From 637fd9e4d1438d616346059db437260ee989f53b Mon Sep 17 00:00:00 2001
From: Martin Mares <mj@ucw.cz>
Date: Wed, 16 Jan 2019 22:51:29 +0100
Subject: [PATCH] =?utf8?q?Planarita:=20Obr=C3=A1zky=20a=20korektury=20k=20?=
 =?utf8?q?nov=C3=A9mu=20d=C5=AFkazu?=
MIME-Version: 1.0
Content-Type: text/plain; charset=utf8
Content-Transfer-Encoding: 8bit

---
 11-planar/11-planar.tex |  73 +++++++++++++++++++++-------------------
 11-planar/Makefile      |   2 ++
 11-planar/minor1.epdf   | Bin 5348 -> 0 bytes
 11-planar/minor1.vrr    |  25 --------------
 11-planar/minor2.epdf   | Bin 5062 -> 0 bytes
 11-planar/minor2.vrr    |  22 ------------
 11-planar/minor3.epdf   | Bin 6178 -> 0 bytes
 11-planar/minor3.vrr    |  29 ----------------
 11-planar/minor4.epdf   | Bin 6345 -> 0 bytes
 11-planar/minor4.vrr    |  35 -------------------
 11-planar/minor5.epdf   | Bin 6354 -> 0 bytes
 11-planar/minor5.vrr    |  30 -----------------
 11-planar/minor6.epdf   | Bin 5935 -> 0 bytes
 11-planar/minor6.vrr    |  29 ----------------
 11-planar/pl-minor1.asy |  39 +++++++++++++++++++++
 11-planar/pl-minor2.asy |  36 ++++++++++++++++++++
 11-planar/pl-minor3.asy |  40 ++++++++++++++++++++++
 11-planar/pl-minor4.asy |  39 +++++++++++++++++++++
 11-planar/pl-minor5.asy |  41 ++++++++++++++++++++++
 11-planar/pl-minor6.asy |  38 +++++++++++++++++++++
 Makerules               |   3 ++
 21 files changed, 276 insertions(+), 205 deletions(-)
 delete mode 100644 11-planar/minor1.epdf
 delete mode 100644 11-planar/minor1.vrr
 delete mode 100644 11-planar/minor2.epdf
 delete mode 100644 11-planar/minor2.vrr
 delete mode 100644 11-planar/minor3.epdf
 delete mode 100644 11-planar/minor3.vrr
 delete mode 100644 11-planar/minor4.epdf
 delete mode 100644 11-planar/minor4.vrr
 delete mode 100644 11-planar/minor5.epdf
 delete mode 100644 11-planar/minor5.vrr
 delete mode 100644 11-planar/minor6.epdf
 delete mode 100644 11-planar/minor6.vrr
 create mode 100644 11-planar/pl-minor1.asy
 create mode 100644 11-planar/pl-minor2.asy
 create mode 100644 11-planar/pl-minor3.asy
 create mode 100644 11-planar/pl-minor4.asy
 create mode 100644 11-planar/pl-minor5.asy
 create mode 100644 11-planar/pl-minor6.asy

diff --git a/11-planar/11-planar.tex b/11-planar/11-planar.tex
index dbf278f..7401c66 100644
--- a/11-planar/11-planar.tex
+++ b/11-planar/11-planar.tex
@@ -315,13 +315,17 @@ graf nebyl rovinný. Tomu věnujeme zbytek kapitoly.
 není rovinný.
 
 \proof Pro spor předpokládejme, že po~zpracování vrcholu~$v$ existuje nějaká
-zpětná hrana~$wv$, kterou algoritmus nenakreslil, čili že přístup z~$v$ k~$w$
+zpětná hrana~$wv$, kterou algoritmus nenakreslil. Tedy přístup z~$v$ k~$w$
 je v~obou směrech blokován externími vrcholy. Rozborem případů ukážeme,
-že pokud jsme se důsledně řídili pravidly \#1 a \#2, graf není rovinný.
+že pokud jsme se důsledně řídili pravidly \#1 a \#2, může se to stát pouze
+v~nerovinném grafu.
 
+{\I Překážející blok.}
 Uvažme posloupnost bloků vedoucí od vrcholu~$v$ k~$w$. V~této posloupnosti se
 musí vyskytovat nějaký blok~$B$, který má na obou stranách hranice aspoň jeden
 externí vrchol -- jinak bychom totiž bloky popřeklápěli tak, abychom se dostali až do~$w$.
+Označme~$x$ externí vrchol na levé straně hranice bloku~$B$, $y$~ten na pravé.
+Vrcholy $x$, $y$, $v$ a~$w$ jsou zjevně navzájem různé.
 
 {\I Nadřazené bloky.}
 Nejprve ukážeme, že kořenem bloku~$B$ musí být přímo vrchol~$v$. Kdyby totiž
@@ -337,80 +341,79 @@ muset zkoumat vnitřek bloku.
 
 {\I Podřízené bloky.}
 Předtím ale rozebereme případ, kdy vrchol~$w$ neleží přímo v~bloku~$B$, nýbrž
-v~nějakém podřízeném bloku, který je připojen pod nějakou artikulaci $w'\in B$.
+v~nějakém podřízeném bloku, který je připojen pod nějakou artikulací $w'\in B$.
 Tento podřízený blok přitom nemůže být externí: kdyby byl, najdeme v~grafu
-minor XXXX obsahující $K_{3,3}$ (do~$w$ jsme zkontrahovali všechny podřízené
-bloky mezi~$w'$ a~$w$). Pokud bychom tedy během obcházení hranice vstoupili do~$w'$,
-pravidlo~\#2 by zaručilo, že dojdeme do~$w$ a díky pravidlu~\#1 bychom vzápětí
-nakreslili hranu~$vw$. Tu jsme nenakreslili, takže jsme nedošli ani do~$w'$.
-Můžeme tedy bez újmy na obecnosti předpokládat, že $w=w'$.
+minor~$P$ obsahující $K_{3,3}$. Tedy není externí, takže vstoupíme-li během obcházení
+do~$w'$, pravidlo~\#2 zaručí, že dojdeme do~$w$ a díky pravidlu~\#1 vzápětí
+nakreslíme hranu~$wv$. Tu jsme nenakreslili, takže jsme nedošli ani do~$w'$.
+Můžeme proto bez újmy na obecnosti předpokládat, že hrana $wv$ vede přímo
+z~bloku~$B$.
+
+\bigskip
+\centerline{\putepdf{}{pl-minor1.pdf}\qquad\putepdf{}{pl-minor2.pdf}}
+\bigskip
+\centerline{\putepdf{}{pl-minor3.pdf}\qquad\putepdf{}{pl-minor4.pdf}}
+\bigskip
+\centerline{\putepdf{}{pl-minor5.pdf}\qquad\putepdf{}{pl-minor6.pdf}}
+\bigskip
 
 {\I Existence plotu.}
 Stačí se tedy omezit na situaci s~jediným blokem~$B$, v~němž leží vrcholy~$v$, $w$,
 $x$ i~$y$. Dokážeme nyní, že uvnitř bloku existuje {\I plot} -- cesta mezi~$x$
-a~$y$, jejíž ostatní vrcholy neleží na hranici bloku.
+a~$y$, jejíž zbývající vrcholy neleží na hranici bloku.
 
 Předpokládejme pro spor, že plot neexistuje. Před nakreslením zpětných hran
 vedoucích do~$v$ ještě blok~$B$ neexistoval a jeho vrcholy patřily do několika
 menších bloků. Speciálně víme, že $w$~byla artikulace oddělující $x$ od~$y$,
 takže každá cesta mezi~$x$ a~$y$ musela procházet přes~$w$. Proto v~pořadí podle DFS
-musí ležet~$w$ před aspoň jedním z~vrcholů $x$ a~$y$. Tento vrchol tedy tehdy musel
+musí ležet~$w$ před aspoň jedním z~vrcholů $x$ a~$y$. Buď~$x$ nebo~$y$ tedy předtím musel
 ležet v~nějakém podřízeném bloku pod~$w$. A~z~nějakého takového bloku také musela
 vést zpětná hrana (tehdy ještě externí) do~$v$, která později uzavřela blok~$B$.
-K~této hraně jsme se tedy museli během obcházení dostat, a~to je možné pouze přes~$w$.
+K~této hraně jsme se museli během obcházení dostat, a~to je možné pouze přes~$w$.
+Vrchol~$w$ tedy musel být navštíven a hrana $wv$ nakreslena, což je spor.
 
 {\I Nejvyšší plot.}
-Plot tedy existuje. Zvolíme mezi všemi ploty ten nejvyšší, tedy nejbližší k~$v$
+Plot tedy existuje. Zvolíme mezi všemi ploty ten nejvyšší, čili nejbližší k~$v$
 (rozmyslete si, že to je v~rovinném nakreslení dobře definované). Označme $p_x$~vrchol,
-v~němž se plot napojuje na cestu z~$v$ do~$w$ přes~$x$, a~obdobně~$p_y$. Rozmyslíme si,
+v~němž se plot napojuje na levou cestu z~$v$ do~$w$, a~obdobně~$p_y$ na pravé cestě. Rozmyslíme si,
 jak může situace vypadat.
 
 {\bf A.} Předně žádný z~vrcholů $p_x$ a~$p_y$ nemůže být blíž k~$v$ než $x$ a~$y$. Pokud by
-bez újmy na obecnosti $p_x$~ležel mezi $v$ a~$x$, našli bychom v~graf uminor AAA obsahující $K_{3,3}$:
+bez újmy na obecnosti $p_x$~ležel mezi $v$ a~$x$, našli bychom v~grafu minor~$N_A$ obsahující $K_{3,3}$:
 cestu mezi $y$ a~$p_y$ jsme zkontrahovali do~$y$, celý plot jsme zkontrahovali do hrany~$p_xy$,
 ostatní vrcholy jsou utvořeny stejně jako v~předchozích minorech.
 
 {\bf B.} Dále ukážeme, že plot může být připojen k~$v$ pouze přes $p_x$ a~$p_y$. V~opačném
-případě by se v~grafu vyskytoval minor BBB obsahujicí $K_{3,3}$: do~$x$ jsme zkontrahovali
+případě by se v~grafu vyskytoval minor~$N_B$ obsahující $K_{3,3}$: do~$x$ jsme zkontrahovali
 cestu mezi $x$ a~$p_x$, podobně do~$y$ cestu mezi $y$ a~$p_y$.
 
-{\bf C.} Nakonec se přesvědčíme, že na \uv{dolní} cestě z~$x$ do~$y$ přes~$w$ nemůže ležet
-žádný externí vrchol. To by totiž způsobovalo minor CCC isomorfní s~$K_5$:
+{\bf C.} Nakonec se přesvědčíme, že na dolní cestě z~$x$ do~$y$ přes~$w$ nemůže ležet
+žádný externí vrchol. To by totiž způsobovalo minor~$N_C$ isomorfní s~$K_5$:
 do~$w$ jsme zkontrahovali cestu mezi externím vrcholem a~$w$ a také všechny
-případné podřízené bloky, v~nichž vede externí hrana.
+případné podřízené bloky až k~externí hraně.
 
 {\I Vnitřek bloku~$B$.}
-Podívejme se, jak graf vypadal před nakreslením vrcholu~$v$. I~tehdy musel plot
-společně s~dolní cestou ležet v~jednom bloku. Ten musel být z~jedné strany připojený
-k~$v$ přes~$p_x$, z~druhé přes~$p_y$ (a~díky vlastnosti~{\bf B} nikudy jinudy).
+Nyní uvažujme, jak graf vypadal před nakreslením vrcholu~$v$. I~tehdy musel plot
+společně s~dolní cestou ležet v~jednom bloku. Tento blok musel být z~jedné strany připojený
+nenakreslenými hranami k~$v$ přes~$p_x$, z~druhé přes~$p_y$ (a~díky vlastnosti~{\bf B} nikudy jinudy).
 Říkejme tomuto bloku~$B'$ a~označme $r_1\in\{p_x,p_y\}$ jeho kořen a $r_2$ druhý
 z~vrcholů $p_x,p_y$.
 
 Jelikož jsme nakonec nakreslili zpětnou hranu uzavírající blok~$B$, museli jsme
-někdy dojít do~$p_x$. V~tomto okamžiku:
+někdy dojít do~$r_1$. V~tomto okamžiku:
 
 \itemize\ibull
 \:Díky {\bf A:} $r_2$~je předkem $x$ nebo~$y$ (případně je takovému vrcholu roven),
 takže je nyní externí.
-\:Díky {\bf B:} půjdeme-li z~$r_1$ po, nejbližší \uv{zajímavý} vrchol bude~$r_2$.
+\:Díky {\bf B:} půjdeme-li z~$r_1$ po plotu, nejbližší \uv{zajímavý} vrchol bude~$r_2$.
 \:Díky {\bf C:} žádný vrchol na dolní cestě není externí.
 \endlist
 
 Při vstupu~$r_1$ tedy plot vede k~externími vrcholu, zatímco dolní cesta
-k~internímu. Podle pravidla \#2 si tedy algoritmus vybere dolní cestu, kde ho
-nic nezastaví, takže dojde až do~$w$ a~nakreslí zpětnou hranu $vw$.
+k~internímu. Podle pravidla \#2 si algoritmus musí vybrat dolní cestu, kde ho
+nic nezastaví, takže dojde až do~$w$ a~nakreslí zpětnou hranu $wv$. To je opět spor.
 \qed
 
-\bigskip
-\centerline{\putepdf{}{minor1.epdf}\qquad\putepdf{}{minor2.epdf}}
-\bigskip
-
-\bigskip
-\centerline{\putepdf{}{minor3.epdf}\qquad\putepdf{}{minor4.epdf}}
-\bigskip
-\centerline{\putepdf{}{minor5.epdf}\qquad\putepdf{}{minor6.epdf}}
-\bigskip
-
 \s{Poznámka:} Podle tohoto důkazu bychom také mohli v~lineárním čase v~každém nerovinném
 grafu nalézt Kuratowského podgraf, dokonce také v~$O(n)$, jelikož když je $m>3n-6$,
 můžeme se omezit na~libovolných $3n-5$ hran, které určitě tvoří nerovinný podgraf.
diff --git a/11-planar/Makefile b/11-planar/Makefile
index 124e770..e286802 100644
--- a/11-planar/Makefile
+++ b/11-planar/Makefile
@@ -1,3 +1,5 @@
 P=11-planar
 
 include ../Makerules
+
+$(P).pdf: $(patsubst %, pl-minor%.pdf, 1 2 3 4 5 6)
diff --git a/11-planar/minor1.epdf b/11-planar/minor1.epdf
deleted file mode 100644
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- (parametric-point 139743272 0.387289702892303 ( (name "noname")))
- (parametric-point 139743272 0.61937540769577 ( (name "noname"))) 139619720 139743272 139742504
- (tex-text (coords -22.7725009918213 10.2322235107422) "$v$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (decorator-point (coords -20.0000019073486 10.0) 0 1.0 6.0491561271192e-7 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -37.1394920349121 -3.80593585968018) "$N_4$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -20.248571395874 -12.9403553009033) "$w$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -6.94939517974854 -1.59444499015808) "$y$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -32.8320045471191 -1.57180035114288) "$x$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -21.8017578125 17.310417175293) "$u$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname"))) 139619464 139742080
- (decorator-point (coords -20.0 15.0) 0 1.0 6.0491561271192e-7 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))))))))
\ No newline at end of file
diff --git a/11-planar/pl-minor1.asy b/11-planar/pl-minor1.asy
new file mode 100644
index 0000000..41fa012
--- /dev/null
+++ b/11-planar/pl-minor1.asy
@@ -0,0 +1,39 @@
+unitsize(1cm);
+currentpen = black + roundcap + 0.2mm;
+
+real vsize = 0.1;
+real xsize = 0.08;
+
+void vert(pair v)
+{
+	filldraw(circle(v, vsize), white, black);
+}
+
+void xert(pair v)
+{
+	filldraw(box(v-(xsize,xsize), v+(xsize,xsize)), white, black);
+}
+
+pair r = (0, 0);
+pair v = (0, 0.5);
+pair u = (0, 1);
+pair w = (0, -2);
+pair x = (-1, -1);
+pair y = (1, -1);
+
+draw(r -- v);
+draw(v -- u);
+filldraw(circle((0,-1), 1), gray(0.75));
+
+draw(x {dir(180)} .. {dir(0)} u);
+draw(y {dir(0)} .. {dir(180)} u);
+draw(w {dir(-45)} .. {dir(180)} v);
+
+vert(r); label("$r$", r, 2S);
+vert(v); label("$v$", v, 1.5W);
+vert(u); label("$u$", u, 2N);
+vert(w); label("$w$", w, 2SW);
+xert(x); label("$x$", x, 1.5SW);
+xert(y); label("$y$", y, 1.5SE);
+
+label("$M$", (-1.3, -1.9));
diff --git a/11-planar/pl-minor2.asy b/11-planar/pl-minor2.asy
new file mode 100644
index 0000000..47b8020
--- /dev/null
+++ b/11-planar/pl-minor2.asy
@@ -0,0 +1,36 @@
+unitsize(1cm);
+currentpen = black + roundcap + 0.2mm;
+
+real vsize = 0.1;
+real xsize = 0.08;
+
+void vert(pair v)
+{
+	filldraw(circle(v, vsize), white, black);
+}
+
+void xert(pair v)
+{
+	filldraw(box(v-(xsize,xsize), v+(xsize,xsize)), white, black);
+}
+
+pair v = (0, 0);
+pair u = (0, 0.5);
+pair w = (0, -2);
+pair x = (-1, -1);
+pair y = (1, -1);
+
+draw(v -- u);
+filldraw(circle((0,-1), 1), gray(0.75));
+
+draw(x {dir(180)} .. {dir(0)} u);
+draw(y {dir(0)} .. {dir(180)} u);
+draw(w {dir(-40)} .. {dir(210)} v);
+
+vert(v); label("$v$", v, 2S);
+vert(u); label("$u$", u, 2N);
+vert(w); label("$w$", w, 2SW);
+xert(x); label("$x$", x, 1.5SW);
+xert(y); label("$y$", y, 1.5SE);
+
+label("$N$", (-1.3, -1.9));
diff --git a/11-planar/pl-minor3.asy b/11-planar/pl-minor3.asy
new file mode 100644
index 0000000..3e91885
--- /dev/null
+++ b/11-planar/pl-minor3.asy
@@ -0,0 +1,40 @@
+unitsize(1cm);
+currentpen = black + roundcap + 0.2mm;
+
+real vsize = 0.1;
+real xsize = 0.08;
+
+void vert(pair v)
+{
+	filldraw(circle(v, vsize), white);
+}
+
+void xert(pair v)
+{
+	filldraw(box(v-(xsize,xsize), v+(xsize,xsize)), white);
+}
+
+pair v = (0, 0);
+pair u = (0, 0.5);
+pair ww = (0, -2);
+pair w = (0, -2.5);
+pair x = (-1, -1);
+pair y = (1, -1);
+
+draw(v -- u);
+draw(ww -- w);
+filldraw(circle((0,-1), 1), gray(0.75));
+
+draw(x {dir(180)} .. {dir(0)} u);
+draw(y {dir(0)} .. {dir(180)} u);
+draw(w {dir(-20)} .. {dir(210)} v);
+draw(w {dir(200)} .. {dir(-20)} u);
+
+vert(v); label("$v$", v, 2S);
+vert(u); label("$u$", u, 2N);
+vert(ww); label("$w'$", ww, 2N);
+xert(w); label("$w$", w, 2S);
+xert(x); label("$x$", x, 1.5SW);
+xert(y); label("$y$", y, 1.5SE);
+
+label("$P$", (-1.3, -1.9));
diff --git a/11-planar/pl-minor4.asy b/11-planar/pl-minor4.asy
new file mode 100644
index 0000000..72a8e07
--- /dev/null
+++ b/11-planar/pl-minor4.asy
@@ -0,0 +1,39 @@
+unitsize(1cm);
+currentpen = black + roundcap + 0.2mm;
+
+real vsize = 0.1;
+real xsize = 0.08;
+
+void vert(pair v)
+{
+	filldraw(circle(v, vsize), white, black);
+}
+
+void xert(pair v)
+{
+	filldraw(box(v-(xsize,xsize), v+(xsize,xsize)), white, black);
+}
+
+pair v = (0, 0);
+pair u = (0, 0.5);
+pair w = (0, -2);
+pair x = (-1, -1);
+pair y = (1, -1);
+pair px = (0,-1) + dir(150);
+
+draw(v -- u);
+filldraw(circle((0,-1), 1), gray(0.75));
+draw(px -- y);
+
+draw(x {dir(180)} .. {dir(0)} u);
+draw(y {dir(0)} .. {dir(180)} u);
+draw(w {dir(-40)} .. {dir(210)} v);
+
+vert(v); label("$v$", v, 2S);
+vert(u); label("$u$", u, 2N);
+vert(w); label("$w$", w, 2SW);
+xert(x); label("$x$", x, 1.5SW);
+xert(y); label("$y$", y, 1.5SE);
+vert(px); label("$p_x$", px, 1.5NNW);
+
+label("$N_A$", (-1.3, -1.9));
diff --git a/11-planar/pl-minor5.asy b/11-planar/pl-minor5.asy
new file mode 100644
index 0000000..657ef49
--- /dev/null
+++ b/11-planar/pl-minor5.asy
@@ -0,0 +1,41 @@
+unitsize(1cm);
+currentpen = black + roundcap + 0.2mm;
+
+real vsize = 0.1;
+real xsize = 0.08;
+
+void vert(pair v)
+{
+	filldraw(circle(v, vsize), white, black);
+}
+
+void xert(pair v)
+{
+	filldraw(box(v-(xsize,xsize), v+(xsize,xsize)), white, black);
+}
+
+pair v = (0, 0);
+pair u = (0, 0.5);
+pair w = (0, -2);
+pair x = (-1, -1);
+pair y = (1, -1);
+pair z = (0,-1);
+
+draw(v -- u);
+filldraw(circle((0,-1), 1), gray(0.75));
+draw(x -- z);
+draw(z -- y);
+draw(v -- z);
+
+draw(x {dir(180)} .. {dir(0)} u);
+draw(y {dir(0)} .. {dir(180)} u);
+draw(w {dir(-40)} .. {dir(210)} v);
+
+vert(v); label("$v$", v, 2*dir(-120));
+vert(u); label("$u$", u, 2N);
+vert(w); label("$w$", w, 2SW);
+xert(x); label("$x$", x, 1.5SW);
+xert(y); label("$y$", y, 1.5SE);
+vert(z); label("$z$", z, 2S);
+
+label("$N_B$", (-1.3, -1.9));
diff --git a/11-planar/pl-minor6.asy b/11-planar/pl-minor6.asy
new file mode 100644
index 0000000..1cb0790
--- /dev/null
+++ b/11-planar/pl-minor6.asy
@@ -0,0 +1,38 @@
+unitsize(1cm);
+currentpen = black + roundcap + 0.2mm;
+
+real vsize = 0.1;
+real xsize = 0.08;
+
+void vert(pair v)
+{
+	filldraw(circle(v, vsize), white, black);
+}
+
+void xert(pair v)
+{
+	filldraw(box(v-(xsize,xsize), v+(xsize,xsize)), white, black);
+}
+
+pair v = (0, 0);
+pair u = (0, 0.5);
+pair w = (0, -2);
+pair x = (-1, -1);
+pair y = (1, -1);
+
+draw(v -- u);
+filldraw(circle((0,-1), 1), gray(0.75));
+draw(x -- y);
+
+draw(x {dir(180)} .. {dir(0)} u);
+draw(y {dir(0)} .. {dir(180)} u);
+draw(w {dir(-40)} .. {dir(210)} v);
+draw(w {dir(220)} .. {dir(-20)} u);
+
+vert(v); label("$v$", v, 2S);
+vert(u); label("$u$", u, 2N);
+xert(w); label("$w$", w, 2S);
+xert(x); label("$x$", x, 1.5SW);
+xert(y); label("$y$", y, 1.5SE);
+
+label("$N_C$", (-1.3, -1.9));
diff --git a/Makerules b/Makerules
index c40d14a..5c4c604 100644
--- a/Makerules
+++ b/Makerules
@@ -14,6 +14,9 @@ all: $P.pdf
 %-uncompressed.pdf: %.pdf
 	qpdf --object-streams=disable --stream-data=uncompress $< $@
 
+%.pdf: %.asy
+	asy -f pdf -tex pdftex $<
+
 mostlyclean:
 	rm -f *.dvi *.log *~ core *.o *.aux *.bbl *.blg *.toc
 
-- 
2.39.2