corners(p)$ must be of the form a), b), c) or d) otherwise $p$ would have a border of size $(1,1)$.
Since any picture in a non-overlapping set is necessarily unbordered, we have that, for any $p \in X$, $corners(p)$ must be of the form a), b), c) or d).
The proof is completed by noting that, for any $p, q \in X$, it must be $corners(p)=corners(q)$; if the sets $corners(p)$ and $corners(q)$ were of two different forms among
a), b), c) or d), then $p$ and $q$ would have an overlap of size $(1,1)$ against the hypothesis that $X$ is non-overlapping.
\end{proof}















\section{The construction}\label{s-constr}


In this section we present some properties on non-expandable non-overlapping  sets, from which it is possible to obtain a
a general method to construct  a NENO set of pictures.
The focus will be firstly on the frames of the pictures in a NENO set and their properties related to the non-expandability of the set (see Theorem  \ref{l-non-exp}).
Subsequently, we will present some further conditions on the internal part of the pictures in a NENO set (see Theorem  \ref{l-non-ov}).
The obtained conditions, all together, will be sufficient to construct a NENO set.




Let us introduce the following definition on string sets.

 \begin{definition}\label{d-full}
Let $S_1$, $S_2 \subseteq \Sigma^n$ and $S_1\cap S_2=\emptyset$.
The pair $(S_1,S_2)$ is {\em full} if for any $s\notin S_1\cup S_2$, there exist
$s_1\in S_1$, $s_2\in S_2$ such that $s$ and $s_1$ overlap, and  $s$ and $s_2$ overlap.
 \end{definition}



\begin{example}\label{e-righe-fanta}
Let $\Sigma=\{ 0, 1\}$ and let $S_1, S_2\subseteq\Sigma^n$ be  the languages
 $S_1=\{ 1^n\}$ and $S_2=\{ 0w0 \ |\ w\in \{ 0, 1\}^{n-2}\}$.
The pair $(S_1,S_2)$ is  cross-non-overlapping, since the  strings in $S_1$ do not contain occurences of symbol $0$.

Let us show that it is also
full.
Let $s$ be any string $s\notin S_1\cup S_2$.
Since $s\notin S_2$, three cases are possible,
$s=0x1$, $s=1y0$, or $s=1z1$.
If $s=0x1$ or $s=1y0$ then $s$ overlaps the string in $S_1$ and any string in $S_2$ with an overlap of length $1$.
If $s=1z1$ then $z$ contains at least one occurence of $0$, because $s\notin S_1$.
Then $s=1^k0r1$, for some $k\geq 1$ and $r\in\Sigma^*$.
Therefore, $s$ and $1^n$ overlap, and also $s$ and $0r10^k$ overlap, with $0r10^k\in S_2$.
\end{example}


\begin{example}\label{e-colonne-fanta}
Let $\Sigma=\{ 0, 1\}$ and let $S_3, S_4\subseteq\Sigma^m$ be the languages
 $S_3=\{ 110^{m-2}\}$ and $S_4=\{ 1w0 \ |\ |w|=m-2, w\neq 0^{m-2}, 1w0$ with no suffix in $110^+\}$.
The pair $(S_3,S_4)$ is  cross-non-overlapping. Indeed, consider $s'=110^{m-2} \in S_3$ and $s''=1w0 \in S_4$ and suppose that
there exists $u$ such that $s'=xu$ and $s''=uy$ or $s'=ux$ and $s''=yu$. In the first case, it must be $|u|=m-1$ and, hence, $w=0^{m-2}$ against the definition of $S_4$.
In the second case, it must be $|u| \geq 3$ and, hence, $w$ would have a suffix in $110^+$ against the definition of $S_4$.

Let us show that the pair $(S_3,S_4)$ is full.
Let $s$ be any string $s\notin S_3\cup S_4$.
If $s\in 0\Sigma^*$ or $s\in \Sigma^* 1$ then $s$ overlap the string in $S_3$ and any string in $S_4$ with an overlap of length $1$.
Consider the case that $s=1w0$.
Since $s\notin S_4$, $w=0^{m-2}$ or $w$ has a suffix in $110^+$.
In the first case $s=10^{m-1}$. Then $s$ and the string in $S_3$ overlap with an overlap of length $m-1$; and $s$
overlaps any string in $S_4$ that has $010$ as a suffix, with an overlap of length $2$.

In the second case
$s=1x110^k$ with $k\geq 1$, and $x\in\Sigma^*$.
Then $s$ and the string in $S_3$ overlap with an overlap of length $k+2$,
and $s$ and any string $10^k y0\in S_4$, for some $y$, overlap with an overlap of length $k+1$.
\end{example}




Recall that the frame of a language $X$
 is the quadruple
$frame(X)$
of the sets of its first and last rows and columns,
$frame(X)=(R_F(X), R_L(X), C_F(X), C_L(X))$.
Note that not any quadruple of string languages  can be the frame of a set of pictures, since their strings need to match    in the lengths and in the corner positions.
Given four string languages $S_1, S_2\subseteq \Sigma^n$ and
 $S_3, S_4\subseteq \Sigma^m$,
we say that the quadruple  $(S_1,S_2,S_3,S_4)$ is {\em frame-compatible} if there exists a picture language $X\subseteq \Sigma^{m,n}$ such that
$frame(X)=(S_1,S_2,S_3,S_4)$.
In the case of non-overlapping sets of pictures,
the constrains are yet stronger.
As previously remarked (see Lemma \ref{l-corner-non-ov-sets}),
if $X$ is a non-overlapping set on $\Sigma=\{ 0, 1\}$, then all the pictures in $X$ have the same set of corners and, for this set, only four cases are possible,
following which symbols appear in the corners of pictures in $X$.
Hence, for example, a quadruple  $(S_1,S_2,S_3,S_4)$ can be the frame of a non-overlapping set $X$ such that, for all $p \in X$, $corners(p)=(0,0,1,1)$,
if $S_1\subseteq 0\Sigma^* 0$,
$S_2\subseteq 1\Sigma^* 1$, and
$S_3, S_4\subseteq 0\Sigma^* 1$.


We give the following definition.


\begin{definition}
Let $X\subseteq \Sigma^{m,n}$.
$X$ is {\em frame-complete} if for any $p, q\in X$, $p$ and $q$ do not frame overlap, and
if for any picture $p \in \Sigma^{m,n}\setminus X$ there exists a picture $q\in X$ such that $p$ and $q$ frame overlap.
\end{definition}


Note that being frame-complete is  a \emph{sufficient condition} for a set $X$ to be non-expandable with respect to the overlapping. Any picture