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\section{Notation and terminology}
\label{sec:notation}

Here we fix notation and terminology for quivers.
%\begin{definition}
%%  We call a pair $Q=(Q_0,Q_1)$
%%  a \defit{quiver}
%%  if $Q_0$ is a finite set and
%%  $Q_1$ is a subset of $Q_0\times Q_0 \setminus \Set{(v,v)|v\in Q_0}$.
%% %
%%  An element of $Q_0$ is called a \defit{vertex} of $Q$,
%%  and an element $(v,w)$ in $Q_1$ is called
%%  an \defit{edge} of $Q$ from $v$ to $w$.
%\end{definition}
%The following classes of quivers play essential roles in this paper:
%\begin{definition}
 A quiver $Q$ is said to be
 \begin{enumerate}
  \item \defit{acyclic} if there do not exist $v_0,\ldots,v_l \in Q_{0}$
	such that
	$l>1$,  $v_l=v_0$, and 
	$(v_t,v_{t+1})\in Q_1$ for $t=0,\ldots,l-1$,

  \item \defit{asymmetric} if $(w,v) \not\in Q_1$ for any $(v,w)\in Q_1$,
	%% \begin{align*}
	%%  (v,w)\in Q_1 \implies (w,v) \not\in Q_1,
	%% \end{align*}
	 and
  \item \defit{symmetric} if $(v,w)\in Q_1$ for all $(w,v) \in Q_1$.
	%% \begin{align*}
	%%  (v,w)\in Q_1 \iff (w,v) \in Q_1.
	%% \end{align*}
 \end{enumerate}
%\end{definition}
Note that a quiver may not be symmetric nor asymmetric.

The sequence $(v_0,v_1,\ldots,v_l)$
is called
a \defit{directed cycle} if
$l>1$,  $v_l=v_0$,
$(v_t,v_{t+1})\in Q_1$ for $t=0,\ldots,l-1$,
and
 $v_{i}\neq v_{j}$ for any pair $(i,j)$ with $0\leq i<j<l$. 
By definition, a quiver is acyclic if and only if it does not contain a
directed cycle.


%\begin{definition}
% We call an undirected graph obtained in the following manner
% the \defit{underlying graph} of $Q$:
 We define the \defit{underlying graph} of a quiver $Q$
 to be
 the (undirected) graph obtained from $Q$ by using all vertices of $Q$ and
 by replacing all directed edges of $Q$ with undirected edges.
%\end{definition}
Underlying graphs may have multiple edges.
The underlying graph of $Q$ is simple
if and only if $Q$ is asymmetric.

In order to describe faces of directed edge polytopes, we need subquivers.
%\begin{definition}
 A quiver $R$ is called a \defit{subquiver} of $Q$
 if $R_0\subset Q_0$ and  $R_1\subset Q_1$.
% We write $R\subset Q$ to denote
% that $R$ is a subquiver of $Q$.
 We say that a subquiver $R$ of $Q$ is
 \begin{enumerate}
  \item \defit{proper}
	if $R_1$ is a proper subset of $Q_1$,
  \item \defit{full}
	if $R_1=\Set{(v,w)\in Q_1| v,w\in R_0}$,
	and
  \item \defit{lluf} if $R_0=Q_0$.
 \end{enumerate}
%\end{definition} 

We make use of (undirected) walks to define connectivity of quivers.
%\begin{definition}
 Let $Q$ be a quiver.
 An \defit{undirected walk} from $v_0$ to $v_l$ in $Q$
 is  a sequence $(v_0,v_1,\ldots,v_l)$ of vertices in $Q$
 such that $(v_t,v_{t+1})\in Q_1$ or $(v_{t+1},v_t)\in Q_1$ for all $t$.
 An undirected walk $(v_0,v_1,\ldots,v_l)$ is called
 \begin{enumerate}
%  \item an \defit{undirected path} if all vertices are distinct,
  \item \defit{closed} if $v_0=v_l$, and
  \item an \defit{undirected cycle} if it is closed and
	$v_i\neq v_j$ for any pair $(i,j)$ with $0\leq i<j<l$. 
 \end{enumerate}
%\end{definition}
%In fact, the number of connected components is 
%\begin{definition}
 We say a quiver $Q$ is \defit{connected} if, for any pair $(v,w)$ of
 vertices of $Q$, there exists an undirected walk from $v$ to $w$ .
%
 A connected maximal subquiver of $Q$
 is called a \defit{connected component} of $Q$. The set of all
 connected components of $Q$ is denoted by $\pi_{0}(Q)$. 
 The number $\numof{Q_{0}}-\numof{\pi_{0}(Q)}$ is denoted by $c(Q)$ and
 is called the \emph{coconnectivity} of $Q$.
%\end{definition}
 Note that a quiver is connected if and only if the underlying graph is
 connected. 
We say that a luff subquiver $R$ of $Q$ is
\defit{component-wise full}
if each connected component $C$ of $R$ is a full subquiver of $Q$.

%% %\begin{definition}
%%  For a connected quiver $Q$, we call a lluf asymmetric subquiver 
%%  $R$ of $Q$ a \defit{spanning polytree} in $Q$ if the underlying graph
%%  of $R$ is a tree, 
%%  i.e.\ an acyclic connected simple undirected graph.
%%  For a quiver $Q$,
%%  we call a lluf subquiver $R$ of $Q$ a \defit{spanning polyforest} in $Q$
%%  if each connected component of $R$
%%  is a spanning polytree in some connected component of $Q$.
%% %\end{definition}


%% We also need directed walks and cycles.
%% %\begin{definition}
%%  A \defit{directed walk} from $v_0$ to $v_l$ in a quiver $Q$
%%  is a sequence $(v_0,v_1,\ldots,v_l)$ of vertices in $Q$
%%  such that $(v_t,v_{t+1})\in Q_1$ for all $t$.
%%  A directed walk $(v_0,v_1,\ldots,v_l)$ is called
%%  a \defit{directed cycle} if $v_{0}=v_{l}$ and
%%  $v_{i}\neq v_{j}$ for any pair $(i,j)$ with $0\leq i<j<l$. 
%% %\end{definition}
%% By definition, a quiver is acyclic if and only if it does not contain a
%% directed cycle.


