diff --git a/om-graphs/Makefile b/om-graphs/Makefile
index f030a71654e4d5eedf2d8414a73eddfebca6bad5..68d5356a65c681e131a6b3e585f72234e16a7868 100644
--- a/om-graphs/Makefile
+++ b/om-graphs/Makefile
@@ -2,7 +2,7 @@ TOP=..
 
 include ../Makerules
 
-IMAGES=range-tree lazy-update heavy-light
+IMAGES=range-tree lazy-update heavy-light expose-idea middle-sons expose-real expose-phases
 
 $(MAIN).pdf: images
 
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+++ b/om-graphs/expose-idea.ipe
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diff --git a/om-graphs/expose-phases.ipe b/om-graphs/expose-phases.ipe
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--- /dev/null
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diff --git a/om-graphs/graphs.tex b/om-graphs/graphs.tex
index 823595534101f33d5ad7956b94660a617c988c43..b71b2880e3badbf67a9ab9313153942d759d1896 100644
--- a/om-graphs/graphs.tex
+++ b/om-graphs/graphs.tex
@@ -40,8 +40,6 @@ $c_1,\dots c_n$ in its leaves (in this order) and inner nodes contain the minimu
 children. Note that each node represents a subpath of~$F$ with leaves being the single
 vertices. 
 
-\TODO perhaps reference the lecture notes on details about range trees
-
 \figure[]{range-tree.pdf}{}{An example of a range tree for path on eight vertices.
 Marked subtrees cover the subpath~$2\to 6$.}
 
@@ -87,7 +85,7 @@ $+7$.\TODO shorter caption}
 \section[hld]{Heavy-light decomposition}
 Now we are ready build data structure for static trees using \em{heavy-light
 decomposition}. We assume our tree $F$ is rooted and we orient all edges
-up, towards the root \NOTE\foot{maybe unnecessary now}.
+up, towards the root.
 
 \defn{
 Let~$F$ be a rooted tree. For any vertex~$v$ we define $s(v)$ to be the size of subtree
@@ -128,7 +126,7 @@ To answer $\LCA(x,y)$ we start at both~$x$ and~$y$ and we jump along heavy paths
 towards the root. Once we discover lowest common heavy path, we compare position of
 ``entry-points'' to decide which one of them is LCA. We have to traverse $\O(\log n)$
 light edges and we can jump over a heavy path in constant time, thus we spend $\O(\log n)$
-time in total. \NOTE\foot{perhaps make it theorem/lemma?}
+time in total.
 
 \subsection{Path queries and updates}
 
@@ -252,10 +250,13 @@ of the fat path.
 Now, assume~$v$ lies on a fat path~$A$. We start
 at~$v$ and jump along~$A$ to its top~$t$. Unless $t$ is the root of the tree (which means
 we are done), $t$ is connected to a fat path~$B$ via thin edge $(t, p)$, see
-Figure~\TODO. We cut~$B$ by turning fat edge below~$p$ into a thin edge. Then we join top
-half of~$B$ with~$A$ by making edge~$(t, p)$ fat. This is one step of the $\Expose$. Now
+Figure~\figref{expose-idea}. We cut~$B$ by turning fat edge below~$p$ into a thin edge.
+Then we join top half of~$B$ with~$A$ by making edge~$(t, p)$ fat. This is one step of the
+$\Expose$. Now
 we jump to the top of the newly created fat path and repeat the whole process. 
-\TODO picture of one step of expose
+
+\figure[expose-idea]{expose-idea.pdf}{}{One step of $\Expose$ in the thin-fat
+decomposition.}
 
 \theorem{
 $\Expose$ operation performs $\O(\log n)$ steps amortized.
@@ -305,12 +306,12 @@ of them is the application in link-cut trees.
 \subsection{Representation of fat paths via splay trees}
 We describe a fat path by a splay tree whose nodes have one-to-one correspondence with the
 vertices of the fat path. Nodes of the tree have no keys, the ordering is given by the
-order of respective vertices on the fat path. That is, inorder traversal of the tree
+order of respective vertices on the fat path. That is, left to right inorder traversal of the tree
 returns the vertices exactly in the order in which they lie on the fat path.
 
 We deal with the costs in the same way as in our data structure for a static path. Each
 node contains the cost of the respective vertex and the path minimum for the subpath the
-whole subtree corresponds to. \TODO maybe a picture here  The only catch is that we need
+whole subtree corresponds to. The only catch is that we need
 to recalculate minima during rotation. But this can be easily done in a constant time.
 
 With this representation, we can easily answer ordering queries like previous vertex on
@@ -343,8 +344,6 @@ represent the thin edges leading into a vertex~$v$. More precisely, if there is
 leading from a fat path~$A$ into the vertex~$v$, then the root of the splay tree representing~$A$
 is a middle son of~$v$. To distinguish between middles sons and left and right son, we
 will call left and right son \em{proper sons}.
-\TODO picture of a middle sons (left part: "true" structure of
-fat paths, right part: representation with splay trees). 
 
 However, a node does not know about its middle sons. We do not store a pointer from a
 node to its middle son. We only store a pointer from the middle son to its parent. Note that we
@@ -357,8 +356,8 @@ This way, each tree is represented by one \em{virtual tree}, where virtual tree
 two types of edges. One type represent the edges in splay trees and the other represents
 thin edges and middle sons. 
 
-
-\TODO picture of the whole tree with middle sons etc.
+\figure[middle-sons]{middle-sons.pdf}{}{Example of a tree decomposed into a system of fat
+paths and a possible corresponding virtual tree.}
 
 \subsection{Implementation of \Expose}
 At the start of $\Expose(v)$ we need to turn a fat edge below vertex~$v$ into a thin edge
@@ -378,10 +377,12 @@ middle son. Now we can join~$A$ and the remnant of~$B$ by making node~$v$ the le
 of~$p$. Then we move to the next step, where vertex~$p$ takes the role of vertex~$v$.
 In the end, when~$v$ and~$\Root(v)$, are within the same splay tree, we splay~$v$.
 
+\figure[expose-real]{expose-real.pdf}{}{One step of $\Expose(v)$ in a virtual tree.}
+
 \subsection{Analysis of \Expose}
 For the sake of analysis we split $\Expose(v)$ into three phases. Let $(r_1, p_1),\dots,
 (r_k, p_k)$ be the thin edges on the path from~$v$ to the root of the virtual tree, see
-Figure~\TODO. Let us
+Figure~\figref{expose-phases}. Let us
 also denote $p_0 = v$. In the first phase, we splay all of $p_0, \dots, p_k$ into the roots
 of their respective splay trees. After first phase, $p_k$ is the root of the virtual tree
 and $p_0, \dots, p_k$ form a path made of thin edges. In the second phase, we turn edges
@@ -390,7 +391,7 @@ of $p_1, \dots, p_k$ into middle sons, just as described in the previous section
 $p_0, \dots, p_k$  form a path inside one splay tree, we splay $p_0 = v$ into the root of
 that tree -- this is the third phase.
 
-\TODO picture of phases
+\figure[expose-phases]{expose-phases.pdf}{}{Three phases of $\Expose(v)$.}
 
 The key phase we need to deal with in our analysis the first phase. Third phase is a
 simple splay in a splay tree, which should have have $\O(\log n)$ amortized complexity,
@@ -439,11 +440,11 @@ the third phase. Thus, third phase will also pay for the $+k$ part of the first
 
 There is one last subtle problem we have to deal with. The problem is that structural
 updates can change the potential of the virtual tree outside of the $\Expose$. 
-The simple case is $\Cut$. Since we remove a subtree from the virtual tree, we only
+The simple case is $\op{Cut}$. Since we remove a subtree from the virtual tree, we only
 decrease the potential in the original virtual tree and the potential of the new tree is
 paid by the potential of the removed subtree.
 
-$\Link(u,v)$ is slightly more complicated, since adding a subtree means increasing
+$\op{Link}(u,v)$ is slightly more complicated, since adding a subtree means increasing
 size of some nodes. However, notice that after $\Expose(u)$ the node~$u$ has no right
 son as it is the last vertex on the fat path and also the root of the virtual tree. Thus,
 by linking $u$ and $v$ we only increase the size of $u$ and we increase it by at most $n$,
diff --git a/om-graphs/middle-sons.ipe b/om-graphs/middle-sons.ipe
new file mode 100644
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