Shading parts of a graph

Question

Although the header looks similar to this question, we could not find anything related to what we want in that answer. we would like to thatch shade a portion of a graph ( graphs with vertices and edges) like in this photo here, We have drawn the graphs using Tkiz like this, . Now we would like to shade various portions of the graph (regions made of square tile-like shapes ), sometime the shadings would overlap so it would be nice if we can choose various angles for the thatches or various densities of the thatching. Thanks in advance for a pointer.

The following is what we have right now,

  \usepackage{tikz}
\usepackage{fullpage}
\usetikzlibrary{decorations.pathreplacing}
\begin{document}
\begin{tikzpicture}[scale=.15]
    
        \draw (6,0)--(4,2)--(6,4)--(8,2)--(6,0);
        \draw [dotted] (6,4)--(8,6);
        \draw [dotted] (8,2)--(10,4);
        \draw (12,10)--(10,8)--(12,6)--(10,4);
        \draw (10,8)--(8,10)--(10,12);
        \draw [dotted](10,12)--(12,14);
        \draw [dotted](12,10)--(14,12);
        \draw (12,14)--(14,16)--(16,18)--(18,16)--(16,14)--(18,12)--(16,10)--(14,12)--(12,14);
        \draw (14,8)--(12,10)--(10,12);
        \draw (16,10)--(14,12)--(12,14);
        \draw (12,6)--(14,8);
        \draw [dotted](14,8)--(16,10);
        \draw (10,4)--(8,6)--(10,8);
        \draw (14,12)--(16,14)--(14,16);
        \draw [dotted](16,18)--(18,20);
        \draw (18,20)--(20,22)--(22,20)--(20,18)--(18,20);
        \draw [dotted](18,16)--(20,18); 
         \draw [decorate,decoration={brace,amplitude=5pt,mirror,raise=.5pt}, yshift=0pt](6.5,-.5)--(12.5,5.5) node [black,midway,xshift=0.5 cm]{\footnotesize $m$};
        \draw [decorate,decoration={brace,amplitude=5pt},xshift=-4pt, yshift=0pt](7.5,10.5)--(13.5,16.5) node [black,midway,xshift=-.5 cm]{\footnotesize $k$};
        \draw [decorate,decoration={brace,amplitude=5pt},xshift=-4pt, yshift=0pt](13.5,16.5)--(19.5,22.5) node [black,midway,xshift=-.5 cm]{\footnotesize $n$}; 

        \begin{scriptsize}

        \fill [color=black] (6,0) circle (4 pt);
        \fill [color=black] (4,2) circle (4 pt);
        \fill [color=black] (8,2) circle (4 pt);
        \fill [color=black] (6,4) circle (4 pt);
        \fill [color=black] (10,4) circle (4 pt);
        \fill [color=black] (8,6) circle (4 pt);
        \fill [color=black] (12,6) circle (4 pt);
        \fill [color=black] (10,8) circle (4 pt);
        \fill [color=black] (8,10) circle (4 pt);
        \fill [color=black] (14,8) circle (4 pt);
        \fill [color=black] (12,10) circle (4 pt);
        \fill [color=black] (10,12) circle (4 pt);
        \fill [color=black] (16,10) circle (4 pt);
        \fill [color=black] (14,12) circle (4 pt);
        \fill [color=black] (12,14) circle (4 pt);
        \fill [color=black] (18,12) circle (4 pt);
        \fill [color=black] (16,14) circle (4 pt);
        \fill [color=black] (14,16) circle (4 pt);
        \fill [color=black] (18,16) circle (4 pt);
        \fill [color=black] (16,18) circle (4 pt);
        \fill [color=black] (20,18) circle (4 pt);
        \fill [color=black] (18,20) circle (4 pt);
        \fill [color=black] (22,20) circle (4 pt);
        \fill [color=black] (20,22) circle (4 pt);
        \end{scriptsize}
        \end{tikzpicture}

    

Answer 1

Like this?

MWE is:

\documentclass[border=3.141592]{standalone}
\usepackage{tikz}
\usetikzlibrary{patterns.meta,
                decorations.pathreplacing,
                    calligraphy,% had to be after decorations.pathreplacing
                positioning
                }

\begin{document}
    \begin{tikzpicture}[
node distance=0mm and 10mm,
BC/.style = {
        decorate,
        decoration={calligraphic brace, amplitude=2mm,
        pre =moveto, pre  length=1pt,
        post=moveto, post length=1pt,
        raise=2mm, #1},
        thick,
        pen colour={black}
        },
dot/.style= {circle, fill, inner sep=1pt, node contents={}},
RN/.style = {draw, minimum size=1cm, outer sep=0pt, 
             node contents={}},
PN/.style = {pattern={Lines[angle=15,distance={3pt},line width=0.2pt]},
             pattern color=gray}
                ]
    \begin{scope}[transform shape, rotate=45]
\node (m1) [RN] {}; 
\node (m2) [RN, right=of m1];
\node (m3) [RN, PN, right=0mm of m2];
\node (m4) [RN, PN, right=of m3];
\node (m5) [RN, PN, above=of m3];
\node (m6) [RN, PN, right=of m5];
\node (m7) [RN, right=0mm of m6];
\node (m8) [RN, right=of m7];
%
\foreach \i [evaluate=\i as \j using int(\i+1)] in {1,3,5,7}
\draw[densely dashed]   (m\i.north east) -- (m\j.north west)
                        (m\i.south east) -- (m\j.south west);
%
\foreach \i in {1,2,...,8}
\path   (m\i.north west)    node[dot]
        (m\i.north east)    node[dot]
        (m\i.south east)    node[dot]
        (m\i.south west)    node[dot];
    \end{scope}
\draw[BC=mirror]    (m1.south west) -- node[below=4mm, sloped] {$m$} (m2.south east);
\draw[BC]           (m5.north west) -- node[above=4mm, sloped] {$k$} (m6.north east);
\draw[BC]           (m7.north west) -- node[above=4mm, sloped] {$n$} (m8.north east);
    
    \end{tikzpicture}
\end{document0}

As you can observe, code is significantly rewritten. Instead of drawing lines, it used nodes, which are drawn in horizontal direction and when completed rotated for 45 degree.

For shading is used patterns.metta (as suggested in @SebGlaw in his comment). For it proces is defined special style, where are define angle of lines, lines thickness and distance between them.

Additional care is given to drawn braces, which are drawn by use of the calligraphy package.