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\begin{document}

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\section*{generalized graph}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{extra_structure}{Extra Structure}\dotfill \pageref*{extra_structure} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{generalized graph} in the sense of \hyperlink{HRY}{HRY} is a generalization of the notion of a pseudograph given at [[graph]] (see in particular \href{https://ncatlab.org/nlab/show/graph#definition_in_terms_of_action_on_a_set_of_halfedges}{Definition in terms action on a set of half edges}). By adding structure like directions for the edges, we can define a [[wheeled graph]] which is a generalization of a [[directed pseudograph]]. What differentiates a generalized graph from a pseudograph and a wheeled graph from a [[directed pseudograph]] is the notion of an ``exceptional cell,'' which should be thought of as a set of half edges which have no vertices.

A generalized graph can be visualized as a set of vertices with half edges attached to them, a rule for attaching half edges to glue the vertices together, and potentially some half edges (flags) that are attached to only one or zero vertices. For instance, a particularly simple generalized graph is one with no vertices and one edge.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

The following is Definition 2.2 of \hyperlink{HRY}{HRY}:

\begin{defn}
\label{}\hypertarget{}{}
A generalized graph $G$ is a finite set $Flag(G)$ with:

\begin{itemize}%
\item a partition of $Flag(G)=\coprod_{\alpha\in A} F_\alpha$ into \textbf{cells} with $A$ finite,


\item a distinguished partition subset $F_\epsilon$ called the \textbf{exceptional cell},


\item an involution $\iota$ satisfying $\iota F_\epsilon\subseteq F_\epsilon$,


\item and a free involution $\pi$ on the set of $\iota$-fixed points of $F_\epsilon$.



\end{itemize}
\end{defn}
Given a generalized graph $G$, Definition 2.3 of \hyperlink{HRY}{HRY} gives some useful terminology:

\begin{defn}
\label{}\hypertarget{}{}
\begin{enumerate}%
\item The elements of $Flag(G)$ are called \textbf{flags}. A flag in a non-exceptional (resp. exceptional) cell is called an \textbf{ordinary flag} (resp. \textbf{exceptional flag}).
\item Call $G$ an \textbf{ordinary graph} if $F_\epsilon$ is empty.
\item Each non-exceptional partition subset $F_\alpha\neq F_\epsilon$ is called a \textbf{vertex}. The set of vertices is denoted by $Vt(G)$. An empty vertex is an \textbf{isolated vertex}. A flag in a vertex is said to be \textbf{adjacent to} or \textbf{attached to} that vertex.
\item An $\iota$-fixed point is a \textbf{leg} of $G$. The set of legs is denoted by $Leg(G)$. An \textbf{ordinary leg} (resp. \textbf{exceptional leg}) is an ordinary (resp. exceptional) flag that is also a leg. For an $\iota$-fixed point $x\in F_\epsilon$, the pair $\{x,\pi x\}$ is called an \textbf{exceptional edge}.
\item A 2-cycle of $\iota$ consisting of ordinary flags is called an \textbf{ordinary edge}. A 2-cycle of $\iota$ contained in a vertex is a \textbf{loop}. A vertex that does not contain any loop is called \textbf{loop free}. A 2-cycle of $\iota$ contained in the exceptional cell is called an \textbf{exceptional loop}.
\item An \textbf{internal edge} is any 2-cycle of $\iota$. An \textbf{edge} is an internal edge, an exceptional edge, or an ordinary leg. The set of edges of $G$ (resp. internal edges) is denoted $Edge(G)$ (resp. Edge\_i(G))\$.
\item An ordinary edge $e=\{e_0,e_1\}$ is said to be \textbf{adjacent to} or \textbf{attached to} a vertex $v$ if either $e_0$, $e_1$ or both are adjacent to $v$.

\end{enumerate}
\end{defn}
\hypertarget{extra_structure}{}\subsection*{{Extra Structure}}\label{extra_structure}

There are a number of important structures that a generalized graph can possess which will be useful in using them to describe [[properads]]. The following is again from \hyperlink{HRY}{HRY}:

\begin{defn}
\label{}\hypertarget{}{}
Suppose $G$ is a generalized graph. Fix a (possibly infinite) set of \emph{colors} $\mathcal{C}$.

\begin{enumerate}%
\item A \textbf{coloring} of $G$ is a function $Flag(G)\overset{\kappa}\to \mathcal{C}$ that is constant on orbits of $\iota$ and $\pi$.


\item A \textbf{direction} for $G$ is a function $Flag(G)\overset{\delta}\to \{-1,1\}$ such that

\begin{itemize}%
\item if $\iota x\neq x$, then $\delta(\iota x)=-\delta(x)$,
\item and if $x \in F_\epsilon$, then $\delta(\pi x)=-\delta (x)$.

\end{itemize}

\item For $G$ with direction, an \textbf{input} (resp. \textbf{output}) of a vertex $v$ is a flag $x\in v$ such that $\delta(x)=1$ (resp. $\delta(x)=-1$). An \textbf{input} (resp. \textbf{output}) of $G$ is a leg $x$ such that $\delta(x)=1$ (resp. $\delta(x)=-1$). For $u\in Vt(G)\cup \{G\}$, the set of inputs (resp. outputs) of $u$ is written $in(u)$ (resp. $out(u)$).


\item A \textbf{listing} for $G$ with direction is a choice for each $u\in Vt(G)\cup \{G\}$ of a bijection of pairs of sets

\begin{displaymath}
(in(u),out(u))\overset{l_u}\to(\{1,\ldots,|in(u)|\},\{1,\ldots,|out(u)|\}).
\end{displaymath}


\end{enumerate}
\end{defn}
Thus, using these properties, we can model [[PROPs]] and [[properads]] with generalized graphs. See there and [[wheeled graph]] for more.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[graph]]


\item [[quiver]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Philip Hackney]], [[Marcy Robertson]] and [[Donald Yau]]. \emph{Infinity Properads and Infinity Wheeled Properads}, Lecture Notes in Mathematics, 2147. Springer, Cham, 2015. \href{http://arxiv.org/pdf/1410.6716v2.pdf}{(arxiv version)}


\item [[Joachim Kock]]. \emph{Graphs, hypergraphs, and properads}, \href{https://arxiv.org/pdf/1407.3744v3.pdf}{arXiv:1407.3744}.



\end{itemize}


\end{document}