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\begin{document}
\title{The Prism of the Acyclic Orientation Graph is Hamiltonian}
\author{{\sc Gara Pruesse}
\thanks{
This material is based upon work supported by the
National Science Foundation under Grant No. NSF OSR-9350540.}\\
{\small \em Department of Computer Science and Electrical Engineering}\\
{\small \em University of Vermont, Burlington, VT, 05405-0156, USA} \\
{\small \em email: \tt gara@cs.uvm.edu} \\
\vspace{1mm}
\and
{\sc Frank Ruskey}
\thanks{Research supported in part by the Natural Sciences and
Engineering Research Council of Canada under Grant A3379.} \\
{\small \em Department of Computer Science} \\
{\small \em University of Victoria, Victoria, B.C., V8W 3P6, Canada} \\
{\small \em email: \tt fruskey@csr.uvic.ca} \\
}
\date{\small Submitted: December 20, 1994; Accepted: March 13, 1995.}
\maketitle
\begin{abstract}
Every connected simple graph $G$ has an acyclic orientation.
Define a graph ${AO}(G)$ whose vertices are the acyclic orientations
of $G$ and whose edges join orientations that differ by reversing
the direction of a single edge.
It was known previously that
${AO}(G)$ is connected but not necessarily Hamiltonian.
However, Squire \cite{Squire}
proved that the square ${AO}(G)^2$ is Hamiltonian.
We prove the slightly stronger result that the prism
${AO}(G) \times e$ is Hamiltonian.
If $G$ is a mixed graph (some edges directed, but not necessarily all),
then ${AO}(G)$ can be defined as before.
The graph ${AO}(G)$ is again
connected but we give examples showing that the prism is not
necessarily Hamiltonian.
\end{abstract}
\section{Introduction}
Every connected simple graph $G$ has an acyclic orientation.
Define a graph ${AO}(G)$ whose vertices are the acyclic orientations
of $G$ and whose edges join orientations that differ by reversing
the direction of a single edge.
Squire, Savage, and West \cite{SquiSavaWest} show that ${AO}(G)$ is
connected and bipartite and discuss the Hamiltonicity of
${AO}(G)$ for well-known types of graphs such as trees, cycles,
wheels, and ladders.
Squire \cite{Squire} proved that the square ${AO}(G)^2$ is Hamiltonian.
Our main goal in this paper is to prove the slightly stronger result
that the prism ${AO}(G) \times e$ is Hamiltonian.
If the prism of a bipartite graph is Hamiltonian, then the square of
the graph is Hamiltonian.
If $G$ is a mixed graph (some edges directed, but not necessarily all),
then ${AO}(G)$ can be defined as before.
Mixed graphs are interesting because special cases of them reduce
to known problems and results.
For example, there is a one-to-one correspondence between the
mixed complete graphs (every edge
is present, either directed or not) and directed acyclic graphs.
If $G$ is a mixed complete graph, then
a topological sort of the maximum directed subgraph of $G$
corresponds to an acyclic orientation of $G$.
Pruesse and Ruskey \cite{sicomp} showed that the prism
of ${AO}(G)$ is Hamiltonian in the case of mixed complete graphs.
It is therefore natural to ask whether the prism is Hamiltonian for
all mixed graphs, since it is true for the cases of no fixed edges
(this paper) and when all edges are present (\cite{sicomp}).
The graph ${AO}(G)$ is again connected and bipartite;
however, the final section of this paper describes an infinite class
whose prisms are not Hamiltonian.
\medskip
We now define a few terms and introduce some notation.
A \emph{poset} ${\cal P} = (S,R)$ is a reflexive, anti-symmetric, transitive
relation $R({\cal P})$ on a set $S({\cal P})$.
We let ${\cal P} - x$ denote the poset on the set $S({\cal P})\setminus x$,
together with all the relations of $R({\cal P})$ that do not involve $x$.
An \emph{ideal} of a poset ${\cal P}$ is a set $I \subseteq S({\cal P})$
for which $y \in I$ and $x \prec y$ imply that $x \in I$.
The \emph{ideal graph} $J({\cal P})$ has vertices which are the ideals
of ${\cal P}$ and edges connecting those ideals that differ by
one element; in other words, it is the Hasse diagram of the lattice
of ideals of a poset, regarded as an undirected graph.
In order to simplify notation we will usually write $x$ for the singleton
set $\{x\}$ and use juxtaposition to indicate union of sets.
E.g., if $E$ is a set and $x$ and element, $Ex$ denotes $E \cup \{x\}$.
For a graph $G$, the \emph{square} of $G$, denoted $G^2$, is the
graph on the same vertex set as $G$ and where $(u,v) \in E(G^2)$
if and only if the distance between $u$ and $v$ in $G$ is 1 or 2.
The \emph{prism} of a graph $G$, denoted $G \times e$, is obtained by taking
two copies of $G$ and adding edges between each corresponding pair
of vertices in the two copies.
We think of the vertices in the prism as being signed; with each vertex
$v$ in $G$ there is associated, in $G \times e$, a plus vertex $+v$
and a minus vertex $-v$.
All of the plus vertices occur in one copy of $G$, the minus vertices
in the other copy of $G$.
The following lemma, from \cite{sicomp}, relates the hamitonicity
of prisms and squares for bipartite graphs.
\begin{lemma}
If $G$ is bipartite and $G \times e$ is hamiltonian, then $G^2$
is hamiltonian.
\label{lemma:prism-square}
\end{lemma}
\medskip
Pruesse and Ruskey \cite{order} proved that the prism of the ideal graph
is Hamiltonian.
\begin{theorem}
For every poset ${\cal P}$
there is a Hamilton path in $J({\cal P}) \times e$ from
$+\emptyset$ to $-\emptyset$.
\label{thm:ideals}
\end{theorem}
It also proves useful to have a Hamilton path in the
prism that ends at $S({\cal P})$.
\begin{theorem}
For every poset ${\cal P}$ there is a
Hamilton path in $J({\cal P}) \times e$ of the form
$+\emptyset,-\emptyset,\ldots,\pm E$, where $E = S({\cal P})$
and the sign of the final vertex depends on the parity of $|E|$.
\label{thm:top}
\end{theorem}
\noindent
{\bf Proof:}
Let $n = |E|$.
We argue by induction on $n$.
If $n = 1$, then $+\emptyset,-\emptyset,-x,+x$ satisfies
the conditions of the theorem.
Note that this path also satisfies the following property:
Between $\pm I$ and $\mp I$ only subsets of $I$ occur.
We argue that this property, called the \emph{subset property},
is maintained inductively.
Let $n > 1$ and let $x$ be a minimal element of ${\cal P}$.
Inductively, there is a Hamilton path $H = +\emptyset,-\emptyset,\ldots,\pm E$
in ${\cal P}-x$ which has the subset property.
The path $P = -x,+x,\ldots,\mp Ex$ in ${\cal P}$, obtained by adding $x$ to
each vertex of $H$ and reversing all signs, includes all ideals
that contain $x$.
We now splice in those ideals that don't contain $x$.
First, replace the edge $[-x,+x]$ with the path
$+\emptyset,-\emptyset,-x,+x$.
Now, starting at the vertex following $+x$,
find the first vertex $\pm Ix$
for which $I$ is an ideal of ${\cal P}$, and replace
the subpath
$\pm Ix = \alpha_1 x,\alpha_2 x,\ldots,\alpha_p x = \mp Ix$
in $P$ by
\begin{equation}
\alpha_1 x,\alpha_1,\alpha_p,\ldots,\alpha_2,\alpha_2 x,\ldots,\alpha_p x.
\label{eq:path}
\end{equation}
Observe that, by the subset property, $\alpha_i$ for $i = 1,2,\ldots,p$ are
all ideals of ${\cal P}$.
Repeat the above process, starting at the vertex following $\mp Ix$,
and so on, and call the resulting Hamilton path $H'$.
Note that if $H$ has the subset property, then so does $H'$,
because each subpath (\ref{eq:path}) has the subset property.
\hfill $\Box$
\medskip
These two theorems will be used in the proof of the next section.
\section{The Prism is Hamiltonian}
\begin{theorem}
If $G$ is a simple graph, then ${AO}(G) \times e$ is Hamiltonian.
\end{theorem}
\noindent
{\bf Proof:}
Our proof is by induction on $n = |V(G)|$. If $n = 1$, then
${AO}(G) \times e$ is a single edge, which we regard as being Hamiltonian,
as it has a Hamilton path whose end points are adjacent.
For $n > 1$, pick an arbitrary vertex $v$ and remove it from $G$.
Inductively, there is a Hamiltonian cycle $H$ in
$({AO}(G-v)) \times e$. Consider a generic acyclic orientation of $G-v$, call
it $\Gamma$, and define a poset ${\cal P}_\Gamma$ as follows.
The elements, $E = S({\cal P}_\Gamma)$, of ${\cal P}_\Gamma$ are
$N(v)$, the vertices in the open neighborhood of $v$.
Note that $E$ depends only on $v$, and not otherwise on $\Gamma$.
The relations of ${\cal P}_\Gamma$ are
\[
R({\cal P}_\Gamma) = \{ x \prec y \mid \mbox{ there is a directed path from
$y$ to $x$ in $\Gamma$} \}.
\]
Note that $\Gamma$ is a partial acyclic orientation of $G$; $\Gamma$
orients all edges not incident with $v$. Orienting the
remaining edges of $G$ is called \emph{extending} $\Gamma$.
The key observation is that there is a one-to-one correspondence
between the ideals of ${\cal P}_\Gamma$ and the acyclic orientations of
$G$ that extend $\Gamma$. Let $I$ be an ideal of ${\cal P}_\Gamma$, and
orient all edges of the form $[u,v]$ either $u \leftarrow v$ or
$u \rightarrow v$ depending on whether $u \in I$ or $u \not\in I$,
respectively; this yields an orientation of $G$ that extends $\Gamma$, and
which we will denote $\langle I \rangle_\Gamma$.
We show that for each ideal $I$ of ${\cal P}_\Gamma$,
$\langle I \rangle_\Gamma$ is acyclic. Since $\Gamma$ is acyclic,
it is sufficient to show that $v$ is not in any cycle.
In the trivial cases $\langle \emptyset \rangle_\Gamma$ means that
all edges are directed into $v$ and $\langle E \rangle_\Gamma$
means that all edges are directed away from $v$, so acyclicity
holds.
In any case,
if $ u_1 u_2 \cdots u_k $ is any directed path in $\Gamma$ with
endpoints in $N(v)$, then $u_1 \in I$ implies $u_k \in I$, and hence
$v$ has a directed edge to $u_1$ only if it also has one to $u_k$.
It follows that $v$ is not in any directed cycle in $\langle I \rangle_\Gamma$.
The argument in the other direction (i.e., that each acyclic orientation
of $\Gamma$ is $\langle I \rangle_\Gamma$ for some $I \in J({\cal P}_\Gamma)$),
is also straightforward.
The addition or removal of a single
element from an ideal to obtain another ideal corresponds to
changing the orientation of an edge. Thus by traversing the ideals
of $J({\cal P}_\Gamma) \times e$ we are following a path in a certain
subgraph of ${AO}(G) \times e$, namely that subgraph induced by all extensions of
$\Gamma$.
We observe that $H$ can be partitioned into
\begin{itemize}
\item
{\bf [positive pairs]}
edges of the form $[+\Gamma,+\Gamma']$, and
\item
{\bf [mates]}
edges of the form $[+\Gamma,-\Gamma]$, and
\item
{\bf [loners]}
vertices of the form $-\Gamma$.
\end{itemize}
One way to obtain such a partitioning is to treat $H$ as a sequence from
which we ``shell'' (remove from the front) pairs and singletons of
vertices. First rotate $H$ as necessary so that the last vertex in $H$ is
negative. Now begin shelling: if the sequence begins with a positive
vertex, shell a pair; if it begins with a negative vertex, shell a ``loner''.
Since the sequence ends with a negative vertex, every element will be shelled,
and hence $H$ is partitioned into positive pairs, mates, and loners.
Our task now is to replace each orientation $\Gamma$ of $(G-v)$ on $H$
with its extensions in $G$.
In the case of ``mates'',
replace the edge $[+\Gamma,-\Gamma]$
by the path $+\langle \emptyset \rangle_\Gamma,\ldots,
-\langle \emptyset \rangle_\Gamma$, where
$+\emptyset,\ldots,-\emptyset$ is the Hamilton cycle in
$J({\cal P}_\Gamma) \times e$ guaranteed by Theorem \ref{thm:ideals}.
Otherwise, $+\Gamma$ occurs in a ``positive pair'', and $-\Gamma$
occurs as a ``loner''. In the path
$-\langle \emptyset \rangle_\Gamma,
+\langle \emptyset \rangle_\Gamma,
\ldots,\mp \langle E\rangle_\Gamma$
in $J({\cal P}_\Gamma) \times e$
(whose existence is guaranteed by Theorem \ref{thm:top}),
we remove the first edge,
obtaining 2 paths, $-\langle \emptyset\rangle_\Gamma$ and
$+\langle \emptyset \rangle_\Gamma,
\ldots,\mp \langle E\rangle_\Gamma$.
For each positive pair,
replace the edge $[+\Gamma,+\Gamma']$ by the path
\[
+\langle \emptyset \rangle_\Gamma, \ldots,
+\langle E \rangle_\Gamma,+\langle E \rangle_{\Gamma'},
\ldots,+\langle \emptyset \rangle_{\Gamma'},
\]
if $|E|$ is even, or
\[
+\langle \emptyset \rangle_\Gamma, \ldots,
-\langle E \rangle_\Gamma,-\langle E \rangle_{\Gamma'},
\ldots,+\langle \emptyset \rangle_{\Gamma'},
\]
if $|E|$ is odd.
In the case of ``loners'', replace the vertex $-\Gamma$ with the
vertex $- \langle \emptyset \rangle_\Gamma$.
After doing the above substitutions $H$ has been transformed into
a Hamilton cycle in ${AO}(G) \times e$.
\hfill $\Box$
\medskip
The above proof does not extend to the case of mixed graphs.
The proof uses the fact that directing all $v$'s edges away from
$v$ extends $\Gamma$ acyclicly. However, this is no longer the
case when some edges incident with $v$ have fixed orientation;
indeed, the cardinality of the maximum set of edges that can
be directed away from $v$ can change greatly by changing the
orientation of one edge of $\Gamma$.
\section{The Example}
Let $K_{n,n}$ denote the complete bipartite graph
and $M$ be a perfect matching in that graph.
Define the mixed graph $G_n$ to be $K_{n,n}$ with all
edges directed from one partite set to the other,
except for the edges in the matching $M$,
which are left undirected.
\begin{lemma}
For all $n \ge 1$, we have ${AO}(G_n) = K_{1,n}$.
\label{lemma:counter}
\end{lemma}
\noindent{\bf Proof:}
Call the two partite sets $A$ and $B$ and assume that the
directed edges are all of the form $a \rightarrow b$,
where $a \in A$ and $b \in B$.
Note that any two edges of $M$ directed
from $B$ to $A$ will induce a 4-cycle.
Thus at most one of the edges in $M$ can be directed from
$B$ to $A$ and still preserve the acyclic property.
Each orientation with an edge from $B$ to
$A$ is only adjacent to the orientation with no edges going from
$B$ to $A$.
\hfill $\Box$
\medskip
For $n \ge 3$, the graph $K_{1,n} \times e$ is not Hamiltonian since
the removal of the two central vertices results in $n$ connected
components.
Since the square of $K_{1,n}$ is Hamiltonian, we can still ask whether
there is a mixed graph $G$ such that ${AO}(G)^2$ is not
Hamiltonian.
Squire \cite{Squire} has extended our Lemma \ref{lemma:counter} to
give such an example.
An open problem is to find an algorithm for generating acyclic orientations
in time proportional to the number of orientations.
If this problem is pursued based on the proof of this paper then a
first step is finding an algorithm for generating the ideals of
a poset in time proportional to the number of ideals, which is
an open problem.
{\small
\begin{thebibliography}{99}
\bibitem{sicomp}
G. Pruesse and F. Ruskey,
\emph{Generating Linear Extensions Fast},
SIAM J. Computing, 23 (1994) 373-386.
\bibitem{order} G. Pruesse and F. Ruskey,
\emph{Gray Codes from Antimatroids},
Order, 10 (1993) 239-252.
\bibitem{Squire} M. Squire,
\emph{Two New Gray Codes for Acyclic Orientations},
94-14, Dept. Computer Science, N.C. State Univ, 1994.
\bibitem{SquiSavaWest} C. Savage, M. Squire, and D. West,
\emph{Gray Code results for Acyclic Orientations},
Congressus Numerantium, 96 (1993) 185-204.
\end{thebibliography}
}
\end{document}