σ-Gamementioning

confidence: 99%

“…In the σ -game on G, we are given a configuration x of ✩ G and aim to determine all those y satisfying x → G y and especially among these those with the smallest light number. Correspondingly, for any x ∈ F V (G) 2 , we put its minimum light number for the σ -game on G to be…”

Section: σ-Game and Lit-only σ-Gamementioning

confidence: 99%

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Lit-only σ-game on pseudo-trees

Wang

2010

Discrete Applied Mathematics

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a b s t r a c tA configuration of a graph is an assignment of one of two states, ON or OFF, to each vertex of it. A regular move at a vertex changes the states of the neighbors of that vertex. A valid move is a regular move at an ON vertex. A pseudo-tree is a graph obtained from a tree by attaching zero or more loops. The following result is proved in this note: given any starting configuration x of a pseudo-tree, if there is a sequence of regular moves which brings x to another configuration in which there are ℓ ON vertices then there must exist a sequence of valid moves which takes x to a configuration with at most ℓ+2 ON vertices. We provide an example to show that the upper bound ℓ + 2 is sharp. Some related problems and conjectures are also reported.

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Section: A Strategy To Play the Lit-only σ-Gamementioning

confidence: 98%

Section: σ-Game and Lit-only σ-Gamementioning

confidence: 99%

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Lit-only σ-game on pseudo-trees

Wang

2010

Discrete Applied Mathematics

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“…Karachanjan and Riordan [14,21] solved the problem (1.1) for multidimensional torus. Bezrukov considered powers of the Diamond graph [4] and powers of cross-sections [5]. We state the results for discrete spaces as corollary.…”

Section: Isoperimetric Spaces and Corollariesmentioning

confidence: 99%

Extremal Lipschitz functions in the deviation inequalities from the mean

Dzindzalieta

2013

Electron. Commun. Probab.

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We obtain an optimal deviation from the mean upper bound \begin{equation} D(x)\=\sup_{f\in \F}\mu\{f-\E_{\mu} f\geq x\},\qquad\ \text{for}\ x\in\R\label{abstr} \end{equation} where $\F$ is the class of the integrable, Lipschitz functions on probability metric (product) spaces. As corollaries we get exact solutions of $\eqref{abstr}$ for Euclidean unit sphere $S^{n-1}$ with a geodesic distance and a normalized Haar measure, for $\R^n$ equipped with a Gaussian measure and for the multidimensional cube, rectangle, torus or Diamond graph equipped with uniform measure and Hamming distance. We also prove that in general probability metric spaces the $\sup$ in $\eqref{abstr}$ is achieved on a family of distance functions.Comment: 7 page

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“…Almost all research and previous literature on Vertex Isoperimetric Parameter and related topics is for undirected graphs. The related notion of Edge Boundary and Edge Isoperimetric Parameter seems to have been studied more extensively than the Vertex Isoperimetric Parameter [11,3,10,5,4,21,2] for undirected graphs. An alternate possible definition of ( , ) is:…”

Section: Related Work and Definitionsmentioning

confidence: 99%

Vertex Isoperimetric Parameter of a Computation Graph

Ranjan

Zubair

2012

Int. J. Found. Comput. Sci.

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Let G = (V,E) be a computation graph, which is a directed graph representing a straight line computation and S ⊂ V. We say a vertex v is an input vertex for S if there is an edge (v, u) such that v ∉ S and u ∈ S. We say a vertex u is an output vertex for S if there is an edge (u, v) such that u ∈ S and v ∉ S. A vertex is called a boundary vertex for a set S if it is either an input vertex or an output vertex for S. We consider the problem of determining the minimum value of boundary size of S over all sets of size M in an infinite directed grid. This problem is related to the vertex isoperimetric parameter of a graph, and is motivated by the need for deriving a lower bound for memory traffic for a computation graph representing a financial application. We first extend the notion of vertex isoperimetric parameter for undirected graphs to computation graphs, and then provide a complete solution for this problem for all M. In particular, we show that a set S of size M = 3k2 + 3k + 1 vertices of an infinite directed grid, the boundary size must be at least 6k + 3, and this is obtained when the vertices in S are arranged in a regular hexagonal shape with side k + 1.

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The vertex isoperimetric problem for the powers of the diamond graph

Cited by 4 publications

References 10 publications

Lit-only σ-game on pseudo-trees

Lit-only σ-game on pseudo-trees

Extremal Lipschitz functions in the deviation inequalities from the mean

Vertex Isoperimetric Parameter of a Computation Graph

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