diff --git a/figures/SAT_mvp.pdf b/figures/SAT_mvp.pdf
index 34e47e8..abc7a6b 100644
Binary files a/figures/SAT_mvp.pdf and b/figures/SAT_mvp.pdf differ
diff --git a/figures/vertex_parallel.pdf b/figures/vertex_parallel.pdf
index d754c9b..4915914 100644
Binary files a/figures/vertex_parallel.pdf and b/figures/vertex_parallel.pdf differ
diff --git a/tikzs/SAT_mvp.tikz b/tikzs/SAT_mvp.tikz
index 2311095..98d3462 100644
--- a/tikzs/SAT_mvp.tikz
+++ b/tikzs/SAT_mvp.tikz
@@ -22,7 +22,7 @@
 		\node [style=none] (20) at (5.25, 5.25) {\Large$\vec v_i$};
 		\node [style=none] (21) at (-3.25, 8.75) {};
 		\node [style=none] (22) at (-3.75, 8.5) {\Large$\vec o_i$};
-		\node [style=none] (23) at (1.75, -2.75) {\Large$i$};
+		\node [style=none] (23) at (1.75, -2.75) {\Large$e_i$};
 	\end{pgfonlayer}
 	\begin{pgfonlayer}{edgelayer}
 		\draw (1.center) to (2.center);
diff --git a/tikzs/pentagon_congruent.tikz b/tikzs/pentagon_congruent.tikz
index 5f545e8..3ef31c6 100644
--- a/tikzs/pentagon_congruent.tikz
+++ b/tikzs/pentagon_congruent.tikz
@@ -24,6 +24,7 @@
 		\tkzMarkAngle[mark=||](4,5,3);
 		\tkzMarkAngle[mark=||](0,5,4);
 		\node [draw] at (5.center) [circle through={(0.center)}]{};
+		% \node [draw, dashed] at (7.center) [circle through={(8.center)}]{};
 
 	\end{pgfonlayer}
 \end{tikzpicture}
diff --git a/tikzs/vertex_intro.tikz b/tikzs/vertex_intro.tikz
index 39a9274..b980bd9 100644
--- a/tikzs/vertex_intro.tikz
+++ b/tikzs/vertex_intro.tikz
@@ -1,24 +1,30 @@
 \begin{tikzpicture}
 	\begin{pgfonlayer}{nodelayer}
-		\node [style=none] (0) at (-3, 3.25) {};
-		\node [style=none] (1) at (2.5, 0) {};
-		\node [style=none] (2) at (-5.5, -2.5) {};
+		\node [style=Medium dot] (0) at (-3, 3.25) {};
+		\node [style=Medium dot] (1) at (2.5, 0) {};
+		\node [style=Medium dot] (2) at (-5.5, -2.5) {};
 		\node [style=none] (3) at (1.25, 5.5) {};
 		\node [style=none] (4) at (0, -4.75) {};
 		\node [style=none] (5) at (5.75, 2.75) {};
 		\node [style=none] (6) at (7.75, -4.25) {};
-		\node [style=none] (7) at (1.25, 0) {};
-		\node [style=none] (8) at (3.25, 0) {};
-		\node [style=none] (9) at (2.5, 0.5) {$v_i$};
-		\node [style=none] (10) at (1.5, 3) {$e_i$};
+		\node [style=none] (9) at (2.5, 0.5) {\Large$V_i$};
+		\node [style=none] (10) at (0.75, 4.5) {\Large$e_j$};
+		\node [style=none] (11) at (-3, 3.75) {\Large$V_{i-1}$};
+		\node [style=none] (12) at (-5.5, -3) {\Large$V_{i+1}$};
+		\node [style=none] (13) at (-6.75, -0.25) {};
+		\node [style=none] (14) at (-5.5, 2) {};
+		\node [style=none] (15) at (-3.25, 0.5) {\huge$A$};
+		\node [style=none] (16) at (3.75, -1) {\huge$B$};
 	\end{pgfonlayer}
 	\begin{pgfonlayer}{edgelayer}
-		\draw (0.center) to (1.center);
-		\draw (1.center) to (2.center);
-		\draw (3.center) to (4.center);
+		\draw (0) to (1);
+		\draw (1) to (2);
+		\draw [style=Very thick] (3.center) to (4.center);
 		\draw (4.center) to (6.center);
 		\draw (6.center) to (5.center);
 		\draw (5.center) to (3.center);
-		\draw (0.center) to (2.center);
+		\draw (2) to (13.center);
+		\draw (13.center) to (14.center);
+		\draw (14.center) to (0);
 	\end{pgfonlayer}
 \end{tikzpicture}
diff --git a/tikzs/vertex_parallel.tikz b/tikzs/vertex_parallel.tikz
index b83a7e4..02d0e5e 100644
--- a/tikzs/vertex_parallel.tikz
+++ b/tikzs/vertex_parallel.tikz
@@ -6,14 +6,15 @@
 		\node [style=none] (4) at (-4, -5) {};
 		\node [style=none] (5) at (2, -5) {};
 		\node [style=none] (6) at (8, 3) {};
-		\node [style=Medium dot] (7) at (-7, -1) {};
+		\node [style=none] (7) at (-7, -1) {};
 		\node [style=none] (8) at (-7, 5.25) {};
 		\node [style=Medium dot] (9) at (0, 6) {};
-		\node [style=none] (10) at (2.25, 3.25) {$V_i$};
-		\node [style=none] (11) at (0.5, 6.5) {$V_{i-1}$};
-		\node [style=none] (12) at (-2, -3.75) {$V_{i+1}$};
-		\node [style=none] (13) at (-7.5, -1.5) {$V_{i+2}$};
-		\node [style=none] (14) at (3.25, 5.25) {$e_j$};
+		\node [style=none] (10) at (2.25, 3.25) {\Large$V_i$};
+		\node [style=none] (11) at (0.5, 6.5) {\Large$V_{i-1}$};
+		\node [style=none] (12) at (-2, -3.75) {\Large$V_{i+1}$};
+		\node [style=none] (14) at (3.5, 5.25) {\Large$e_j$};
+		\node [style=none] (15) at (-3.5, 2.25) {\huge$A$};
+		\node [style=none] (16) at (2.75, 0.25) {\huge$B$};
 	\end{pgfonlayer}
 	\begin{pgfonlayer}{edgelayer}
 		\draw [style=Very thick] (2) to (1);
@@ -23,7 +24,7 @@
 		\draw (5.center) to (4.center);
 		\draw (8.center) to (9);
 		\draw (9) to (1);
-		\draw (2) to (7);
-		\draw (7) to (8.center);
+		\draw (2) to (7.center);
+		\draw (7.center) to (8.center);
 	\end{pgfonlayer}
 \end{tikzpicture}