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\begin{center}
\vskip 1cm{\LARGE\bf 
Introduction to the ``Prisoners and Guards'' \\
\vskip .1in
Game
}
\vskip 1cm
\large
Timothy Howard and Eugen J. Ionascu\\
Department of Mathematics\\
Columbus State University\\
4225 University Avenue \\
Columbus, GA 31907 \\
USA \\
\href{mailto:thoward@colstate.edu}{\tt thoward@colstate.edu} \\
\href{mailto:ionascu_eugen@colstate.edu}{\tt ionascu\_eugen@colstate.edu} \\
\ \\
David  Woolbright \\
TSYS Department of Computer Science\\
Columbus State University\\
4225 University Avenue \\
Columbus, GA 31907 \\
USA \\
\href{mailto:woolbright@colstate.edu}{\tt woolbright@colstate.edu} \\
\end{center}

\vskip .2 in

\begin{abstract}
We study the half-dependent problem for the king graph
$K_n$. We give proofs to establish the
values $h(K_n)$ for $n\in \{1,2,3,4,5,6\}$ and an
upper bound for $h(K_n)$ in general. These proofs are independent of
computer assisted results. Also, we introduce a two-player game
whose winning strategy is tightly related with the values $h(K_n)$.
This strategy is analyzed here for $n=3$ and some facts are given
for the case $n=4$. Although the rules of the game are very simple,
the winning strategy is highly complex even for $n=4$.
\end{abstract}

\newtheorem{theorem}{Theorem}
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%\newtheorem{theorem}{Theorem}[section]
%\newtheorem{proposition}{Proposition}[section]
%\newtheorem{corollary}{Corollary}[section]
%\newtheorem{lemma}{Lemma}[section]
%\newtheorem{definition}{Definition}[section]
%\newtheorem{problem}{Problem}[section]



\section{\label{intro}Introduction}

Suppose that you are competing in a two-player game in which you and
your opponent attempt to pack as many ``prisoners'' as possible on
the squares of an $n\times n$ checkerboard; each prisoner has to be
``protected'' by an appropriate number of guards.  Initially, the
board is covered entirely with guards.  The players -- designated as
``red'' and ``blue'', with red going first -- take turns adjusting
the board configuration using one of the following rules in each
turn:
\begin{itemize}
\item [I.]  Replace one guard with a prisoner of the player's
color.
\item [II.]  Replace one prisoner of either color with a guard
and replace two other guards with prisoners of the player's color.
\end{itemize}

\noindent That is, each player takes a turn increasing the total
number of prisoners by one.  We require that, at every stage of the
game, {\it the number of guards adjacent to a given prisoner is not
less than the number of prisoners lying adjacent to that prisoner}.
%each prisoner lies adjacent to at least as many guards as the number
%of the other prisoners adjacent it}
 The squares {\it adjacent} to a given square are those squares,
situated directly above, below, to the left, to the right, or
diagonal to the square in question. An arrangement of prisoners and
guards that satisfies this requirement and has exactly one occupant
per square is called a {\it valid board}. The game ends when neither
player can further adjust the board using rules I and II while
maintaining a valid board.  The player whose color represents more
prisoners is the winner.  This is the game of Prisoners and Guards
-- a game that can be played and analyzed without extensive
knowledge of mathematics.
%Figure~\ref{blackmon-rd-players} depicts students from Blackmon Road
%Middle School, in Columbus, Georgia, trying their hands at the game.
We invite the reader to play the game online by running the Java
Applet available at  \href{http://csc.colstate.edu/woolbright/}
{\underline{http://csc.colstate.edu/woolbright/}}.
%\begin{figure}[h]
%\centering
%\includegraphics[scale=0.25]{blackmon-rd-players.eps}
%\caption{Blackmon Road Middle School Students Play PvG}
%\label{blackmon-rd-players}
%\end{figure}

The guards in this game are related to the half domination set in
the king's graph as introduced in  a paper by Dunbar, Hoffman,
Laskar, and Markus \cite{dhlm}. Similar domination problems
have been studied by Bode, Harborth, and Harporth \cite{bhh},
Dutton, Lee,  and Brigham \cite{dlb}, Watkins, Ricci, and
McVeigh \cite{wrm} and many others. Two concepts in the
domination literature very closely related to ours are those of {\it
unfriendly partition} \cite{amp}  and {\it global offensive
alliance} \cite{sr}. At the end of Section~\ref{deficiency} we show
how some of our estimates relate to a general result in \cite{sr}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The Prisoners and Guards game originated as a puzzle created by the
third author with a focus on minimizing the size of the dominating
set (the guards).

In the two-player game, one fundamental question that naturally
arises is ``How do we decide when the game is over?''  The short
answer is that the game is over when the board configuration has
reached a maximal state.  A valid board to which no adjustments can
be made to increase the total number of prisoners is called a {\it
maximal} board. One can also define a {\it maxi\underline{mum}}
configuration as being an arrangement of prisoners and guards that
has the greatest number of prisoners of all valid boards. Clearly,
every maximum configuration is a maximal configuration. For $n\in
\left\{1,2,3\right\}$ all maximal boards are also maximum
configurations.  We will see examples of $4\times 4$ boards that are
maximal but do not contain maximum configurations.  Let $P(n)$
denote the number of prisoners in a maximum configuration. Finding
the exact values of the sequence $\left\{P(n)\right\}_{n=1}^\infty$
will help us determine when to end the game.  It also proves an
interesting avenue for exploration on its own.

Since the lone square on a $1\times 1$ board has no adjacent
squares, we can place a prisoner in it and be sure that there are at
least as many guards as prisoners lying in all adjacent squares --
none. Therefore, we have $P(1)=1$.  By exhaustively checking all
sixteen $2\times 2$ cases, we find eleven valid boards, each having
zero, one, or two prisoners.  Thus, $P(2)=2$.  We analyze the cases
$n=3$, $4$, $5$, and $6$ in sections \ref{3x3and4x4cases} and
\ref{5x5and6x6cases}.  Exact values for $P(n)$, $n\in
\left\{7,8,9,10,11\right\}$, are $P(7)=28$, $P(8)=39$, $P(9)=49$,
$P(10)=59$ and $P(11)=73$, and for the corresponding maximum
configuration one can consult a paper by Ionascu, Pritikin, and
Wright \cite{ipw}, who employ among various methods binary
linear programming techniques in the study of $P(n)$.  However, in
this paper we include proofs for the above mentioned cases which are
independent of computer searches and short enough to be read with
ease. In Section~\ref{deficiency} we also obtain an upper bound on
$P(n)$; this is about the best that we can say for $n\ge 12$. In
Section~\ref{gridgraphs} we show how this technique from
Section~\ref{deficiency} can be used in order to completely answer
the best density 1/2-domination problem in grid type graphs.

\section{\label{3x3and4x4cases} The game analysis for $n=3$ and $n=4$}

Playing Prisoners and Guards on a $1\times 1$ board or on a $2\times
2$ board is not all that interesting.  When we increase the board
size slightly and consider the game on a $3\times 3$ board, strategy
becomes more of a factor.  We will see that the arrangement in
Figure~\ref{3x3maximumboard} is a maximum configuration, as we
establish in Theorem~\ref{3x3maxtheorem}.  We use the diamond to
represent prisoners and blank squares represent guards.

\begin{figure}[ht]
\begin{center}
%\begin{equation}\label{eq1}
\begin{tabular}{|r|r|r|}
\hline
\b &amp; \ \  &amp; \b \\
\hline
\b &amp; \ \  &amp; \b \\
\hline
\b &amp;\ \  &amp; \b \\
\hline
\end{tabular}.
%\end{equation}
\caption{Maximum 3x3 Board} \label{3x3maximumboard}
\end{center}
\end{figure}

\noindent In fact, this is the only maximal arrangement (up to a
rotation).  Let us observe that a maximal board permits no
adjustments using either Rule I or Rule II.  First we consider
arrangements that are maximal with respect to Rule I (i.e. one
cannot simply add more prisoners in the existing configuration).

Perhaps it is not difficult to convince oneself that any valid board
having zero, one, or two prisoners can be adjusted using Rule I;
after factoring out rotations and  reflections, there are ten unique
cases to check.  Therefore, a maximal board must have at least three
prisoners.  Figure~\ref{3x3maximalboardswrtruleI} depicts all valid
boards (up to rotations and reflections) that contain three, four,
or five prisoners and are maximal with respect to Rule I.

\begin{figure}[ht]
\begin{center}
\begin{tabular}{|r|r|r|}
\hline
\ \ &amp; \b \  &amp; \ \ \\
\hline
\ \ &amp; \b \  &amp; \ \ \\
\hline
\ \ &amp;\b \  &amp; \ \ \\
\hline
\end{tabular} \ \
\begin{tabular}{|r|r|r|}
\hline
\  &amp; \b \  &amp; \ \\
\hline
\b &amp; \ \  &amp; \b \\
\hline
\ &amp;\b \  &amp; \ \\
\hline
\end{tabular} \ \
\begin{tabular}{|r|r|r|}
\hline
\ \ &amp; \b \  &amp; \ \ \\
\hline
\ \ &amp; \b \  &amp; \ \ \\
\hline
\b \ &amp;\ \  &amp; \b \ \\
\hline
\end{tabular}\ \
\begin{tabular}{|r|r|r|}
\hline
\ \ &amp; \b \  &amp; \ \ \\
\hline
\ \ &amp; \b \  &amp; \b \ \\
\hline
\b \ &amp;\ \  &amp; \ \ \\
\hline
\end{tabular}\ \
\begin{tabular}{|r|r|r|}
\hline
\b &amp; \ \  &amp; \b \\
\hline
\ &amp; \b \  &amp; \ \\
\hline
\b &amp;\ \  &amp; \b \\
\hline
\end{tabular}\ \
\begin{tabular}{|r|r|r|}
\hline
\ &amp; \b \  &amp; \ \\
\hline
\b &amp; \ \  &amp; \b \\
\hline
\b &amp;\ \  &amp; \b \\
\hline
\end{tabular}\ \ \
\begin{tabular}{|r|r|r|}
\hline
\b \ &amp; \b \  &amp; \ \ \\
\hline
\ \ &amp; \ \  &amp; \b \ \\
\hline
\b \ &amp;\ \  &amp; \b \ \\
\hline
\end{tabular}
\caption{$3\times 3$ Boards that are Maximal w.r.t. Rule I}
\label{3x3maximalboardswrtruleI}
\end{center}
\end{figure}

Each one of these configurations can be adjusted to match the
arrangement in Figure~\ref{3x3maximumboard} by using Rule II (and
one or more adjustments using Rules I and II in some of the cases).
It follows that a maximal $3\times 3$ board must contain at least
six prisoners.  We record this fact in the following lemma.

\begin{lemma} Any maximal $3\times 3$  board has at least six prisoners.
\label{3x3sixprisoners}\end{lemma}

With Figure~\ref{3x3maximumboard}, we see that a valid $3\times 3$
configuration can have six prisoners.  Does there exist a valid
configuration with more than six prisoners? Suppose that a $3\times
3$ board arrangement contains seven prisoners (and two guards).
Since there are four non-corner edge squares, a prisoner must occupy
at least one of them.  Since this prisoner lies adjacent to at most
two guards, the board cannot be valid.  These observations,
Lemma~\ref{3x3sixprisoners}, and the fact that
Figure~\ref{3x3maximumboard} depicts a valid configuration with six
prisoners lead us to the following conclusion.

\begin{theorem} A maximum $3\times 3$ board contains six prisoners, {\it i.e.}
$P(3)=6$. \label{3x3maxtheorem}\end{theorem}

As a matter of fact, the configuration shown in
Figure~\ref{3x3maximumboard} is the \underline{only} maximal
$3\times 3$ board (up to a rotation of the board).  From this we
learn that the second player has a good chance to win by using Rule
II all of the time. The first player may force a tie if she can lead
the board configuration in such a manner that will require her
opponent to use Rule I. In fact, this is manageable if she plays
into the pattern in Figure~\ref{3x3maximumboard}, forcing the second
player to use Rule I in the last step and so the final board will
have an equal number of prisoners of each color.

We now turn our attention to $4\times 4$ boards.  This board size
proves interesting because there exist many maximal arrangements
that are not maximum configurations.  We include some maximal
arrangements with eight prisoners in Figure~\ref{maximal4x4boards}
that we found but there may be others.

\begin{figure}[ht]
\begin{center}
\begin{tabular}{|r|r|r|r|}
\hline
\    \       &amp; \b   &amp; \b &amp; \   \\
\hline
\b   &amp; \  \  &amp; \  \  &amp; \b \ \\
\hline
\b &amp; \  \  &amp; \   &amp;    \b  \     \\
\hline
\ &amp; \b \  &amp; \b   &amp; \  \  \   \\
\hline\end{tabular}\ \ \
\begin{tabular}{|r|r|r|r|}
\hline
\b    \       &amp; \   &amp; \ &amp; \b  \\
\hline
\   &amp; \b  \  &amp; \b  \  &amp; \ \ \\
\hline
\ &amp; \b  \  &amp; \b   &amp;    \ \     \\
\hline
\b &amp; \ \  &amp; \   &amp; \b  \  \   \\
\hline\end{tabular}\ \ \
\begin{tabular}{|r|r|r|r|}
\hline
\b   \ &amp; \  \ &amp; \ \  &amp; \b   \\
\hline
\b  \ &amp; \  \  &amp; \  \  &amp; \b \ \\
\hline
\b \ &amp; \  \  &amp; \  \  &amp;    \b  \     \\
\hline
\b \ &amp; \ \  &amp; \ \  &amp; \b  \    \\
\hline\end{tabular}\ \ \
\begin{tabular}{|r|r|r|r|}
\hline
\b   \ &amp; \  \ &amp; \ \  &amp; \b   \\
\hline
\  \ &amp; \b  \  &amp; \b  \  &amp; \ \ \\
\hline
\b \ &amp; \  \  &amp; \  \  &amp;    \b  \     \\
\hline
\ \ &amp; \b \  &amp; \b \  &amp; \  \    \\
\hline\end{tabular}

\vspace{0.1 in}
\begin{tabular}{|r|r|r|r|}
\hline
\   \ &amp; \b  \ &amp; \b \  &amp; \   \\
\hline
\b  \ &amp; \  \  &amp; \  \  &amp; \b \ \\
\hline
\b \ &amp; \  \  &amp; \  \  &amp;    \b  \     \\
\hline
\b \ &amp; \ \  &amp; \ \  &amp; \b  \    \\
\hline\end{tabular}\ \
\begin{tabular}{|r|r|r|r|}
\hline
\b   \       &amp; \   &amp; \b &amp; \   \\
\hline
\  &amp; \b  \  &amp; \  \  &amp; \b \ \\
\hline
\b &amp; \  \  &amp; \b   &amp;    \  \     \\
\hline
\ &amp; \b \  &amp; \   &amp; \b  \  \   \\
\hline\end{tabular}\ \
\begin{tabular}{|r|r|r|r|}
\hline
\b   \ &amp; \  \ &amp; \ \  &amp; \b   \\
\hline
\  \ &amp; \b  \  &amp; \b  \  &amp; \ \ \\
\hline
\b \ &amp; \  \  &amp; \b  \  &amp;    \  \     \\
\hline
\ \ &amp; \b \  &amp; \ \  &amp; \b  \    \\
\hline\end{tabular} \ \
\begin{tabular}{|r|r|r|r|}
\hline
\b   \ &amp; \  \ &amp; \b \  &amp; \b   \\
\hline
\b  \ &amp; \  \  &amp; \  \  &amp; \ \ \\
\hline
\ \ &amp; \  \  &amp; \  \  &amp;    \b  \     \\
\hline
\b \ &amp; \b \  &amp; \ \  &amp; \b  \    \\
\hline\end{tabular} \caption{Some Maximal $4\times 4$ Boards}
\label{maximal4x4boards}
\end{center}
\end{figure}

If we factor out rotations and reflections of the board, there are
three maximum arrangements as depicted in
Figure~\ref{4x4maximumboards}. We obtained these via an exhaustive
search of all $4\times 4$ valid boards and verified that there are
no other equivalence classes.

\begin{figure}[ht]
\begin{center}
\begin{tabular}{|r|r|r|r|}
\hline
\ \       &amp; \b   &amp; \b &amp; \b \\
\hline
\b   &amp; \ \  &amp; \ \  &amp; \ \ \\
\hline
\b &amp; \ \  &amp; \b   &amp; \b \\
\hline
\b &amp; \ \  &amp; \b   &amp; \ \  \\
\hline
\end{tabular},\  \  \
\begin{tabular}{|r|r|r|r|}
\hline
\b  &amp; \b   &amp; \b &amp; \b \\
\hline
\ \ &amp; \ \  &amp; \ \  &amp; \ \  \\
\hline
\b &amp; \ \   &amp; \b  &amp; \b \\
\hline
\b &amp; \ \   &amp; \b   &amp; \ \  \\
\hline
\end{tabular},\  \   \
\begin{tabular}{|r|r|r|r|}
\hline
\b       &amp; \ \   &amp; \b &amp; \b \\
\hline
\ \       &amp; \b        &amp; \ \  &amp; \ \             \\
\hline
\b &amp; \ \         &amp; \b  &amp; \b \\
\hline
\b &amp; \ \         &amp; \b  &amp; \ \  \\
\hline
\end{tabular}.
%\end{equation}
\caption{Maximum $4\times 4$ Boards} \label{4x4maximumboards}
\end{center}
\end{figure}

To prove something about maximum $4\times 4$ board configurations,
it helps to dissect the board and consider what can happen in the
vicinity of the corner squares.  Suppose that we have a $2\times 2$
block $C$ of squares situated in one corner of the board.  If the
corner square within $C$ does not contain a guard, then it contains
a prisoner.  If the latter is the case, then $C$ must contain at
least two guards.  Thus we have established the following fact.

\begin{lemma}\label{twobytwo}
If $C$ is a $2\times 2$ corner block within a valid board ($n \ge
2$), then it must contain at least one guard.
\end{lemma}

With this in mind, we are equipped to consider maximum $4\times 4$
boards by partitioning them into four $2\times 2$ corner blocks and
following through with the consequences.  This will lead us to the
conclusion of the next proposition.

\begin{proposition}\label{maximum4x4boards} $P(4)=9$.  That is, every maximum $4\times 4$
valid board has nine prisoners.
\end{proposition}

\begin{proof}
Since the configurations in
Figure~\ref{4x4maximumboards} are valid and each contains nine
prisoners, it follows that $P(4)\ge 9$. It is enough to show that
$P(4)\le 9$. We therefore assume that there exists a valid board $B$
with ten or more prisoners. By dropping prisoners if necessary, we
can say there are exactly ten.  We shall see that this leads to a
contradiction. We partition $B$ into four $2\times 2$ blocks as
indicated in Figure~\ref{blocksof4x4}(a).


\begin{figure}[ht]
\[\underset{(a)} {\begin{tabular}{|r|r||r|r|}
\hline
\ \       &amp; \    &amp; \  &amp; \  \\
\hline
\   &amp; \ \  &amp; \ \  &amp; \ \ \\
\hline \hline
\ &amp; \ \  &amp; \   &amp; \ \\
\hline
\ &amp; \ \  &amp; \   &amp; \ \  \\
\hline
\end{tabular},} \ \ \
\underset{(b)}{
\begin{tabular}{|r|r||r|r|}
\hline
\  \       &amp; \b    &amp; G  &amp; \  \\
\hline
\b   &amp; \b \  &amp; G   &amp; \ \ \\
\hline \hline
G &amp; G   &amp; \   &amp; \ \\
\hline
\ &amp; \ \  &amp; \   &amp; \ \  \\
\hline
\end{tabular}}, \ \underset{(c)}{\begin{tabular}{|r|r||r|r|}
\hline
\  \       &amp; \b    &amp; \   &amp; \  \\
\hline
\b   &amp; \b \  &amp; \   &amp; \ \ \\
\hline \hline
\ &amp; \ \  &amp; \b   &amp; \b \\
\hline
\ &amp; \ \  &amp; \b   &amp; \ \  \\
\hline
\end{tabular}.}
\]
\caption{Block Partitions of a $4\times 4$ Board}
\label{blocksof4x4}
\end{figure}

Since by assumption the board contains ten prisoners, it follows
from Lemma~\ref{twobytwo} that at least two of these blocks must
contain three prisoners each. Without loss of generality, assume
that the upper left block is one of them. We see in the proof of
Lemma~\ref{twobytwo} that the lone guard must lie in square
$b_{11}$, as depicted in Figure~\ref{blocksof4x4}(b).

For the board to be valid, the non-corner edge prisoners in $b_{12}$
and $b_{21}$ must each lie adjacent to three guards. This is only
possible if guards lie in the squares $b_{13}$, $b_{23}$, $b_{31}$,
and $b_{32}$, as Figure~\ref{blocksof4x4}(b) indicates. As
previously noted, at least two of the   blocks must contain three
prisoners each. The only way to achieve this will be for the lower
right block to have three prisoners, with a guard in $b_{44}$ as
shown in Figure~\ref{blocksof4x4}(c).

Now we see that the prisoners situated in squares $b_{34}$ and
$b_{43}$ necessitate the presence of guards in squares $b_{24}$ and
$b_{42 }$. By placing prisoners in all squares not yet committed, we
will have a total of only eight prisoners on the board,
contradicting our assumption that the board has ten prisoners. Thus,
our assumption was invalid.
\end{proof}

\section{\label{5x5and6x6cases}Analysis of the $5\times 5$ and $6\times 6$ Cases}

In our analysis of $5\times 5$ and $6\times 6$ board configurations,
we will partition the boards into $3\times 3$ blocks.  The following
lemma will help in the examination of these blocks.

\begin{lemma}\label{threebythree}
If $C$ is a $3\times 3$ corner block within a valid board ($n &gt; 3$),
then it must contain at least three guards. Moreover, if $C$
contains exactly three guards, then it must contain a prisoner
diagonally opposite (within $C$) to the corner square.
\end{lemma}

\begin{proof}
Assume that there exists a valid $n\times n$ ($n &gt; 3$) board
configuration with a $3\times 3$ corner block $C$ that contains only
two guards.  Without loss of generality, suppose that $C$ is
situated in the upper left corner of the board.  Since four guards
are required to protect a prisoner residing on an interior square,
and we have two guards, a guard must occupy $c_{22}$. Since, by
assumption, there remains only one more guard, there must lie a
prisoner in   $c_{12}$ or $c_{21}$.  However, three guards are
required to cover a prisoner that is situated in a non-corner edge
square. Hence, the board cannot be valid and we have a
contradiction.

To establish the last part of our lemma, let us observe that since
one guard must occupy $c_{22}$, if another one of the guards were
located in $c_{33}$ then we would have a prisoner in either $c_{12}$
or $c_{21}$ without a sufficient number of adjacent guards. 
\end{proof}

Possible arrangements, up to a reflection about the main diagonal,
appear in Figure~\ref{possiblecornerblocks}.  As we can see, all
these configurations have a prisoner in $c_{33}$.



\begin{figure}[ht]
\begin{center}
\begin{tabular}{|r|r|r|}
\hline
\ &amp; \ \  &amp; \b \\
\hline
\b &amp; \ \  &amp; \b \\
\hline
\b &amp;\b \  &amp; \b \\
\hline
\end{tabular}\ \
\begin{tabular}{|r|r|r|}
\hline
\b &amp; \ \  &amp; \b \\
\hline
\b &amp; \ \  &amp; \b \\
\hline
\ &amp;\b \  &amp; \b \\
\hline
\end{tabular}\ \
%\begin{tabular}{|c|c|c|}
%\hline
%\b  &amp; \ \  &amp; \ \\
%\hline
%\b &amp; \  &amp; \b \\
%\hline
%\b &amp;\b &amp; \b \\
%\hline
%\end{tabular}\ \
%\begin{tabular}{|c|c|c|}
%\hline
%\b  &amp; \ &amp; \b  \\
%\hline
%\b &amp; \   &amp; \  \\
%\hline
%\b &amp;\b &amp; \b \\
%\hline
%\end{tabular}\ \
\begin{tabular}{|r|r|r|}
\hline
\b  &amp; \ \  &amp; \b \\
\hline
\b &amp; \ \  &amp; \b \\
\hline
\b &amp;\ \  &amp; \b \\
\hline
\end{tabular}
\begin{tabular}{|r|r|r|}
\hline
\b &amp; \ \  &amp; \b \\
\hline
\ &amp; \ \  &amp; \b \\
\hline
\b &amp;\b \  &amp; \b \\
\hline
\end{tabular}\ \
\caption{Possible $3\times 3$ UL Corner Blocks With 3 Guards, ($n&gt;
3$)} \label{possiblecornerblocks}
\end{center}
\end{figure}

Now, we are ready to consider the $5\times 5$ case.  The only
maximum configuration (up to rotations) is illustrated in
Figure~\ref{5x5maxboard}.  Proposition~\ref{p5} establishes that
this is a maximum $5\times 5$ board configuration.

\begin{figure}[ht]
\begin{center}
\begin{tabular}{|r|r|r|r|r|}
\hline
\b  \       &amp; \  \  &amp; \b  &amp; \ \ &amp; \b  \\
\hline
\b  \       &amp; \  \  &amp; \b  &amp; \ \ &amp; \b  \\
\hline
\b  \       &amp; \  \  &amp; \b  &amp; \ \ &amp; \b  \\
\hline
\b  \       &amp; \  \  &amp; \b  &amp; \ \ &amp; \b  \\
\hline
\b  \       &amp; \ \   &amp; \b  &amp; \ \ &amp; \b  \\
\hline
\end{tabular}
\caption{The Maximum $5\times 5$ Board Configuration}
\label{5x5maxboard}
\end{center}
\end{figure}

\begin{proposition}\label{p5}
A maximum $5\times 5$ board configuration contains fifteen
prisoners; that is, $P(5)=15$.
\end{proposition}

\begin{proof}
Assume that there exists a valid $5\times 5$ board
configuration with 16 or more prisoners.  We will see that this
leads to a contradiction.

Divide the $5\times 5$ board into two opposite (overlapping) corner
$3\times 3$ blocks, $A$ and $C$, that have a square in common and
two $2\times 2$ opposite corner blocks, $B$ and $D$, as illustrated
in Figure~\ref{5x5boards}(a).
\medskip

\begin{figure}[h]
\[\underset{(a)}{ \begin{tabular}{|c|c|r|c|c|} \hline
A &amp; A &amp; A \ \ \   &amp; B &amp; B \\
\hline
A &amp; A &amp; A \ \ \   &amp; B &amp; B   \\
\hline
A &amp; A &amp; A/C  &amp; C &amp; C  \\
\hline
D &amp; D &amp; \ C   &amp; C &amp; C   \\
\hline
D &amp; D &amp; \ C &amp; C &amp; C  \\
\hline
\end{tabular}}\ \ \ \
\underset{(b)}{
\begin{tabular}{|c|c|r|c|c|} \hline
A &amp; A &amp; \ &amp; \b &amp; \ \\
\hline
A &amp; A &amp; \ &amp; \b &amp; \b \\
\hline
A &amp; A &amp; A/C  &amp; \ &amp; \  \\
\hline
D &amp; D &amp; \ C   &amp; C &amp; C   \\
\hline
D &amp; D &amp; \ C &amp; C &amp; C  \\
\hline
\end{tabular}}\ \ \ \
\underset{(c)}{
\begin{tabular}{|c|c|r|c|c|} \hline
A &amp; A &amp; \ &amp; \b &amp; \ \\
\hline
A &amp; A &amp; \ &amp; \b &amp; \b \\
\hline
\ &amp; \ &amp; A/C  &amp; \ &amp; \  \\
\hline
\b &amp; \b &amp; \ &amp; C &amp; C   \\
\hline
\ &amp; \b &amp; \ &amp; C &amp; C  \\
\hline
\end{tabular}}
\]
\caption{Partitions of the $5\times 5$ Board} \label{5x5boards}
\end{figure}


According to Lemma~\ref{threebythree}, the two $3\times 3$ blocks
$A$ and $C$ collectively contain at most $2(6)-1=11$ prisoners since
the shared square (common to blocks $A$ and $C$) must contain a
prisoner if at least one of blocks A and C has six prisoners. Hence
at least one of the $2\times 2$ blocks must have three prisoners.

Recall that Lemma~\ref{twobytwo} establishes that each of blocks $B$
and $D$ contains at most three prisoners.  Thus, for the board to
contain a total of at least sixteen prisoners we must find either
ten or eleven prisoners shared in blocks $A$ and $C$.

%(They cannot share less than ten prisoners since that would force a
%$2\times 2$ block to hold four prisoners).

{\bf Case 1.  Blocks $A$ and $C$ share 11 prisoners.} In this case,
one of the $2\times 2$ blocks holds three prisoners and the other
holds at least two prisoners.  Also the blocks $A$ and $C$ must have
six prisoners each. Without loss of generality, let us suppose that
the $B$ block has three prisoners; then the one guard must lie in
the (1,5) position. To cover the three prisoners in block $B$,
guards must be placed in the (1,3), (2,3), (3,4), and (3,5)
positions (refer to Figure~\ref{5x5boards}(b)).  For the board to be
valid, block $A$ (or its diagonal reflection) must match one of the
corner blocks depicted in Figure~\ref{possiblecornerblocks}. None of
these allows guards in both the (1,3) and the (2,3) positions.
Therefore, the board is not valid and we have a contradiction in
this case.

{\bf Case 2.  Blocks $A$ and $C$ share 10 prisoners.} In this case,
each of the $2\times 2$ blocks $B$ and $D$ holds three prisoners. In
order to maintain a valid board configuration, we are then forced to
place guards in the (1,3), (1,5), (2,3), (3,1), (3,2), (3,4), (3,5),
(4,3), and (5,3) positions as indicated in
Figure~\ref{5x5boards}(c).  But then there remain only nine
uncommitted squares in which to place the ten prisoners that blocks
$A$ and $C$ are supposed to share.  Thus, we also find a
contradiction in this case.  
\end{proof}
\bigskip

For $n=6$ all maximum boards amount to rotations or small
perturbations of the arrangement in Figure~\ref{6x6maximumboard},
the validity of which yields the lower bound $P(6)\ge 22$. We will
show that, in fact, $P(6)=22$ by an analysis of manageable size.
Dunbar, Hoffman, Laskar, and Markus assert (without proof) a fact
about 1/2-domination in the king's graph dimension $6\time 6$ which,
if true, implies that $P(6)=22$ \cite{dhlm}. This is indeed
the case, as we shall establish next. We use a more specific version
of Lemma~\ref{threebythree} in order to obtain this fact.

\begin{figure}[ht]
\begin{center}
\begin{tabular}{|r|r|r|r|r|r|}
\hline
\b &amp; \ \  &amp; \b  &amp; \b &amp; \ \ &amp; \b \\
\hline
\b &amp; \ \   &amp; \b  &amp; \ \ &amp;\ \  &amp;\b      \\
\hline
\b &amp; \ \   &amp; \b   &amp; \b &amp; \ \ &amp;\b \\
\hline
\b &amp; \ \   &amp; \b  &amp; \b &amp;\ \ &amp;\b  \\
\hline
\b &amp; \ \   &amp; \b  &amp; \ \ &amp;\ \ &amp;\b \\
\hline
\b  &amp; \ \  &amp; \b  &amp; \b &amp;\ \ &amp;\b  \\
\hline
\end{tabular}
\caption{A Maximum $6\times 6$ Board Configuration}
\end{center}
\label{6x6maximumboard}
\end{figure}

\begin{lemma}\label{threebythreees}
If $C$ is a $3\times 3$ corner block holding six prisoners within a
valid board ($n &gt; 3$), where $c_{11}$ is the corner square, then up
to a diagonal symmetry the block has one of the four arrangements in
Figure~\ref{possiblecornerblocks}.
\end{lemma}

\begin{proof}
By Lemma~\ref{threebythree}, the position $c_{22}$ must
have a guard as we have seen and also one of the positions $c_{12}$
or $c_{21}$ must have a guard or we will require four guards to
cover it. By symmetry we can assume we have a guard at $c_{12}$. If
we have a prisoner at $c_{21}$ then this leaves three possibilities
for the third guard. Adding in the case with guards in $c_{21}$ and
$c_{21}$, we see that there are only four configurations (up to
rotation and/or diagonal reflection), as shown in
Figure~\ref{possiblecornerblocks}, of $3\times 3$ corner blocks that
contain six prisoners.
\end{proof}

\begin{proposition}\label{p6}
A maximum $6\times 6$ board contains twenty two prisoners.  That is,
$P(6)=22$.
\end{proposition}

\begin{proof}
We have observed that $P(6)\ge 22$. To verify that
$P(6)\le 22$ let us assume the existence of a valid arrangement $C$
with 23 prisoners; we shall see that this leads to a contradiction.
By Lemma~\ref{threebythree}, three of the four $3\times 3$ corner
blocks have six prisoners and one has five prisoners. Without loss
of generality, we may assume that the block with five prisoners is
the lower right one. By Lemma~\ref{threebythreees} and by symmetry,
we can assume that the block in the upper left corner is one of
those in Figure~\ref{possiblecornerblocks}. Possibilities for the
upper right $3\times 3$ corner block can be generated from
arrangements found in Figure~\ref{possiblecornerblocks} first by
reflecting them about their vertical axis. Secondly, three more
possibilities are generated by reflecting about counter-diagonal of
what is obtained after the first reflection. We summarize the
possibilities for the upper right block in
Figure~\ref{corneroptions}.

\begin{figure}[ht]
\begin{tabular}{|r|r|r|}
\hline
\b  &amp; \ \  &amp; \b \\
\hline
\b &amp; \ \  &amp; \b \\
\hline
\b &amp;\ \  &amp; \b \\
\hline
\end{tabular} \ \ \
\begin{tabular}{|r|r|r|}
\hline
\b &amp; \b \  &amp; \b \\
\hline
\ &amp; \ \  &amp; \ \\
\hline
\b &amp;\b \  &amp; \b \\
\hline
\end{tabular}\ \ \
\begin{tabular}{|r|r|r|}
\hline
\ &amp; \b \  &amp; \b \\
\hline
\b &amp; \ \  &amp; \ \\
\hline
\b &amp;\b \  &amp; \b \\
\hline
\end{tabular}\ \ \
\begin{tabular}{|r|r|r|}
\hline
\b &amp; \ \  &amp; \b \\
\hline
\b &amp; \ \  &amp; \b \\
\hline
\b &amp;\b \  &amp; \ \\
\hline
\end{tabular}\ \ \
\begin{tabular}{|r|r|r|}
\hline
\b &amp; \ \  &amp; \ \\
\hline
\b &amp; \ \  &amp; \b \\
\hline
\b &amp;\b \  &amp; \b \\
\hline
\end{tabular}\ \ \
\begin{tabular}{|r|r|r|}
\hline
\b &amp; \b \  &amp; \ \\
\hline
\b &amp; \ \  &amp; \ \\
\hline
\b &amp;\b \  &amp; \b \\
\hline
\end{tabular}\ \ \
\begin{tabular}{|r|r|r|}
\hline
\b &amp; \ \  &amp; \b \\
\hline
\b &amp; \ \  &amp; \ \\
\hline
\b &amp;\b \  &amp; \b \\
\hline
\end{tabular}
\caption{Possible Upper Right $3\times 3$ Corner Blocks}
\label{corneroptions}
\end{figure}

Technically we need to analyze $28$ possibilities but let us observe
that by taking any of the four arrangements in
Figure~\ref{possiblecornerblocks} as the upper left corner and any
of the arrangements in Figure~\ref{corneroptions} with the exception
of the third one, in the upper right corner, will put a prisoner in
the $c_{14}$ position which will not be adequately covered by
guards. If we choose the third configuration from
Figure~\ref{corneroptions} in the upper right corner, then this will
put a prisoner at $c_{24}$ which will not have enough guards around
it. This contradicts the existence of a configuration with 23
prisoners or more. 
\end{proof}

We suspect that this block partition approach can be adapted to
compute or bound $P(n)$ for larger sizes of $n$, although this
approach could turn out to be quite lengthy. These proofs may very
well be pursued as undergraduate research projects.

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\section{\label{deficiency}Upper bound for $P(n)$ and the deficiency function}

As the board size grows larger, establishing the exact number of
prisoners on a maximum board becomes increasingly difficult.  The
proof of Proposition~\ref{maximum4x4boards} foreshadows the
importance of finding useful upper bounds on $P(n)$. In this
section, we construct a tool that will help in establishing these
bounds -- the deficiency matrix.  We then use the deficiency matrix
to determine a general upper bound for $P(n)$.

Suppose that we have fixed the board size at $n\times n$ ($n \ge
3$). With each configuration, we associate a binary matrix
$X=\left(x_{ij}\right)$ defined by

$$x_{ij}=\left\{
\begin{array}{cl}
1, &amp; \mbox{if a prisoner lies in the $(i,j)$ position;}\\
0, &amp; \mbox{if a guard lies in the $(i,j)$ position.}
\end{array}
\right.$$

\noindent  Many who work in combinatorics and graph theory, such as
Hedetniemi, Hedetniemi, and Reynolds \cite{hedetniemi} have
employed this idea. In any local measure of optimality we must be
attentive to the number of prisoners lying in squares adjacent to a
particular square; we let $x_{ij}^*$ denote the number of prisoners
lying in squares adjacent to the $(i,j)$ square.

The deficiency matrix serves as an ad-hoc, local measure of the
optimality of a given board configuration.  Its construction arises
from our observations and conjectures of maximum board
configurations.  We define the deficiency matrix
$\delta=\left(\delta_{ij}\right)$ by
$$\delta_{ij}=\mbox{expectation}-x_{ij}^*, \mbox{where}$$
$$\mbox{expectation}=\left\{
\begin{array}{cll}
1, &amp; \mbox{if $(i,j)$ is a corner square with} &amp; x_{ij}=1;\\
2, &amp; \mbox{if $(i,j)$ is a corner square with} &amp; x_{ij}=0;\\
2, &amp; \mbox{if $(i,j)$ is an edge square with} &amp; x_{ij}=1;\\
4, &amp; \mbox{if $(i,j)$ is an edge square with} &amp; x_{ij}=0;\\
4, &amp; \mbox{if $(i,j)$ is an interior square with} &amp; x_{ij}=1;\\
6, &amp; \mbox{if $(i,j)$ is an interior square with} &amp; x_{ij}=0.
\end{array}\right.$$

Figure~\ref{4x4deficiency_example} depicts a $4\times 4$ non-maximal
board configuration and its corresponding deficiency matrix.  The
positive entries in the deficiency matrix indicate areas of the
board that are thought to be less than optimal; the 2's indicate
that the ``worst'' deficiencies occur on the corresponding interior
squares.
\bigskip

\begin{figure}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
\b &amp; \   &amp; \b &amp; \ \\
\hline
\ &amp; \b &amp; \ &amp; \b \\
\hline
\b &amp; \   &amp; \b &amp; \ \\
\hline
\ &amp; \b &amp; \ &amp; \b \\
\hline
\end{tabular} \  \  \
\begin{tabular}{|c|c|c|c|}
\hline
$0$ &amp; $1$ &amp; $0$ &amp; $0$ \\
\hline
$1$ &amp; $0$ &amp; $2$ &amp; $0$ \\
\hline
$0$ &amp; $2$ &amp; $0$ &amp; $1$ \\
\hline
$0$ &amp; $0$ &amp; $1$ &amp; $0$ \\
\hline
\end{tabular}
\caption{Non-maximal $4\times 4$ Board and Its Deficiency Matrix}
\label{4x4deficiency_example}
\end{center}
\end{figure}
\bigskip

Since we use these values to obtain an upper bound on $P(n)$, it
helps to first consider bounds on $\delta_{ij}$ for $1\le i,j\le n$.
Suppose that the $(i,j)$ square contains a prisoner.  If $(i,j)$ is
a corner square, then at least two of the three adjacent squares
must contain guards; therefore $x_{ij}^*\le 1$ and in checking our
expectation value above we see that $\delta_{ij}\ge 0$.  Likewise,
if $(i,j)$ is an edge square containing a prisoner or an interior
square with a prisoner, we find that $\delta_{ij}\ge 0$.

Suppose that we find a guard in a corner square $(i,j)$.  Then three
squares lie adjacent to this square, so we find at most three
prisoners in the neighboring squares.  Therefore, $x_{ij}^*\le 3$
and so $\delta_{ij}\ge -1$.  Via similar considerations, we find
that if a guard occupies an edge square then $\delta_{ij}\ge -1$ and
for an interior square we get $\delta_{ij}\ge -2$.

We define the {\it net deficiency of a board configuration} as the
sum of all entries in the deficiency matrix,
$$\Delta=\sum_{i,j=1}^n\delta_{ij}.$$ \noindent For instance, the
board configuration in Figure~\ref{4x4deficiency_example} has a net
deficiency of 8.  We expect maximum board configurations to
correspond to minimum net deficiency values.  We will relate
$\Delta$ to the overall number of guards in a given board
configuration.  Let $P_C$ and $G_C$ denote the total number of
prisoners and guards, respectively, found in the corner squares.
Similarly, $P_E$ and $G_E$ refer to the prisoners and guards in edge
squares, and $P_I$ and $G_I$ refer to prisoners and guards in
interior squares.  With this notation we have
$$\begin{array}{rll}
\Delta &amp; = &amp; \sum_{\mbox{corners}}\delta_{ij}+
\sum_{\mbox{edges}}\delta_{ij}+
\sum_{\mbox{interior}}\delta_{ij} \\
\ &amp; \ge &amp; \left(0\cdot P_C-1\cdot G_C\right) +\left(0\cdot P_E-1\cdot G_E\right)+\left(0\cdot P_I - 2\cdot G_I\right)\\
\  &amp; = &amp; -G_C-G_E-2\cdot G_I.
  \end{array}
$$

\noindent This establishes the next lemma.

\begin{lemma}  The net deficiency of a given configuration satisfies the inequality
$\Delta \ge -G_C-G_E-2\cdot G_I$.\label{deficiencybound}\end{lemma}

Now we are ready to think about bounding the size of $P(n)$.
Observe that
\begin{equation}\label{ineeq}
\begin{array}{l} 4x_{ij}+x_{ij}^{*}=
\begin{cases} 8-\delta_{ij}, \mbox{if $x_{ij}=1$  and $(i,j)$ is an interior square; } \\
6-\delta_{ij}, \mbox{if $x_{ij}=0$ and $(i,j)$ is an interior
square.}
\end{cases}\\  \\ 3x_{ij}+x_{ij}^{*}=
\begin{cases} 5-\delta_{ij},
\mbox{ if $x_{ij}=1$ and $(i,j)$ is an edge square;} \\
 4-\delta_{ij}, \mbox{if $x_{ij}=0$ and $(i,j)$ is an edge square.}
\end{cases}
\\  \\ 2x_{ij}+x_{ij}^{*}=
\begin{cases} 3-\delta_{ij}, \mbox{if $x_{ij}=1$ and $(i,j)$ is a corner square;} \\
 2-\delta_{ij}, \mbox{if $x_{ij}=0$ and $(i,j)$ is a corner square.}
\end{cases}
\end{array}
\end{equation}


\begin{theorem}\label{main}  The number of prisoners in a valid configuration is given by
\begin{equation}\label{exacteqnk}
P=\frac{3n^2}{5}-\frac{4n}{5}+\frac{1}{10}(3P_E+6P_C-\Delta ).
\end{equation}
\end{theorem}

\begin{proof}
Summing the left hand sides of the equations in
(\ref{ineeq}) over all squares of the board, we obtain $$4\cdot P_I
+3\cdot P_E+2\cdot P_C+\sum_{1\le i,j\le n}x_{ij}^*=4\cdot P_I
+3\cdot P_E+2\cdot P_C+8\cdot P_I +5\cdot P_E+3\cdot P_C=12\cdot P_I
+8\cdot P_E+5\cdot P_C.$$

We will equate this result with the sum of the right hand sides. In
summing over the interior squares that contain prisoners, this
contributes $8-\delta_{ij}=6+2-\delta_{ij}$ for each such square,
whereas the interior square that contain guards contribute only
$6-\delta_{ij}$ per square.  There are $(n-2)^2$ interior squares,
so altogether these sum to $6(n-2)^2+2\cdot P_I-\sum_{\text{int.
sqrs.}}\delta_{ij}$.  Similarly summing the right hand sides over
all edge squares we get $ 4\left[4(n-2)\right]+1\cdot P_E-
\sum_{\text{edge sqrs.}}\delta_{ij}$.  Summing over the corners
yields $8+1\cdot P_C-\sum_{\text{corner sqrs.}}\delta_{ij}$.
Combining these right-hand sums and equating with the left-hand sum,
we obtain the equation

$$12P_I+8P_E+5P_C=6(n-2)^2+2P_I+16(n-2)+P_E+8+P_C-\Delta $$
or
$$10P_I+7P_E+4P_C=6n^2-8n-\Delta.$$
Then since $P=P_I+P_E+P_C$ we then obtain

$$10P=6n^2-8n+3P_E+6P_C-\Delta,$$
which leads to (\ref{exacteqnk}).
\end{proof}

By combining the inequality in Lemma~\ref{deficiencybound} with this
theorem, we obtain a crude upper bound on $P(n)$.

\begin{corollary}\label{firstupb} In a maximum configuration of prisoners and guards
on a $n\times n$ board the number of prisoners obeys the inequality
\begin{equation}\label{upperbfkg}
P(n) \le \frac{2n^2+n}{3}.
\end{equation}
\end{corollary}

\begin{proof}
Using Lemma~\ref{deficiencybound} and (\ref{exacteqnk}) we
obtain
\begin{eqnarray*}
\ds P(n) &amp; \le &amp; \frac{3n^2}{5}-\frac{4n}{5}+\frac{1}{10}(3P_E+6P_C+2G_I+G_E+G_C)\\
&amp; = &amp; \ds \frac{3n^2}{5} - \frac{4n}{5} + \frac{1}{10}\Big[2P_E+5P_C+2(n-2)^2-2P_I+4(n-2)+4\Big]\\
&amp; = &amp; \ds \frac{3n^2}{5} - \frac{4n}{5} +
\frac{1}{10}\Big[2P_E+5P_C+2(n-2)^2-2P+2P_E+2P_C+4n-4\Big].
\end{eqnarray*}

\n This implies
\begin{eqnarray*}
\ds \left(1+\frac{1}{5}\right )P(n) &amp; \le &amp; \ds \frac{3n^2}{5} - \frac{4n}{5} + \frac{1}{10}(4P_E+7P_C+2n^2-4n+4)\\
&amp; \le &amp; \ds
\frac{3n^2}{5}-\frac{4n}{5}+\frac{1}{10}\Big[(4)(4)(n-2)+(7)(4)+2n^2-4n+4\Big]\\
&amp; = &amp; \frac{4n^2+2n}{5}.
\end{eqnarray*}

\n Therefore $\ds P(n)\le
\left(\frac{5}{6}\right)\left(\frac{4n^2+2n}{5}\right)=\frac{2n^2+n}{3}$.
\end{proof}

\noindent From Proposition 14 in \cite{dhlm} one can heuristically
obtain the upper bound of $P(n)$ as being about $\frac{2}{3}n^2$ as
we have just seen, by neglecting the boundary vertices. We believe
that $\Delta \le O(n)$ in general. This fact is equivalent to $P(n)
\le 3n^2/5 +O(n)$. However we can tighten the upper bound
(\ref{upperbfkg}) by getting a better estimate for $\Delta$.

\begin{lemma}\label{secondub} In a valid configuration the net deficiency satisfies
$$\Delta\ge -1\cdot G = -1(G_I+G_E+G_C).$$
\end{lemma}

\begin{proof}
Recall our previous observations about the
possible range of values for $\delta_{ij}$.  If the $(i,j)$ board
position contains a prisoner then from the definition it follows
that $\delta_{ij}\ge 0$.  If the square is a corner or edge square
containing a guard then $\delta_{ij}\ge -1$.  For an interior square
containing a guard, we have noted that $\delta_{ij}\ge -2$.  Let us
focus on this last case.

Suppose that a guard occupies the $(i,j)$ interior position in a
valid board configuration and that $\delta_{ij}=-2$.  Then it must
be the case that all adjacent squares contain prisoners, as depicted
in Figure~\ref{configurationnearguard}(a).  The g's denote guards
that are then forced into the arrangement in order for the
configuration to be valid.  We see that each of the prisoners in the
squares diagonally adjacent to this position lies adjacent to five
or six guards (depending on the occupants in the squares marked with
asterisks).  The possible deficiency values for neighboring squares
appear in Figure~\ref{configurationnearguard}(b).  Summing these
deficiency values, we find that the net contribution of the $3\times
3$ block satisfies $2\le \Delta_{local} \le 6$.  Notice that, as the
g's in Figure~\ref{configurationnearguard}(a) suggest, it is not
possible for two such $3\times 3$ blocks around guards with
deficiency $-2$ to overlap. Thus, we see that each guard on the
board contributes a net deficiency value not less than $-1$.

\begin{figure}[ht]
\[\underset{(a)}{
\begin{tabular}{|c|c|c|c|c|}
\hline
$\ast$ &amp; g &amp; g &amp; g &amp; $\ast$  \\
\hline
g &amp; \b  &amp; \b \ &amp; \b &amp;g \\
\hline
g &amp; \b  &amp; \ \ &amp; \b &amp;g \\
\hline
g &amp; \b  &amp; \b \ &amp; \b &amp;g \\
\hline
$\ast$ &amp; g &amp; g &amp; g &amp; $\ast$  \\
\hline
\end{tabular}}
\ \ \ \ \underset{(b)}{
\begin{tabular}{|c|c|c|c|c|}
\hline
\ &amp; \ &amp; \ &amp; \ &amp; \  \\
\hline
\ \ \ &amp; {\Small 1, 2} &amp; 0 &amp; {\Small 1, 2} &amp; \ \ \ \\
\hline
\ &amp; 0 &amp; -2 &amp; 0 &amp; \ \\
\hline
\ &amp; {\Small 1, 2} &amp; 0 &amp; {\Small 1, 2} &amp; \ \\
\hline
\ &amp; \ &amp; \ &amp; \ &amp; \  \\
\hline
\end{tabular}}
\]
\caption{Local Configuration Near a Guard with Deficiency -2}
\label{configurationnearguard}
\end{figure}

Summing the $\delta_{ij}$'s over all board positions, we have
$\Delta = \sum_{prisoners}\delta_{ij} + \sum_{guards}\delta_{ij} \ge
-1\cdot G$.  
\end{proof}

Using this bound on $\Delta$ in Theorem~\ref{main}, we obtain a
better upper bound for $P(n)$.  The calculations parallel those used
in the proof of Corollary~\ref{firstupb}.

\begin{theorem}\label{secondubforp} For an $n\times n$ maximum arrangement of prisoners
and guards, the number of prisoners, $P(n)$, satisfies the
inequality
\begin{equation}\label{mainineq}
P(n)\le \frac{7n^2+4n}{11}.
\end{equation}
\end{theorem}

\begin{proof}
By Lemma~\ref{secondub}, $-\Delta\le G$.  Applying this
upper bound in Theorem~\ref{main}, we get
\begin{eqnarray*}
P &amp; \le &amp; \frac{3}{5}n^2 - \frac{4}{5}n +
\frac{1}{10}\Big[3P_E+6P_C+G \Big] =
\frac{3}{5}n^2 - \frac{4}{5}n + \frac{1}{10}\Big[3P + 3P_C-3P_I+G \Big]\\
&amp; = &amp; \frac{3}{5}n^2 - \frac{4}{5}n +
\frac{1}{10}\Big[2P+3\big(P_C-P_I\big)+n^2 \Big].
\end{eqnarray*}
\noindent Subtracting $\frac{2}{10}P$ from both sides and combining
the $n^2$ terms, we see that this implies
\begin{eqnarray*}
\frac{8}{10}P &amp;\le &amp; \frac{7}{10}n^2 - \frac{4}{5}n + \frac{3}{10}\big(P_C-P_I\big)\\
&amp;\le &amp;  \frac{7}{10}n^2 - \frac{4}{5}n + \frac{3}{10}\big(4-P+P_E+P_C\big)\\
&amp;\le &amp; \frac{7}{10}n^2 - \frac{4}{5}n +
\frac{3}{10}\big(4-P+4n-4\big).
\end{eqnarray*}
\noindent The claim now follows after a bit of arithmetic.
\end{proof}

Let us mention here that according to \cite{sr}, the {\it global
offensive number} $\gamma_0$ of a graph is the minimum cardinality
of a global offensive alliance in that graph. An {\it offensive
alliance in a graph} is a set of vertices, say $O$,  with the
property that a majority of the vertices in the neighborhood of
every vertex in the boundary of $O$ is in $O$. An offensive alliance
$O$ is said to be global if it affects all vertices not in $O$. It
is easy to see that in the king's graph a minimum cardinality
offensive alliance must be global. If we think of the vertices in
such a global offensive alliance as guards and the rest of the
vertices as prisoners we observe that the restriction on the
prisoners is a little stronger than in our problem. This means that
every minimum cardinality offensive alliance in the king's graph
gives a valid configuration of prisoners versus guards in our
domination problem.  Hence $n^2-\gamma_o\le P(n)$. Since the number
of edges in the king's graph is $m=4n^2-6n+2$ for an $n\times n$
board, and the maximum degree is $8$, we get from Theorem~7 in
\cite{sr} that

$$\gamma_o(K_n)\ge \ds \lceil \frac{9n^2-12n+4}{25} \rceil .$$

For every $n\ge 2$, this inequality follows from (\ref{mainineq}) as
a corollary.

\begin{corollary} The global offensive alliance number $\gamma_o$
for the king's graph associated with an $n\times n$ board satisfies
$$\gamma_o(K_n)\ge \lceil \frac{4n^2-4n}{11} \rceil.$$
\end{corollary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{\label{gridgraphs} Grid type graphs and half-dependent best density arrangements}

Thus far we have considered the Prisoners and Guards game using
the king's graph to determine which board squares are adjacent.  The
``grid graph" provides another variation that graph theorists
contemplate in domination problems; in this context we consider
adjacent squares to be those that lie directly above/below or
left/right of a particular square.  This rule for adjacency
resembles the movements of a rook on a chessboard, assuming the
rook's move is limited to one square at a time.

If $P_n$ denotes a path with $n$ vertices and $C_n$ is a cycle with
$n$ vertices, we will consider the naturally defined half-dependent
problem in each of the three customarily grid type graphs:
$G_n:=P_n\times P_n$ (the usual grid graph), $GC_n:=C_n\times P_n$
(cylindrical grid graph), and $GT_n:=C_n\times C_n$ (toroidal grid
graph). We are going to use the model of an $n\times n$ chess board
in order to think about these graphs. The half-dependent problem in
each one of these graphs is to determine the maximum cardinality of
a set of prisoners such that each one has at least as many guards
around (neighbors) as other prisoners.  Let us denote by
$P_{grid}(n)$, $P_{grid\ cylinder}(n)$, and $P_{grid\ torus}(n)$
respectively, the maximum cardinality of a (1/2)-dominating set of
prisoners in each of the corresponding graphs as above.

As shown in \cite{ipw} we can easily derive that the following
limits exist and in fact

$$\lim_{n\to \infty} \frac{P_{grid}(n)}{n^2}=\lim_{n\to \infty} \frac{P_{grid \ cylinder}(n)}{n^2}=
\lim_{n\to \infty} \frac{P_{grid\
torus}(n)}{n^2}:=\rho_{1/2}(grid).$$

\n This is based on the fact that $\limsup_{n\to
\infty}\frac{P_{torus}(n)}{n^2}$ exists and one shows that

$$\rho_{1/2}(grid)=\limsup_{n\to \infty}\frac{P_{grid\ torus}(n)}{n^2}.$$


 \noindent From Figure~\ref{grid5by5}, we notice that in the toroidal case the following
 arrangement in $GT_3$ (the shaded squares are prisoners and the unshaded
represent guards) gives
$$\rho_{1/2}(grid)\ge 2/3.$$


\begin{figure}[h]
\centering
\includegraphics[scale=1]{gridbest.eps}
\caption{{\it 6 \ prisoners\ out \ of \ 9} } \label{grid5by5}
\end{figure}

\noindent We next show that the inequality


\begin{equation}\label{theotherway}
\rho_{1/2}(grid)\le 2/3 \end{equation}


\noindent  must hold true. As before, we associate a binary matrix
$X=\left(x_{ij}\right)$ defined by

$$x_{ij}=\left\{
\begin{array}{cl}
1, &amp; \mbox{if a prisoner lies in the $(i,j)$ position;}\\
0, &amp; \mbox{if a guard lies in the $(i,j)$ position.}
\end{array}
\right.$$


\noindent The problem in the toroidal case is described by

$$2x_{ij}+x_{ij}^*\le 4, \ \ \ \ 1\le i,j\le n,$$


\noindent  where $x_{ij}^*$ is the number of prisoners lying in
squares adjacent to the $(i,j)$ square. If we add the above
inequalities we get

$$2\sum_{i,j}x_{ij}+4\sum_{i,j}x_{ij}\le 4n^2,$$

\noindent which gives $P_{grid\ torus}(n)\le \frac{2}{3}n^2$. Hence,
we must have


\begin{proposition}\label{driddensity}
$$\rho_{1/2}(grid)=\frac{2}{3}.$$
\end{proposition}


We calculated some of the values of the sequences $\{P_{grid}(n)\}$,
$\{P_{grid\ cylinder}(n)\}$,  and $\{P_{grid\ torus}(n)\}$ using
LPSolve IDE. These values are listed in the table below. One may try
to use our techniques from previous sections in order to prove that
the numbers listed below are valid:


\vspace{0.2in}

 \centerline{
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c| }
  \hline
  n &amp; 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 &amp; 6 &amp; 7 &amp; 8 &amp; 9 &amp; 10 &amp; 11&amp; 12\\   \hline
  $P_{grid}(n)$ &amp; 1 &amp; 2 &amp; 5 &amp; 9 &amp; 14 &amp; 20 &amp; 28 &amp; 37 &amp; 47 &amp;  &amp; &amp;\\
  \hline
 $P_{grid \ cylinder}(n)$ &amp; 0 &amp; 2 &amp; 5 &amp; 8 &amp; 14 &amp; 20 &amp; 28 &amp; 37 &amp; 48 &amp;
 &amp;  &amp;
 \\ \hline
 $P_{grid\ torus}(n)$ &amp; 0 &amp; 2 &amp; 6 &amp; 9 &amp; 15 &amp; 24 &amp; 30 &amp; 40 &amp; 54 &amp; 63 &amp; 77&amp; 96\\
  \hline
\end{tabular}}


\vspace{0.2in}


Some arrangements that give the maximum number of prisoners in the
usual grid graph for the half-dependent problem are included in
Figure~\ref{regularGrid}.


\begin{figure}[h]
\centering
\includegraphics[scale=0.6]{regularGridArrangements.eps}
\caption{Best half-dependent arrangements for $G_n$, $n=2\ldots 9$ }
\label{regularGrid}
\end{figure}











%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\label{whatelse}Other Results, Conjectures and Open Questions}

We believe that the method of finding an upper bound we implemented
in Section 4, can be further sharpened.  However, we have not found
a complete analysis to show that for instance $\Delta \ge -O(n)$
which, we think, would be the sharpest result for this type of
domination problem. If this is true, the first approximation for
$P(n)$ will be $P(n)= \frac{3n^2}{5}+O(n)$. There is another
interesting phenomenon here that we would like to mention. The
boundary conditions are in a sense less restrictive than the
constraints in the center of the board. As a result it is expected
to have quite a good proportion of prisoners on the boundary in a
maximum configuration. In support of this we present two examples we
found that give ``best" (so far) arrangements/proportions in the
cases $n=15$ (Figure~\ref{15by15}) and $n=21$ (Figure~\ref{21by21}).
It seems to be possible to construct a sequence of arrangements for
which
$$\liminf _{n\to \infty}\frac{P(n)-\frac{3n^2}{5}}{n}&gt;0.$$


\begin{figure}[h]
\centering
\includegraphics[scale=0.5]{15by15.eps}
\caption{136 \ prisoners, $\frac{3}{5}(15^2)=135$ } \label{15by15}\
\par
\includegraphics[scale=0.6]{21by21.eps}
\caption{266 \ prisoners, $\frac{3}{5}(21^2)=264.6$ } \label{21by21}
\end{figure}



Ionascu, Pritikin, and Wright have established values of $P(n)$ for
$n\in \{7, 8, 9, 10, 11\}$ \cite{ipw}.  Most of their
arrangements were obtained using the LPSolve IDE program in the
Lesser GNU public domain for solving integer linear programming
problems with Branch-and-Bound and Simplex Methods. The second
author used CPLEX while visiting at the Georgia Institute of
Technology in the Faculty Development Program in 2005-2006; with the
help of Professor William Cook he analyzed the case $n=11$.

Many interesting questions remain to be answered.  What are the
values of $P(n)$ for integers $n$ larger than 11?  With error-free
play, does one particular player enjoy an advantage?  Perhaps the
advantage varies with the board size.


If $P(n)$ is odd, we conjecture that the game favors the red player,
but it is not clear that a winning strategy exists. When $P(n)$ is
even, we suspect that error-free play by both players will lead to a
tie.

Given that we find several maximal $4\times 4$ board configurations
with eight prisoners (an even number), it seems that the second
player (blue) will find opportunities to win unless s/he is forced
to use Rule I. The question is: can the red player always achieve a
win or a tie? We believe there is a strategy for the red player to
win despite all of these chances for the blue player.  In general,
it appears that the final maximal configuration is an important
factor in the game, since the number of prisoners in it determines
the fate of the game. So it is in the red player's interest to end
in a maximal arrangement with an odd number of prisoners on the
board. Similarly it is part of the blue player's strategy to divert
the end configuration to a maximal one that has an even number of
prisoners. Each player may change the configuration at only one
place at which the opponent has previously placed his two prisoners
and leave one of them as it is. As a result, almost half the
prisoners on the final board configuration are where each player
wanted them to be. So from this perspective the end game is dictated
by the parity and the number of maximal configurations with
$P(n)-1$, $P(n)-2$, ... prisoners.

In this paper, we have shown that $P(n)$ is bounded above by
$\ds\frac{7n^2+4n}{11}$, but we conjecture that $P(n)=3n^2/5+O(n)$.
Computer assisted methods as employed in \cite{ipw} show that this
bound can be improved and that our conjecture is very plausible. It
would be great to see a case analysis proof similar to those used
here of our conjecture.

\begin{figure}[h]
\centering
\includegraphics[scale=0.6]{36by36.eps}
\caption{777 \ prisoners, $\frac{3}{5}(36^2)=777.6$ } \label{36by36}
\end{figure}

\section{Acknowledgments}

First, we thank the referee of this paper who
has pointed us in the right direction to relate our work with other
papers in the domination literature and whose encouragement to look
at different graphs lead to the inclusion of
Section~\ref{gridgraphs}.

Special thanks go to Annalisa Crannell, of Franklin and Marshall
College, for her many helpful comments and suggestions for this
paper. We also thank  Mike McCoy, a Columbus State University
student who spent hours designing a program that has a very good
run-time, even for large size boards, and is producing arrangements
that we think are ``very close" to  maximum arrangements. We include
in Figure~\ref{36by36} one of them as a curiosity in the case
$36\times 36$ (containing $777$ prisoners).



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\end{thebibliography}


\bigskip
\hrule
\bigskip

\noindent 2000 {\it Mathematics Subject Classification}: Primary
05D99; Secondary 91A05, 91A24.\\

\noindent {\it Keywords}: upper bounds, winning strategies, maximal
configurations, domination in graphs.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequence \seqnum{A103139}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received August 12 2008;
revised version received October 6 2008; October 18 2008.  
Published in {\it Journal of Integer Sequences}, December 14 2008.

\bigskip
\hrule
\bigskip

\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
\vskip .1in


\end{document}

                                                                                

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