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\title{ProximityG: A Python package for proximity graphs in the plane}
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\author[1]{Héctor Maravillo\thanks{Corresponding author.}}
%\thanks{ORCID: 0000-0000-0000-0000.}}
\author[2]{Diego Villareal}
\author[2]{Heriberto Espino}

\affil[1]{Universidad Autónoma de la Ciudad de México, Mexico}
\affil[2]{Universidad de las Américas Puebla, México}

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\maketitle

\section*{Tags}
Python; proximity graphs; biological graphs; 

\section*{Summary}

 %   Introducción to proximity graphs (MST, RNG, GG, $\beta$-skeletons, Delaunay, SIG, NNG, Unit Disk) \cite{Mitchell2018, Mathieson2019}

 %   RNG, GG y $\beta$-skeletons, ($k$-RNG) \cite{Jaromczyk2002} Internal and external shape descriptors ($\alph$-shape and hull, $\beta$-skeletons, NNG) \cite{Radke1988} \cite{Radke 1988}

 %   Empty region graphs \cite{Cardinal2009}

 %   Delaunay triangulation \cite{deBerg2008}, algorithms, properties and subgraphs of DT \cite{Aurenhammer2000}

 %   $\gamma$-neighborhood graph \cite{Veltkamp1992}

 %   $\alpha$-shapes, $\alpha$-hull \cite{Edelsbrunner1983}

 %   $\beta$-skeletons \cite{Kirkpatrick1985} 

 %   Gabriel Graph \cite{Gabriel1969, Matula1980}

 %   Unit disk graph \cite{Clark1990}

 %   Empty Elliptic graph \cite{Devillers2008}

 %   Ellitic gabriel graph \cite{Park2006}

  %  RNG \cite{Toussaint1980, Jaromczyk2002}

 %   Stepping Stone Graph \cite{Kannangara2018, Kannangara2019}

 %   Sigma-local graph \cite{Bose2010}

 %   sphere of influence \cite{Toussaint1988,Toussaint2014}
 
 
 Let $V$ be a (finite or discrete) point set in  $\mathbb{R}^d$, with $d\geq 2$. For each pair of different points $\{p,q\}\in V\times V$ is associated a region or neighborhood $N(p,q) \subset \mathbb{R}^d$ and let $\mathcal{P}$ be a geometrical property defined in the set   $\mathcal{N} = \left\{ N(p,q) : \{p,q\} \in V \times V \right\}$. A proximity graph, also called a neighborhood graph \cite{Jaromczyk2002}, with regions in $\mathcal{N}$ and the property $\mathcal{P}$, is a graph $G_{\mathcal{N},\mathcal{P}}(V,E)$ with the node set $V$ and the set of edges $E$ is such that
 \begin{equation}
 	\{p,q\} \in E \quad \Leftrightarrow \quad  N(p,q)
 	\ \ \text{holds } \mathcal{P}
 \end{equation}
 In general, the property 
 $\mathcal{P}$ is defined trying to represent some notion of closeness between a pair of points.  If the property $\mathcal{P}$ is 
 \begin{equation}
 	N(p,q) \cap V \setminus \{p,q\} = \emptyset
\end{equation}
 it is called a empty neighborhood graph \cite{Cardinal2009}.
 
 Two widely known proximity graphs defined to $d=2$ are the Gabriel graph (GG) and the relative neighborhood graph (RNG). In the first, the region is defined as the (open) circle whose boundary trough $p$ and $q$. In RNG, the neighborhood is the intersection of two (open) circles, with centers $p$ and $q$, respectively, and radius the (euclidean) distance between $p$ and $q$.
 

\section*{Statement of need}

Applications: clustering and manifold learning \cite{Zemel2004}

Machine learning \cite{Yang2002}

Asymptotic analysis of proximity graphs  \cite{Devroye1988, Chalker1999}

Probabilistic properties of the $k$-nearest-neighbors graph \cite{Eppstein1997}

recomender systems \cite{Mathieson2019}

Street network characterization \cite{Watanabe2010, Osaragi2014, Maniadakis2016}

See references about applications of the SIG (computer vision, cluster analysis, pattern recognition, SIG, modeling visual illusions, streaming process in music perception) in \cite{Toussaint2014SIG}

See references about applications of RNG, GG and $\beta$-skeletons (Morphology and computer vision, geographic analysis, pattern classification) \cite{Jaromczyk2002}

See references about applications of the RNG (vision, cluster analysis, shape boundary detection, visualization and computer graphics, archaeology, natural computing, cosmology, urban planning, machine learning, percolation theory, wireless ad-hoc networks, graph theory ) \cite{Toussaint2014RNG}

graph theory about proximity graphs \cite{Cimikowski1992, Bose2012}

transport and fungi networks \cite{Adamatzky2011}

computational experiment about Asymptotic properties of $\beta$-skeletons \cite{Adamatzky2014, Alonso2019}

\section*{Statement of need}




\section*{Package overview}

\section*{Usage example}


\section*{Acknowledgements}


\section*{References}

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