Ipelets

A collection of ipelets that save me a lot of time when drawing figures. To use, add the .ipe files into your ./ipelets folder.

Dense Graphs

Some functions to draw dense graphs. These are included in k_clique.lua. The following graphs can be drawn:

• Complete graphs.
• Complete multipartite graphs.
• Complete split graphs.

Usage

Complete graphs

A complete graph is a graph with $n$ vertices where there is an edge for any pair of distinct vertices.

To draw a complete graph embedded into a circle for any number of vertices, follow:

1. Draw a circle using one of the three options of ipe.
2. Select the circle
3. Run “ipelets -> Dense Graphs -> Complete graph”
4. Enter the number of vertices and accept.

Example:

An example of a complete graph for $n=10$:

Complete multipartite graphs

A complete multipartite graph consists of non-empty independent sets $S_i$ and $(x,y)$ is an edge whenever $x\in S_i$ and $y\in S_j$ with $i\neq j$.

To draw a multipartite graph with independent sets $S_1,\ldots,S_r$ where $|S_i| = n_i$ for all $i$, embedded into $r$ lines, follow:

1. Draw $r$ paths (lines, curves) $L_1,\ldots,L_r$.
2. Select them in labelling order: First $L_1$, then $L_2$, $L_3$, and so on.
3. Run “ipelets -> Dense Graphs -> Complete Multipartite Graph”
4. Enter the sizes of $S_1,\ldots,S_r$ in this format: $s_1,s_2,\ldots,s_r$ (no spaces).

Example:

An example with three independent sets where $n_1,n_2,n_3 = 3,4,5$:

Complete Split Graph

A complete split graph is a graph that can be partitioned in an independent set and a clique such every vertex in the independent set is adjacent to every vertex in the clique.

To draw a complete split graph with a clique $A$ with $n$ vertices and independent set $B$ with $m$ vertices, follow:

1. Draw a circle and a path.
2. Select both.
3. Run “ipelets -> Dense Graphs -> Complete split graph”
4. Enter the sizes $n,m$ (no spaces).

Example:

An example with $n = 3$ and $m = 4$:

Intersection Graphs

Functions to draw intersection graphs. The ipelet is intersection_graphs.lua.

An (geometric) intersection graph is a graph that describes the intersection of a set of geometric objects. That is, two objects intersect if and only if their corresponding vertices share an edge.

To draw an [object] (objects: disk, open disk, rectangle, segment), do as follows:

1. Draw some objects (of only one class)
2. Select all of them
3. Run “ipelets -> Dense Graphs -> [object]”

Examples:

Unwrap Circles

A function to unwrap a circle into a segment, preserving (scaled) length. The circle may have some points in its boundary. When unwrapping, the position of points is also translated correspondingly. The ipelet is unwrap_circle.lua.

Usage

To unwrap a circle, we do it deciding two things: the cut point and the unwrapping direction: clockwise or counterclockwise:

1. Draw a circle $C$ with some points $p_1,\ldots,p_n$ intersecting its boundary.
2. Draw the cut point $x$ where desired.
3. Select in this order:
   1. The circle $C$,
   2. the points $p_1,\ldots,p_n$ and,
   3. the cut point $x$ (the function cuts the circle at the last point selected!).
4. Run “ipelets -> Unwrap Circle”
5. Enter scale (default is one) and choose direction.

Example

An example with six points $p_1,\ldots,p_6$ on a circle $C$. The grey arrow indicates clockwise direction. The unwrapped line above is clockwise; the one below is counterclockwise. Both lines have 1 of scale factor.