Get the graph sq4,3. (You will find it under Grids. The graph looks like
Fig. 37. except I have colored the seven regions defined by the graph
(the outside counts as a region too).
Fig. 37, The seven regions of sq4,3
The dual of a plane graph G is obtained as
follows: for each region of the graph G we have a vertex in dual(G). If two
regions
of the graph G have a common edge as a border then the corresponding
vertices in dual(G) will be adjacent. If you click Properties and then
Planar Graphs you will find an item Find Dual Graph. Select that and you will get
the dual graph of sq4,3. In order to keep the dual graph, click on
Swap so that the current graph becomes the dual of sq4,3. The
swap menu item is on the small screen where you have both the graph and its dual displayed. The
program
tries to put the points of the dual graph in positions that correspond to the center
of the region they represent. The point that represents the outside region (in
this case the vertex 2) is difficult to place so use the mouse to move vertex 2 to a
suitable location like in fig. 38. Also you might want to enlarge the picture a bit by clicking
Picture and Size and then choosing larger.
Fig. 38, The dual of sq4,3
Actually we can do better because this graph is planar. With a little patience you can get the
plane depiction in fig. 40. Note that since our program uses straight lines it is possible to have a
planar graph that has no planar depiction using straight lines.
Fig. 40, A plane depiction of the dual of sq4,3
Well, now that you have a plane depiction you can find the dual of the dual. See how many duals you
can do before you fail to get a plane depiction using straight lines or run out of patience!. You will
have to be shrewd about
how you move the vertices. After you have had some fun here are some questions:
Get the plan depiction of the cube. What is its dual? Hint: It is another platonic graph.
Actually, the dual of a platonic graph is always a platonic graph. Fill in the table below with the
appropriate
platonic graph. Hint: after you find get the dual click on swap to keep it as the current graph. Then use
the statistics to give you an idea as to which platonic graph it is isomorphic to. Save your dual graph;
get the suspected platonic graph and check if it is isomorphic to your saved graph.
Lesson 22:
Dual Graphs
