diff --git a/Arrangement_on_surface_2/doc/Arrangement_on_surface_2/fig_src/arr_segs.tex b/Arrangement_on_surface_2/doc/Arrangement_on_surface_2/fig_src/arr_segs.tex
index e6198387b5f..e25fa3fd2be 100644
--- a/Arrangement_on_surface_2/doc/Arrangement_on_surface_2/fig_src/arr_segs.tex
+++ b/Arrangement_on_surface_2/doc/Arrangement_on_surface_2/fig_src/arr_segs.tex
@@ -2,98 +2,95 @@
 \input{header}
 \pagestyle{empty}
 \begin{document}
-\psscalebox{1.8}{\arrset{0.8}
-  \pspicture[](-0.2,-0.2)(12.2,12.2)
-  %\psgrid(0,0)(12,12)
-  \arrMainVertex( 1, 3){a}
-  \arrMainVertex( 2, 4){b}
-  \arrMainVertex( 0, 6){c}
-  \arrv[90](5,11){1}
-  \arrMainVertex(9,11){f}
-  \arrMainVertex(12, 8){g}
-  \arrv[-60](8,8){2}
-  \arrMainVertex( 8, 4){i}
-  \arrMainVertex( 6, 4){j}
-  \arrMainVertex( 6, 2){k}
-  \arrMainVertex( 4, 0){m}
-  \arrMainVertex( 9, 0){n}
-  \arrMainVertex(11, 2){o}
-  \arrMainVertex(12, 4){p}
-  \arrMainVertex(12, 0){q}
-  \arrMainVertex( 4, 2){r}
-  \arrMainVertex( 3, 3){s}
-  \arrMainVertex( 5, 3){t}
+\begin{tikzpicture}[scale=1.0,thick]
+  \node[point] at ( 1, 3) (a) {};
+  \node[point] at ( 2, 4) (b) {};
+  \node[point] at ( 0, 6) (c) {};
+  \node [point,label={[label distance=0pt]90:{$v_1$}}] at ( 5,11) (1) {};
+  \node[point] at ( 9,11) (f) {};
+  \node[point] at (12, 8) (g) {};
+  \node [point,label={[label distance=0pt]-45:{$v_2$}}] at ( 8, 8) (2) {};
+  \node[point] at ( 8, 4) (i) {};
+  \node[point] at ( 6, 4) (j) {};
+  \node[point] at ( 6, 2) (k) {};
+  \node[point] at ( 4, 0) (m) {};
+  \node[point] at ( 9, 0) (n) {};
+  \node[point] at (11, 2) (o) {};
+  \node[point] at (12, 4) (p) {};
+  \node[point] at (12, 0) (q) {};
+  \node[point] at ( 4, 2) (r) {};
+  \node[point] at ( 3, 3) (s) {};
+  \node[point] at ( 5, 3) (t) {};
   %
-  \ncline[linewidth=0.5pt,linecolor=blue]{a}{b}
-  \ncline[linewidth=0.5pt,linecolor=blue]{b}{c}
-  \ncline[linewidth=0.5pt,linecolor=blue]{c}{1}
-  \ncline[linewidth=0.5pt,linecolor=blue]{1}{f}
-  \ncline[linewidth=0.5pt,offset=3pt,nodesepB=6pt]{->}{f}{1}\Aput[1pt]{$e_{\mathrm{prev}}$}
-  \ncline[linewidth=0.5pt,linecolor=blue]{f}{g}
-  \ncline[linewidth=0.5pt,linecolor=blue]{g}{2}
-  \ncline[linewidth=0.5pt,linecolor=blue]{g}{2}
-  \ncline[linewidth=0.5pt,offset=3pt,nodesepB=4pt]{->}{2}{g}\Aput[1pt]{$e_{\mathrm{next}}$}
-  \ncline[linewidth=0.5pt,linecolor=blue]{2}{i}
-  \ncline[linewidth=0.5pt,linecolor=blue]{i}{j}
-  \ncline[linewidth=0.5pt,linecolor=blue]{j}{k}
-  \ncline[linewidth=0.5pt,linecolor=blue]{k}{m}
-  \ncline[linewidth=0.5pt,linecolor=blue]{m}{a}
-  \ncline[linewidth=0.5pt,linecolor=blue]{1}{2}
-  \ncline[linewidth=0.5pt,linecolor=blue]{g}{i}
-  \ncline[linewidth=0.5pt,linecolor=blue]{m}{n}
-  \ncline[linewidth=0.5pt,linecolor=blue]{n}{i}
-  \ncline[linewidth=0.5pt,linecolor=blue]{g}{n}
-  \ncline[linewidth=0.5pt,linecolor=blue]{n}{o}
-  \ncline[linewidth=0.5pt,linecolor=blue]{o}{p}
-  \ncline[linewidth=0.5pt,linecolor=blue]{p}{q}
-  \ncline[linewidth=0.5pt,linecolor=blue]{q}{o}
-  \ncline[linewidth=0.5pt,linecolor=blue]{r}{s}
-  \ncline[linewidth=0.5pt,offset=3pt]{->}{r}{s}
-  \ncline[linewidth=0.5pt,offset=3pt]{->}{s}{r}
-  \ncline[linewidth=0.5pt,linecolor=blue]{r}{t}
-  \ncline[linewidth=0.5pt,offset=3pt]{->}{r}{t}
-  \ncline[linewidth=0.5pt,offset=3pt]{->}{t}{r}
+  \draw[blue] (a)--(b);
+  \draw[blue] (b)--(c);
+  \draw[blue] (c)--(1);
+  \draw[blue] (1)--(f);
+  \draw [halfedge,shorten >=4pt] (f) to node[sloped,below]{$e_{\mathrm{prev}}$} (1);
+  \draw[blue] (f)--(g);
+  \draw[blue] (g)--(2);
+  \draw [halfedge,shorten >=4pt] (2) to node[sloped,above]{$e_{\mathrm{next}}$} (g);
+  \draw[blue] (2)--(i);
+  \draw[blue] (i)--(j);
+  \draw[blue] (j)--(k);
+  \draw[blue] (k)--(m);
+  \draw[blue] (m)--(a);
+  \draw[blue] (1)--(2);
+  \draw[blue] (g)--(i);
+  \draw[blue] (m)--(n);
+  \draw[blue] (n)--(i);
+  \draw[blue] (g)--(n);
+  \draw[blue] (n)--(o);
+  \draw[blue] (o)--(p);
+  \draw[blue] (p)--(q);
+  \draw[blue] (q)--(o);
+  \draw[blue] (r)--(s);
   %
-  \ncline[linewidth=0.5pt,offset=3pt]{->}{b}{a}
-  \ncline[linewidth=0.5pt,offset=3pt]{->}{c}{b}
-  \ncline[linewidth=0.5pt,offset=3pt]{->}{1}{c}
-  \ncline[linewidth=0.5pt,offset=3pt,nodesepA=4pt]{->}{g}{f}
-  \ncline[linewidth=0.5pt,offset=3pt]{->}{i}{2}
-  \ncline[linewidth=0.5pt,offset=3pt]{->}{j}{i}
-  \ncline[linewidth=0.5pt,offset=3pt]{->}{k}{j}
-  \ncline[linewidth=0.5pt,offset=3pt]{->}{m}{k}
-  \ncline[linewidth=0.5pt,offset=3pt]{->}{a}{m}
-  \ncline[linewidth=0.5pt,offset=3pt,nodesepA=6pt]{->}{1}{2}\Aput[1pt]{$e$}
-  \ncline[linewidth=0.5pt,offset=3pt]{->}{2}{1}\Aput[1pt]{$e'$}
+  \path [halfedge] (r) edge (s);
+  \path [halfedge] (s) edge (r);
+  \draw [blue] (r)--(t);
+  \path [halfedge] (r) edge (t);
+  \path [halfedge] (t) edge (r);
   %
-  \rput(2,10){$f_0$}
-  \rput(3,4){$f_1$}
-  \rput(8.5,9.5){$f_2$}
-  \rput(5,7){$f_3$}
-  \rput(4.5,5.6){$f_4$}
+  \path [halfedge] (b) edge (a);
+  \path [halfedge] (c) edge (b);
+  \path [halfedge] (1) edge (c);
+  \path [halfedge,shorten <=4pt] (g) edge (f);
+  \path [halfedge] (i) edge (2);
+  \path [halfedge] (j) edge (i);
+  \path [halfedge] (k) edge (j);
+  \path [halfedge] (m) edge (k);
+  \path [halfedge] (a) edge (m);
+  \draw [halfedge,shorten <=4pt] (1) to node[sloped,above=-1pt]{$e$} (2);
+  \draw [halfedge] (2) to node[sloped,below=-1pt]{$e'$} (1);
   %
-  \arru[90](2,6){1}
-  \arru[90](5,9){2}
+  \node at (2,10){$f_0$};
+  \node at (3,4){$f_1$};
+  \node at (8.5,9.5){$f_2$};
+  \node at (5,7){$f_3$};
+  \node at (4.5,5.6){$f_4$};
   %
-  \arrMainVertex(4,5){a}
-  \arrMainVertex(4,7){b}
-  \arrMainVertex(5,8){c}
-  \arrMainVertex(5,4){f}
-  \arrMainVertex(5,6){e}
-  \arrMainVertex(6,7){d}
-  \ncline[linewidth=0.5pt,linecolor=blue]{a}{b}
-  \ncline[linewidth=0.5pt,linecolor=blue]{b}{c}
-  \ncline[linewidth=0.5pt,linecolor=blue]{c}{d}
-  \ncline[linewidth=0.5pt,linecolor=blue]{d}{e}
-  \ncline[linewidth=0.5pt,linecolor=blue]{e}{f}
-  \ncline[linewidth=0.5pt,linecolor=blue]{f}{a}
-  \ncline[linewidth=0.5pt,linecolor=blue]{b}{e}
-  \ncline[linewidth=0.5pt,offset=3pt]{->}{a}{b}
-  \ncline[linewidth=0.5pt,offset=3pt]{->}{b}{c}
-  \ncline[linewidth=0.5pt,offset=3pt]{->}{c}{d}
-  \ncline[linewidth=0.5pt,offset=3pt]{->}{d}{e}
-  \ncline[linewidth=0.5pt,offset=3pt]{->}{e}{f}
-  \ncline[linewidth=0.5pt,offset=3pt]{->}{f}{a}
-  \endpspicture
-}
+  \node [point,label={[label distance=0pt]90:{$u_1$}}] at (2,6) (1) {};
+  \node [point,label={[label distance=0pt]90:{$u_2$}}] at (5,9) (1) {};
+  %
+  \node[point] at (4,5) (a) {};
+  \node[point] at (4,7) (b) {};
+  \node[point] at (5,8) (c) {};
+  \node[point] at (5,4) (f) {};
+  \node[point] at (5,6) (e) {};
+  \node[point] at (6,7) (d) {};
+  \draw[blue] (a)--(b);
+  \draw[blue] (b)--(c);
+  \draw[blue] (c)--(d);
+  \draw[blue] (d)--(e);
+  \draw[blue] (e)--(f);
+  \draw[blue] (f)--(a);
+  \draw[blue] (b)--(e);
+  \path [halfedge] (a) edge (b);
+  \path [halfedge] (b) edge (c);
+  \path [halfedge] (c) edge (d);
+  \path [halfedge] (d) edge (e);
+  \path [halfedge] (e) edge (f);
+  \path [halfedge] (f) edge (a);
+\end{tikzpicture}%
 \end{document}
diff --git a/Arrangement_on_surface_2/doc/Arrangement_on_surface_2/fig_src/bezier_curves.tex b/Arrangement_on_surface_2/doc/Arrangement_on_surface_2/fig_src/bezier_curves.tex
index e0013b99d72..6266e6524d0 100644
--- a/Arrangement_on_surface_2/doc/Arrangement_on_surface_2/fig_src/bezier_curves.tex
+++ b/Arrangement_on_surface_2/doc/Arrangement_on_surface_2/fig_src/bezier_curves.tex
@@ -2,7 +2,7 @@
 \input{header}
 \pagestyle{empty}
 \begin{document}
-\psscalebox{1.8}{\arrset{1}%
+\psscalebox{1}{\arrset{1}%
 \def\bodyBezier{
 \psline[linecolor=blue](2295,5175)(2492.01,5062.83)(2677.89,4983.16)(2854.44,4929.86)(3023.31,4897.62)(3185.98,4881.79)(3343.77,4878.41)(3497.83,4884.1)(3649.2,4896.04)(3798.76,4911.9)(3947.28,4929.78)(4095.39,4948.19)(4243.63,4965.94)(4392.4,4982.15)(4542.04,4996.13)(4692.77,5007.39)(4844.74,5015.54)(4998.01,5020.26)(5152.6,5021.23)(5308.42,5018.09)(5465.37,5010.38)(5623.29,4997.49)(5781.96,4978.58)(5941.15,4952.58)(6100.6,4918.08)(6260.03,4873.33)(6419.14,4816.11)(6577.65,4743.77)(6735.25,4653.1)(6891.68,4540.32)(7046.68,4400.99)(7200,4230)
 \psline[linecolor=blue](8460,6165)(8622.25,6003.86)(8725.83,5853.19)(8782.58,5711.47)(8802.7,5577.35)(8794.88,5449.63)(8766.41,5327.29)(8723.29,5209.44)(8670.37,5095.32)(8611.44,4984.3)(8549.35,4875.84)(8486.17,4769.53)(8423.24,4665.02)(8361.36,4562.04)(8300.84,4460.4)(8241.66,4359.96)(8183.58,4260.61)(8126.25,4162.29)(8069.34,4064.94)(8012.64,3968.53)(7956.2,3873.02)(7900.41,3778.35)(7846.17,3684.44)(7794.97,3591.19)(7749.02,3498.44)(7711.36,3405.97)(7685.99,3313.49)(7677.98,3220.65)(7693.59,3127)(7740.38,3031.97)(7827.34,2934.9)(7965,2835)
diff --git a/Arrangement_on_surface_2/doc/Arrangement_on_surface_2/fig_src/header.tex b/Arrangement_on_surface_2/doc/Arrangement_on_surface_2/fig_src/header.tex
index b75164d2a5c..26980faab6f 100644
--- a/Arrangement_on_surface_2/doc/Arrangement_on_surface_2/fig_src/header.tex
+++ b/Arrangement_on_surface_2/doc/Arrangement_on_surface_2/fig_src/header.tex
@@ -43,10 +43,25 @@
 \usepackage{tikz}
 \usepackage{pgfplots}
 \usepackage{mathptmx}
-\usetikzlibrary{calc,patterns,decorations.pathmorphing,decorations.markings,matrix,fit,decorations.pathreplacing,arrows,automata,positioning,shapes,chains,spy}
-%\usetikzlibrary{quotes,angles,graph}
-\usetikzlibrary{intersections,through,backgrounds}
+\usetikzlibrary{calc}
+\usetikzlibrary{backgrounds}
+\usetikzlibrary{decorations.pathmorphing}
+\usetikzlibrary{decorations.markings}
+\usetikzlibrary{decorations.pathreplacing}
+\usetikzlibrary{patterns}
+\usetikzlibrary{matrix}
+\usetikzlibrary{fit}
+\usetikzlibrary{arrows,arrows.meta}
+\usetikzlibrary{automata}
+\usetikzlibrary{positioning}
+\usetikzlibrary{shapes}
+\usetikzlibrary{chains}
+\usetikzlibrary{spy}
+\usetikzlibrary{intersections}
+\usetikzlibrary{through}
+\usetikzlibrary{cd}
 \usepackage{tikz-3dplot}
+%\usetikzlibrary{quotes,angles,graph}
 \tikzset{%
   invisible/.style={opacity=0},
   only/.code args={<#1>#2}{\only<#1>{\pgfkeysalso{#2}}},
@@ -54,8 +69,49 @@
   temporal/.code args={<#1>#2#3#4}{%
     \temporal<#1>{\pgfkeysalso{#2}}{\pgfkeysalso{#3}}{\pgfkeysalso{#4}}},
   point/.style={circle,inner sep=1.5pt,minimum size=1.5pt,draw,fill=#1},
-  point/.default=red
+  point/.default=red,
+  halfedge/.style={-{Stealth[left,scale=1.5]},commutative diagrams/shift left=#1},
+  halfedge/.default=3pt
 }
+% ------------------------------------------------------------------------------
+% #1 half length of tick
+% #2 min x
+% #3 min y
+% #4 max x
+% #5 max y
+\NewDocumentCommand{\myAxes}{O{4pt}mmmmO{0.001}}{%
+  %axis
+  \draw[-{Stealth[scale=1.5]}] (#2,0)--(#4,0);
+  \draw[-{Stealth[scale=1.5]}] (0,#3)--(0,#5);
+  %ticks
+  \pgfmathtruncatemacro{\xs}{int(#2)};
+  \pgfmathtruncatemacro{\xe}{int(#4-#6)};
+  \pgfmathtruncatemacro{\ys}{int(#3)};
+  \pgfmathtruncatemacro{\ye}{int(#5-#6)};
+  \foreach \x in {\xs,...,\xe} \draw (\x,#1)--(\x,-#1);
+  \foreach \y in {\ys,...,\ye} \draw (#1,\y)--(-#1,\y);%
+}
+% ------------------------------------------------------------------------------
+%Syntax: [draw options] (center) (initial angle:final angle:radius)
+\def\centerarc[#1](#2)(#3:#4:#5){\draw[#1]([shift=(#3:#5)]#2) arc (#3:#4:#5)}
+% ------------------------------------------------------------------------------
+\def\arrCrossVertexZ(#1)#2#3{\node[cross=#3] at (#1) (#2) {};}
+\def\arrQueryVertexZ(#1)#2{\arrCrossVertexZ(#1){#2}{red}}
+\def\arrCrossLabeledVertexZ[#1](#2)#3#4#5{\node [cross=#5,label={[label distance=-3pt]#1:{#4}}] at (#2) (#3) {};}
+\def\arrQueryLabeledVertexZ[#1](#2)#3#4{\arrCrossLabeledVertexZ[#1](#2){#3}{#4}{red}}
+%
+\def\arrColorVertexZ(#1)#2#3{\node[point=#3] at (#1) (#2) {};}
+\def\arrMinorVertexZ(#1)#2{\arrColorVertexZ(#1){#2}{cyan}}
+\def\arrMainVertexZ(#1)#2{\arrColorVertexZ(#1){#2}{red}}
+\def\arrIntersectionVertexZ(#1)#2{\arrColorVertexZ(#1){#2}{white}}
+\def\arrColoredLabeledVertexZ[#1](#2)#3#4#5{\node [point=#5,label={[label distance=-3pt]#1:{#4}}] at (#2) (#3) {};}
+\def\arrMainLabeledVertexZ[#1](#2)#3#4{\arrColoredLabeledVertexZ[#1](#2){#3}{#4}{red}}
+\def\arrMinorLabeledVertexZ[#1](#2)#3#4{\arrColoredLabeledVertexZ[#1](#2){#3}{#4}{cyan}}
+% ------------------------------------------------------------------------------
+\def\arrvZ[#1](#2)#3{\arrColoredLabeledVertexZ[#1](#2){#3}{$v_{#3}$}{red}}
+\def\arruZ[#1](#2)#3{\arrColoredLabeledVertexZ[#1](#2){#3}{$u_{#3}$}{red}}
+\def\arrpZ[#1](#2)#3{\arrColoredLabeledVertexZ[#1](#2){#3}{$p_{#3}$}{red}}
+% ==============================================================================
 \makeatletter
 \def\nodesDef{\@ifnextchar[{\@nodesDefWith}{\@nodesDefWithout}}
 \def\@nodesDefWith[#1]#2{\foreach \c in #2 { \node[point=#1] (n\c) at (\c) {}; }}
@@ -81,6 +137,7 @@
 % \usepackage{picins}
 \usepackage[absolute,overlay]{textpos}
 \usepackage{ifthen}
+\usepackage{xparse}
 \usepackage{etoolbox}
 %
 
diff --git a/Arrangement_on_surface_2/doc/Arrangement_on_surface_2/fig_src/insert.tex b/Arrangement_on_surface_2/doc/Arrangement_on_surface_2/fig_src/insert.tex
index 44f58b9c38a..060f751f133 100644
--- a/Arrangement_on_surface_2/doc/Arrangement_on_surface_2/fig_src/insert.tex
+++ b/Arrangement_on_surface_2/doc/Arrangement_on_surface_2/fig_src/insert.tex
@@ -2,96 +2,92 @@
 \input{header}
 \pagestyle{empty}
 \begin{document}
-\psscalebox{1.8}{\arrset{0.5}%
-  \begin{tabular}{ccc}
-    \pspicture[](0.5,0)(10,7)
-    \arrMainVertex(1,4){a}
-    \arrMainVertex(5,6){b}
-    \arrMainVertex(9,4){c}
-    \arrMainVertex(7,2){d}
-    \arrMainVertex(3,1){e}
-    \arrMainVertex(3,2){f}
-    \ncline[linewidth=0.5pt,linecolor=blue]{a}{b}
-    \ncline[linewidth=0.5pt,linecolor=blue]{b}{c}
-    \ncline[linewidth=0.5pt,linecolor=blue]{c}{d}
-    \ncline[linewidth=0.5pt,linecolor=blue]{d}{e}
-    \ncline[linewidth=0.5pt,linecolor=blue]{e}{f}
-    \ncline[linewidth=0.5pt,linecolor=blue]{f}{a}
-    \arrMinorVertex(3,3){1}\uput[135]{0}(3,3){$v_1$}
-    \arrMinorVertex(6,4){2}\uput[-45]{0}(6,4){$v_2$}
-    \ncline[linestyle=dashed,linewidth=0.5pt,linecolor=blue]{1}{2}
-    \ncline[linewidth=0.5pt,offset=3pt]{->}{1}{2}
-    \ncline[linewidth=0.5pt,offset=3pt]{->}{2}{1}
-    \rput(5,5){$f$}
-    \psframe[linewidth=0.5pt](0.5,0)(10,7)
-    \endpspicture &
-    %
-    \pspicture[](0.5,0)(10,7)
-    \arrMainVertex(1,3){a}
-    \arrMainVertex(2,5){b}
-    \arrMainVertex(5,6){c}
-    \arrMainVertex(7,3){u}
-    \uput[180]{0}(7,3){$u$}
-    \arrMainVertex(5,1){e}
-    \arrMainVertex(3,2){f}
-    \arrMainVertex(2,1){g}
-    \ncline[linewidth=0.5pt,linecolor=blue]{a}{b}
-    \ncline[linewidth=0.5pt,linecolor=blue]{b}{c}
-    \ncline[linewidth=0.5pt,linecolor=blue]{c}{u}
-    \ncline[linewidth=0.5pt,linecolor=blue]{u}{e}
-    \ncline[linewidth=0.5pt,linecolor=blue]{e}{f}
-    \ncline[linewidth=0.5pt,linecolor=blue]{f}{g}
-    \ncline[linewidth=0.5pt,linecolor=blue]{g}{a}
-    \arrMinorVertex(9,4){v}
-    \uput[90]{0}(9,4){$v$}
-    \ncline[linestyle=dashed,linewidth=0.5pt,linecolor=blue]{u}{v}
-    \ncline[linestyle=dashed,linewidth=0.5pt,offset=3pt]{->}{c}{u}
-    \ncline[linestyle=dashed,linewidth=0.5pt,offset=3pt]{->}{u}{e}
-    \ncline[linewidth=0.5pt,offset=3pt]{->}{u}{v}
-    \ncline[linewidth=0.5pt,offset=3pt]{->}{v}{u}
-    \psframe[linewidth=0.5pt](0.5,0)(10,7)
-    \endpspicture &
-    %
-    \pspicture[](0.5,0)(10,7)
-    \arrMainVertex(1,3){a}
-    \arrMainVertex(2,6){b}
-    \arru[45](5,6){1}
-    \arru[-45](7,1){2}
-    \arrMainVertex(4,2){e}
-    \arrMainVertex(2,1){f}
-    \ncline[linewidth=0.5pt,linecolor=blue]{a}{b}
-    \ncline[linewidth=0.5pt,linecolor=blue]{b}{1}
-    \ncline[linestyle=dashed,linewidth=0.5pt,offset=3pt]{->}{b}{1}
-    \ncline[linestyle=dashed,linewidth=0.5pt,offset=3pt]{->}{1}{b}
-    \ncline[linestyle=dashed,linewidth=0.5pt,linecolor=blue]{1}{2}
-    \ncline[linewidth=0.5pt,offset=3pt]{->}{1}{2}
-    \ncline[linewidth=0.5pt,offset=3pt,nodesepA=6pt]{->}{2}{1}
-    \ncline[linewidth=0.5pt,linecolor=blue]{2}{e}
-    \ncline[linestyle=dashed,linewidth=0.5pt,offset=3pt]{->}{2}{e}
-    \ncline[linestyle=dashed,linewidth=0.5pt,offset=3pt,nodesepB=4pt]{->}{e}{2}
-    \ncline[linewidth=0.5pt,linecolor=blue]{e}{f}
-    \ncline[linewidth=0.5pt,linecolor=blue]{f}{a}
-    \arrMainVertex(2.5,4){a}
-    \arrMainVertex(3.5,5){b}
-    \arrMainVertex(4.5,4){c}
-    \arrMainVertex(3.5,3){d}
-    \ncline[linewidth=0.5pt,linecolor=blue]{a}{b}
-    \ncline[linewidth=0.5pt,linecolor=blue]{b}{c}
-    \ncline[linewidth=0.5pt,linecolor=blue]{c}{d}\Aput[1pt]{$h_1$}
-    \ncline[linewidth=0.5pt,linecolor=blue]{d}{a}
-    \ncline[linewidth=0.5pt,linecolor=blue]{a}{c}
-    \rput(1,5){$f$}
-    \rput(2.5,5){$f'$}
-    \arrMainVertex(7,4){a}
-    \arrMainVertex(8,4){b}
-    \arrMainVertex(8,5){c}
-    \arrMainVertex(9,3){d}
-    \ncline[linewidth=0.5pt,linecolor=blue]{a}{b}
-    \ncline[linewidth=0.5pt,linecolor=blue]{b}{c}
-    \ncline[linewidth=0.5pt,linecolor=blue]{c}{a}
-    \ncline[linewidth=0.5pt,linecolor=blue]{b}{d}\Aput[1pt]{$h_2$}
-    \psframe[linewidth=0.5pt](0.5,0)(10,7)
-    \endpspicture
-  \end{tabular}
-}
+\begin{tabular}{ccc}
+  \begin{tikzpicture}
+    \node[point] at (1,4) (a) {};
+    \node[point] at (5,6) (b) {};
+    \node[point] at (9,4) (c) {};
+    \node[point] at (7,2) (d) {};
+    \node[point] at (3,1) (e) {};
+    \node[point] at (3,2) (f) {};
+    \draw[blue,very thick](a)--(b);
+    \draw[blue,very thick](b)--(c);
+    \draw[blue,very thick](c)--(d);
+    \draw[blue,very thick](d)--(e);
+    \draw[blue,very thick](e)--(f);
+    \draw[blue,very thick](f)--(a);
+    \node[point=cyan,label={[label distance=0pt]135:{$v_1$}}] at (3,3) (1) {};
+    \node[point=cyan,label={[label distance=0pt]135:{$v_2$}}] at (6,4) (2) {};
+    \draw[dashed,blue,very thick] (1)--(2);
+    \draw[halfedge] (1) to (2);
+    \draw[halfedge] (2) to (1);
+    \node at (5,5) {$f$};
+    \draw[thin](0.5,0) rectangle (10,7);
+  \end{tikzpicture} &
+  %
+  \begin{tikzpicture}
+    \node[point] at (1,3) (a) {};
+    \node[point] at (2,5) (b) {};
+    \node[point] at (5,6) (c) {};
+    \node[point,label={[label distance=0pt]180:{$u$}}] at (7,3) (u) {};
+    \node[point] at (5,1) (e) {};
+    \node[point] at (3,2) (f) {};
+    \node[point] at (2,1) (g) {};
+    \draw[blue,very thick](a)--(b);
+    \draw[blue,very thick](b)--(c);
+    \draw[blue,very thick](c)--(u);
+    \draw[blue,very thick](u)--(e);
+    \draw[blue,very thick](e)--(f);
+    \draw[blue,very thick](f)--(g);
+    \draw[blue,very thick](g)--(a);
+    \node[point=cyan,label={[label distance=0pt]90:{$v$}}] at (9,4) (v) {};
+    \draw[dashed,blue,very thick] (u)--(v);
+    \draw[dashed,halfedge] (c) to (u);
+    \draw[dashed,halfedge] (u) to (e);
+    \draw[halfedge] (u) to (v);
+    \draw[halfedge] (v) to (u);
+    \draw[thin] (0.5,0) rectangle (10,7);
+  \end{tikzpicture} &
+  %
+  \begin{tikzpicture}
+    \node[point] at (1,3) (a) {};
+    \node[point] at (2,6) (b) {};
+    \node[point,label={[label distance=0pt]45:{$u_1$}}] at (5,6) (1) {};
+    \node[point,label={[label distance=0pt]45:{$u_2$}}] at (7,1) (2) {};
+    \node[point] at (4,2) (e) {};
+    \node[point] at (2,1) (f) {};
+    \draw[blue,very thick](a)--(b);
+    \draw[blue,very thick](b)--(1);
+    \draw[dashed,thick,halfedge] (b) to (1);
+    \draw[dashed,thick,halfedge] (1) to (b);
+    \draw[dashed,blue,very thick](1)--(2);
+    \draw[thick,halfedge] (1) to (2);
+    \draw[thick,halfedge,shorten <=4pt] (2) to (1);
+    \draw[blue,very thick] (2)--(e);
+    \draw[dashed,halfedge] (2) to (e);
+    \draw[dashed,halfedge,shorten >=4pt] (e) to (2);
+    \draw[blue,very thick] (e)--(f);
+    \draw[blue,very thick] (f)--(a);
+    \node[point] at (2.5,4) (a) {};
+    \node[point] at (3.5,5) (b) {};
+    \node[point] at (4.5,4) (c) {};
+    \node[point] at (3.5,3) (d) {};
+    \draw[blue,very thick](a)--(b);
+    \draw[blue,very thick](b)--(c);
+    \draw[blue,very thick](c)-- node[midway,below,sloped] {$h_1$} (d);
+    \draw[blue,very thick](d)--(a);
+    \draw[blue,very thick](a)--(c);
+    \node at (1,5) {$f$};
+    \node at (2.5,5) {$f'$};
+    \node[point] at (7,4) (a) {};
+    \node[point] at (8,4) (b) {};
+    \node[point] at (8,5) (c) {};
+    \node[point] at (9,3) (d) {};
+    \draw[blue,very thick] (a)--(b);
+    \draw[blue,very thick] (b)--(c);
+    \draw[blue,very thick] (c)--(a);
+    \draw[blue,very thick] (b)-- node[midway,below,sloped] {$h_2$} (d);
+    \draw[thin] (0.5,0) rectangle (10,7);
+  \end{tikzpicture}
+\end{tabular}
 \end{document}