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Jump (Click me) \subsection{Notation for sets and graphs I} \subsubsection{Disjoint Set notations (Click Me)} $P_{S}(T)$ denotes the set of subsets of set $T$ whose orders lie in list $S \subset \N$. Example: $P_{1,2}(\{a,b,c \}) = \{ \{ a \}, \{ b \} , \{ c \}, \{ a,b \} , \{ a,c \} , \{ b,c \} \}$. %$\mathbb{M,N}$ \subsubsection{Graphs } An (undirected) graph is a triple $g=(V,E,f)$ where $V,E$ are sets (vertex set and edge set respectively) and $f:E \rightarrow P_{1,2}(V)$. (A handy reference using a slightly different notation is here .) \subsubsection{Graphs and pictures} It will be useful to represent a graph $g$ on a countable vertex set by a picture. For an exposition, see here or here . %Exercise: %It will be useful to % represent a graph $g$ on a countable vertex set by a picture. A plane drawn picture $d = \ppic$ is a closed rectangular interval of the plane $R$ together with a label set $V$, an injective map $\lambda : V \rightarrow R$, and a subset $S$ of $R$ consisting of smoothly embedded line intervals terminating at certain of the marked points $\lambda: V$.
(Obviously such a thing can only be approximated by a `drawn picture' in the artistic sense, but we will use PDPs in such a way that such approximations are generally sufficient. With this caveat we will speak of `drawing' PDPs, and indeed speak of pictures such as the one on the left as if they are PDPs.)

An alcove of such a picture $\ppic$ is a connected component of $R \setminus S$. (See also here.) A set of line interval embeddings is generic if when two lines meet at an unmarked point the tangents do not coincide; (we may also require that no third line passes through such a point;) and no line meets a marked point except at an endpoint.

In order to represent a graph $g=(V,E,f)$ by such a picture $\ppic$, one proceeds as follows. First one draws marked points such that each may be identified with a vertex in $V$. Each edge $\epsilon$ in $E$ is then represented by a line interval between the vertices in $f(\epsilon)$, and labelled by $\epsilon$. There is much freedom of choice in this, but one wants that a complete single line can be determined unambiguously from the drawn set of lines. To this end the set of line interval embeddings is chosen to be
  generic.
(This is sophistry, in that it is not really possible to determine tangents to lines by looking at drawn pictures, except within some tolerance. Thus if we use pictures in practice, tangents must not only be different, but sufficiently different.)

In such a picture we shall call the boundary of the rectangular region the frame . A vertex may lie on the frame (call such a vertex exterior), but by convention a line interval may touch the frame only at its endpoints.