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Peter J. Slater - One of the best experts on this subject based on the ideXlab platform.
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On Open Neighborhood locating-dominating in graphs
Electronic Journal of Graph Theory and Applications, 2014Co-Authors: Mustapha Chellali, Peter J. SlaterAbstract:A set D of vertices in a graph G = (V (G), E(G)) is an Open Neighborhood locating-dominating set (OLD-set) for G if for every two vertices u, v of V (G) the sets N(u) ∩ D and N(v) ∩ D are non-empty and different. The Open Neighborhood locating-dominating number OLD(G) is the minimum cardinality of an OLD-set for G. In this paper we characterize graphs G of order n with OLD(G) = 2, 3, or n and graphs with minimum degree (G) ≥ 2 that are C4-free with OLD(G) = n-1.
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an introduction to closed Open Neighborhood sums minimax maximin and spread
Mathematics in Computer Science, 2011Co-Authors: Allen Oneal, Peter J. SlaterAbstract:For a graph G of order |V(G)| = n and a real-valued mapping $${f:V(G)\rightarrow\mathbb{R}}$$ , if $${S\subset V(G)}$$ then $${f(S)=\sum_{w\in S} f(w)}$$ is called the weight of S under f. The closed (respectively, Open) Neighborhood sum of f is the maximum weight of a closed (respectively, Open) Neighborhood under f, that is, $${NS[f]={\rm max}\{f(N[v])|v \in V(G)\}}$$ and $${NS(f)={\rm max}\{f(N(v))|v \in V(G)\}}$$ . The closed (respectively, Open) lower Neighborhood sum of f is the minimum weight of a closed (respectively, Open) Neighborhood under f, that is, $${NS^{-}[f]={\rm min}\{f(N[v])|v\in V(G)\}}$$ and $${NS^{-}(f)={\rm min}\{f(N(v))|v\in V(G)\}}$$ . For $${W\subset \mathbb{R}}$$ , the closed and Open Neighborhood sum parameters are $${NS_W[G]={\rm min}\{NS[f]|f:V(G)\rightarrow W}$$ is a bijection} and $${NS_W(G)={\rm min}\{NS(f)|f:V(G)\rightarrow W}$$ is a bijection}. The lower neighbor sum parameters are $${NS^{-}_W[G]={\rm max}NS^{-}[f]|f:V(G)\rightarrow W}$$ is a bijection} and $${NS^{-}_W(G)={\rm max}NS^{-}(f)|f:V(G)\rightarrow W}$$ is a bijection}. For bijections $${f:V(G)\rightarrow \{1,2,\ldots,n\}}$$ we consider the parameters NS[G], NS(G), NS −[G] and NS −(G), as well as two parameters minimizing the maximum difference in Neighborhood sums.
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Open Neighborhood locating domination for infinite cylinders
ACM Southeast Regional Conference, 2011Co-Authors: Suk Seo, Peter J. SlaterAbstract:For a graph G that models a facility or a multi-processor network, detection devices can be placed at the vertices so as to identify the location of an intruder such as a thief or saboteur or a faulty processor. Open Neighborhood locating-dominating sets are of interest when the intruder/fault at a vertex precludes its detection by a device at that location. For example, a processor might be able to determine whether another processor directly connected to it is faulty, but it cannot be assumed to detect its own fault. In this paper we illustrate these concepts using cylindrical graphs as examples of multi-processor networks.
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Open Neighborhood locating-dominating in trees
Discrete Applied Mathematics, 2011Co-Authors: Peter J. SlaterAbstract:For a graph G that models a facility or a multiprocessor network, detection devices can be placed at the vertices so as to identify the location of an intruder such as a thief or saboteur or a faulty processor. Open Neighborhood locating-dominating sets are of interest when the intruder/fault at a vertex precludes its detection at that location. The parameter OLD(G) denotes the minimum cardinality of a vertex set S@?V(G) such that for each vertex v in V(G) its Open Neighborhood N(v) has a unique non-empty intersection with S. For a tree T"n of order n we have @?n/2@?+1@?OLD(T"n)@?n-1. We characterize the trees that achieve these extremal values.
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Open Neighborhood locating dominating sets
Australas. J Comb., 2010Co-Authors: Suk Seo, Peter J. SlaterAbstract:For a graph G that models a facility, various detection devices can be placed at the vertices so as to identify the location of an intruder such as a thief or saboteur. Here we introduce the Open Neighborhood locatingdominating set problem. This deals with problems in which the intruder at a vertex can interfere with the detection device located there. We seek a minimum cardinality vertex set S with the property that for each vertex v its Open Neighborhood N(v) has a unique non-empty intersection with S. Such a set is an OLD-set for G. Among other things, we describe minimum density OLD-sets for various infinite grid graphs.
Otero M. - One of the best experts on this subject based on the ideXlab platform.
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An Extension Result for Maps Admitting an Algebraic Addition Theorem
'Springer Science and Business Media LLC', 2019Co-Authors: Baro E., De Vicente J., Otero M.Abstract:This is a post-peer-review, pre-copyedit version of an article published in Journal of Geometric Analysis. The final authenticated version is available online at: https://doi.org/10.1007/s12220-018-9992-7We prove that if an analytic map f: U→ Cn, where U⊂ Cn is an Open Neighborhood of the origin, admits an algebraic addition theorem, then there exists a meromorphic map g: Cn⤏ Cn admitting an algebraic addition theorem such that each coordinate function of f is algebraic over C(g) on U (this was proved by Weierstrass for n= 1). Furthermore, g admits a rational addition theorem.All the authors were supported by Spanish MTM2014-55565 and MTM2017-82105-P. The second author was also supported by a Grant of the International Program of Excellence in Mathematics at Universidad Autónoma de Madri
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An extension result for maps admitting and algebraic addition theorem
'Springer Science and Business Media LLC', 2018Co-Authors: Baro E., J De Vicente, Otero M.Abstract:We prove that if an analytic map : U [flecha] Cn, where U [incluye] C [elevado a ] n is an Open Neighborhood of the origin, admits an algebraic addition theorem then, there exists a meromorphic map g : C [elevado a]n [flecha] C [elevado a]n admitting an algebraic addition theorem such that each coordinate function of f is algebraic over C(g) on U (this was proved by K. Weierstrass for n = 1). Furthermore, g admits a rational addition theorem
B. Sooryanarayana - One of the best experts on this subject based on the ideXlab platform.
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Open Neighborhood chromatic number of an antiprism graph
2015Co-Authors: Narahari Narasimha Swamy, B. Sooryanarayana, Geetha Kempanapura, Nanjunda SwamyAbstract:An Open Neighborhood k-coloring of a simple connected undirected graph G(V,E) is a k-coloring c : V ! {1,2,··· ,k}, such that, for every w 2 V and for all u,v 2 N(w), c(u) 6 c(v). The minimum value of k for which G admits an Open Neighborhood k-coloring is called the Open Neighborhood chromatic num
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Open Neighborhood Coloring of Prisms
Journal of Mathematical and Fundamental Sciences, 2014Co-Authors: Geetha Kempanapura Nanjunda Swamy, K. N. Meera, Narahari Narasimha Swamy, B. SooryanarayanaAbstract:For a simple, connected, undirected graph neighbor-hood coloring of the graph for each w V ∈ and ∀ ∈ ( ), f w ( ) w V G ∀ ∈ is called the span of the Open Neighborhood The minimum span of f the Open Neighborhood chromatic number of paper, we determine the Open Neighborhood chromatic number of prism which is a generalized Petersen graph
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Open Neighborhood coloring of graphs
The International Journal of Contemporary Mathematical Sciences, 2013Co-Authors: K N Geetha, K. N. Meera, N Narahari, B. SooryanarayanaAbstract:For a simple, connected, undirected graph G(V,E) an Open Neighborhood coloring of the graph G is a mapping f : V (G) → Z+ such that for each w ∈ V , and ∀u, v ∈ N(w), f(u) = f(v). The maximum value of f(w),∀w ∈ V (G) is called the span of the Open Neighborhood coloring f . The minimum value of span of f over all Open Neighborhood colorings f is called Open Neighborhood chromatic number of G, denoted by χonc(G). In this paper we determine the Open Neighborhood chromatic number of some standard graphs, trees and the infinite triangular lattice. 676 Geetha K. N., K. N. Meera, Narahari N. and B. Sooryanarayana Mathematics Subject Classification: 05C15
Waters Alden - One of the best experts on this subject based on the ideXlab platform.
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Analytic properties of heat equation solutions and reachable sets
2020Co-Authors: Strohmaier Alexander, Waters AldenAbstract:For the heat equation on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ we show that at positive time all solutions to the heat equation are analytically extendable to a geometrically determined subdomain $\mathcal{E}(\Omega)$ of $\mathbb{C}^d$ containing $\Omega$. This result is sharp in the sense that there is no larger domain for which this is true. If $\Omega$ is a ball we prove an almost converse of this theorem. Any function that is analytic in an Open Neighborhood of $\mathcal{E}(\Omega)$ is reachable in the sense that it can be obtained from a solution of the heat equation at positive time. This is based on an analysis of the convergence of heat equation solutions in the complex domain using the boundary layer potential method for the heat equation. The converse theorem is obtained using a simple Wick rotation into the complex domain that is justified by our results. This gives a simple explanation for the shapes appearing in the one-dimensional analysis of the problem in the literature.Comment: 12 pages 1 figur
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Analytic properties of heat equation solutions and reachable sets
2020Co-Authors: Strohmaier Alexander, Waters AldenAbstract:There recently has been some interest in the space of functions on an interval satisfying the heat equation for positive time in the interior of this interval. Such functions were characterised as being analytic on a square with the original interval as its diagonal. In this short note we provide a direct argument that the analogue of this result holds in any dimension. For the heat equation on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ at positive time all solutions are analytically extendable to a geometrically determined subdomain $\mathcal{E}(\Omega)$ of $\mathbb{C}^d$ containing $\Omega$. This domain is sharp in the sense that there is no larger domain for which this is true. If $\Omega$ is a ball we prove an almost converse of this theorem. Any function that is analytic in an Open Neighborhood of $\mathcal{E}(\Omega)$ is reachable in the sense that it can be obtained from a solution of the heat equation at positive time. This is based on an analysis of the convergence of heat equation solutions in the complex domain using the boundary layer potential method for the heat equation. The converse theorem is obtained using a Wick rotation into the complex domain that is justified by our results. This gives a simple explanation for the shapes appearing in the one-dimensional analysis of the problem in the literature. It also provides a new short and conceptual proof in that case.Comment: 15 pages 1 figure, some additional explanations and references adde
Baro E. - One of the best experts on this subject based on the ideXlab platform.
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An Extension Result for Maps Admitting an Algebraic Addition Theorem
'Springer Science and Business Media LLC', 2019Co-Authors: Baro E., De Vicente J., Otero M.Abstract:This is a post-peer-review, pre-copyedit version of an article published in Journal of Geometric Analysis. The final authenticated version is available online at: https://doi.org/10.1007/s12220-018-9992-7We prove that if an analytic map f: U→ Cn, where U⊂ Cn is an Open Neighborhood of the origin, admits an algebraic addition theorem, then there exists a meromorphic map g: Cn⤏ Cn admitting an algebraic addition theorem such that each coordinate function of f is algebraic over C(g) on U (this was proved by Weierstrass for n= 1). Furthermore, g admits a rational addition theorem.All the authors were supported by Spanish MTM2014-55565 and MTM2017-82105-P. The second author was also supported by a Grant of the International Program of Excellence in Mathematics at Universidad Autónoma de Madri
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An extension result for maps admitting and algebraic addition theorem
'Springer Science and Business Media LLC', 2018Co-Authors: Baro E., J De Vicente, Otero M.Abstract:We prove that if an analytic map : U [flecha] Cn, where U [incluye] C [elevado a ] n is an Open Neighborhood of the origin, admits an algebraic addition theorem then, there exists a meromorphic map g : C [elevado a]n [flecha] C [elevado a]n admitting an algebraic addition theorem such that each coordinate function of f is algebraic over C(g) on U (this was proved by K. Weierstrass for n = 1). Furthermore, g admits a rational addition theorem
