How can I draw the following graph with tikz?

Question

Some irregular curves and their surrounding shadows are all needed to draw a graph in graph theory. I'm not very good at drawing this with Tikz, but I want to do my best to draw the following graph.

I used some code from Drawing Königsberg landscape showing the bridges and it seems that no good and concise. But I didn't draw it well enough, and the code wasn't clean enough.

\documentclass{article}
\usepackage{xcolor}
\usepackage{tikz}
\usetikzlibrary{decorations.pathmorphing, calc}
\definecolor{babypink}{rgb}{0.96, 0.76, 0.76}
\tikzset{%
    contour/.style={dashed,%
        very thick,%
        decoration={%
            random steps,%
            segment length=4pt,%
            amplitude=0.5pt%
        },%
        rounded corners=1pt,%
        decorate%
    }%
}

\begin{document}
    \begin{tikzpicture}[x=10cm, y=9.19cm]
        \filldraw[babypink] ($(0, 1) + (0.241, -0.622)$) -- ($(0, 1) + (0.235,
        -0.587)$)
        --
        ($(0, 1) + (0.240, -0.540)$) -- ($(0, 1) + (0.249, -0.524)$) --
        ($(0, 1) + (0.252, -0.498)$) -- ($(0, 1) + (0.266, -0.482)$) --
        ($(0, 1) + (0.271, -0.462)$) -- ($(0, 1) + (0.288, -0.454)$) --
        ($(0, 1) + (0.300, -0.434)$) -- ($(0, 1) + (0.308, -0.418)$) --
        ($(0, 1) + (0.320, -0.412)$) -- ($(0, 1) + (0.328, -0.404)$) --
        ($(0, 1) + (0.399, -0.399)$) -- ($(0, 1) + (0.453, -0.393)$) --
        ($(0, 1) + (0.518, -0.386)$) -- ($(0, 1) + (0.549, -0.388)$) --
        ($(0, 1) + (0.609, -0.404)$) -- ($(0, 1) + (0.624, -0.410)$) --
        ($(0, 1) + (0.644, -0.438)$) -- ($(0, 1) + (0.663, -0.486)$) --
        ($(0, 1) + (0.670, -0.519)$) -- ($(0, 1) + (0.668, -0.546)$) --
        ($(0, 1) + (0.658, -0.590)$) -- ($(0, 1) + (0.648, -0.612)$) --
        ($(0, 1) + (0.636, -0.648)$) -- ($(0, 1) + (0.633, -0.666)$) --
        ($(0, 1) + (0.617, -0.677)$) -- ($(0, 1) + (0.596, -0.700)$) --
        ($(0, 1) + (0.535, -0.708)$) -- ($(0, 1) + (0.500, -0.709)$) --
        ($(0, 1) + (0.457, -0.717)$) -- ($(0, 1) + (0.412, -0.708)$) --
        ($(0, 1) + (0.372, -0.702)$) -- ($(0, 1) + (0.336, -0.695)$) --
        ($(0, 1) + (0.291, -0.679)$) -- ($(0, 1) + (0.268, -0.652)$) --
        cycle;
        \draw[contour] ($(0, 1) + (0.241, -0.622)$) -- ($(0, 1) + (0.235, -0.587)$) --
        ($(0, 1) + (0.240, -0.540)$) -- ($(0, 1) + (0.249, -0.524)$) --
        ($(0, 1) + (0.252, -0.498)$) -- ($(0, 1) + (0.266, -0.482)$) --
        ($(0, 1) + (0.271, -0.462)$) -- ($(0, 1) + (0.288, -0.454)$) --
        ($(0, 1) + (0.300, -0.434)$) -- ($(0, 1) + (0.308, -0.418)$) --
        ($(0, 1) + (0.320, -0.412)$) -- ($(0, 1) + (0.328, -0.404)$) --
        ($(0, 1) + (0.399, -0.399)$) -- ($(0, 1) + (0.453, -0.393)$) --
        ($(0, 1) + (0.518, -0.386)$) -- ($(0, 1) + (0.549, -0.388)$) --
        ($(0, 1) + (0.609, -0.404)$) -- ($(0, 1) + (0.624, -0.410)$) --
        ($(0, 1) + (0.644, -0.438)$) -- ($(0, 1) + (0.663, -0.486)$) --
        ($(0, 1) + (0.670, -0.519)$) -- ($(0, 1) + (0.668, -0.546)$) --
        ($(0, 1) + (0.658, -0.590)$) -- ($(0, 1) + (0.648, -0.612)$) --
        ($(0, 1) + (0.636, -0.648)$) -- ($(0, 1) + (0.633, -0.666)$) --
        ($(0, 1) + (0.617, -0.677)$) -- ($(0, 1) + (0.596, -0.700)$) --
        ($(0, 1) + (0.535, -0.708)$) -- ($(0, 1) + (0.500, -0.709)$) --
        ($(0, 1) + (0.457, -0.717)$) -- ($(0, 1) + (0.412, -0.708)$) --
        ($(0, 1) + (0.372, -0.702)$) -- ($(0, 1) + (0.336, -0.695)$) --
        ($(0, 1) + (0.291, -0.679)$) -- ($(0, 1) + (0.268, -0.652)$) --
        ($(0, 1) + (0.241, -0.622)$);
    \node[draw,circle] (u) at ($(0, 1) + (0.5, -0.388)$)[label={$u$}]{};
 \node[draw,circle] (u2)  at ($(0, 1) + (0.658,
 -0.582)$)[label=right:{$u_2$}]{};
\node[draw,circle] (v) at ($(0, 1) + (0.5, -0.71)$)[label=below:{$v$}]{};
\node[draw,circle] (u1) at ($(0, 1) + (0.23, -0.582)$)[label=left:{$u_1$}]{};
\draw [in=-165, out=165, looseness=5.00](u) to (v);
\node[] at (0.5,0.5) {$f$};
\end{tikzpicture}
\end{document}

Holding a learning attitude, I‘d like to learn more concise tikz code which can draw the graph I want. For those irregular curves in the original graph, I don't know if there is any software that can assist in generating them.

Answer 1

That's a very painful way of drawing something like that. Unfortunately, the random steps command is not very easy to use in this context so maybe drawing this step by step with rounded corners could be a solution:

\documentclass[tikz,border=3.14mm]{standalone}
\usetikzlibrary{positioning}

\begin{document}
    \tikzset
        {
        dot/.style={circle,draw,thick,fill=white,inner sep=2pt},
        rdm/.style={thick,dotted,rounded corners=5pt},
        }
    \begin{tikzpicture}[node distance=2cm]
        \draw[cyan] (-2,-2) grid (4,2);
        \node[dot,label=below left:$z_1$] (z1) {};
        \node[dot,above right = of z1,label=above right:$u$] (u) {};
        \node[dot,below right = of z1,label=below right:$v$] (v) {};
        \node[dot,above right = of v,label=below right:$z_2$] (z2) {};
        
        \draw[thick] (u) to[out=160,in=190,looseness=3] node[midway,left] {$e$} (v) ;
        
        \def\a{0.35}
        \draw[rdm] (z1) --++ (.5*\a,\a) --++ (\a,0) --++ (0,1.5*\a) node[left] {$P_1$} --++ (\a,0) --++ (0,\a) -- (u);
        \draw[rdm] (z1) --++ (0,-\a) --++ (\a,0) --++ (0,-1.5*\a) --++ (\a,0) --++ (0,-\a) -- (v);
        \draw[rdm] (v) --++ (\a,0.5*\a) --++ (\a,0) --++ (0,\a) --++ (\a,0) --++ (0,\a) --++ (\a,0) -- (z2);
        \draw[rdm] (u) --++ (.75*\a,0) --++ (0,-\a) --++ (1.75*\a,0) --++ (0,-\a) --++ (1.5*\a,0) node[right] {$P_2$} -- (z2);
        \path (u) -- (v) node[midway] {$f$};
    \end{tikzpicture}
\end{document}

This is obviously strongly customizable.

Answer 2

I start by apologize for giving a MetaPost/MetaFun+ConTeXt answer, even though you were asking for a TikZ one. This is a method I have used before to draw some random smooth curves. I can probably be used also for TikZ, but I do not know how.

The idea is to randomize and smoothen a path. Look at the following picture:

Here we have started with a circle (defined by circularpath(10), which gives a circle with 40 points), drawn in blue. We then randomize it by using randomized (the light pink curve). This curve is not smooth. We get a smooth curve by applying curved to the result (the dark pink curve). The pink curves are built by the same points (shown in dark red), but with different control points.

For your example, the full code could look like this (this is the code I used, it is compiled with context). Note that since it uses randomization, your result might look a bit different.

\startMPpage[offset=3bp]
u:=1cm;
pickup pencircle scaled 1.5bp;

path blob,handle;
pair blobtop, blobleft, blobbottom, blobright;

blob = curved(circularpath(10) scaled 6u randomized 0.5u);

blobtop := (ulcorner blob -- urcorner blob) intersectionpoint blob;
blobleft := (ulcorner blob -- llcorner blob) intersectionpoint blob;
blobbottom := (llcorner blob -- lrcorner blob) intersectionpoint blob;
blobright := (lrcorner blob -- urcorner blob) intersectionpoint blob;

handle = blobtop{dir 160} .. {dir 25}blobbottom;

fill blob withcolor 0.95white;
draw blob dashed withdots scaled 0.75bp;
draw handle;

for i = blobtop,blobleft,blobbottom,blobright:
  unfill fullcircle scaled 8bp shifted i;
  draw fullcircle scaled 8bp shifted i;
endfor;


label.rt("$e$", point 0.5 of handle);
label("$f$", center blob);
label.lft("$P_1$", point 33 of blob);
label.rt("$P_2$", point 9 of blob);
labeloffset := 6bp;
label.urt("$u$",blobtop);
label.lft("$z_1$",blobleft);
label.lrt("$v$",blobbottom);
label.rt("$z_2$",blobright);
\stopMPpage