[Inquiry] Re: Theme One Program -- Exposition

[Jon Awbrey]

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TOP.  Expository Note 3

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2.  Painted And Rooted Cacti And Conifers (cont.)

Figure 2 illustrates one way to visualize the correspondence between
cactus graphs and cactus strings, exemplified in the instance of the
cactus from Figure 1.  The polygons of the cactus graph are referred
to as its "lobes".  An edge not a part of a larger polygon counts as
a "bi-gon", and when it is terminal it forms a specialized lobe that
is known as a "spike".

       o
   a  (|)        d
   o---o         o
   (\ /)  b c   (|)
     o--,--o--,--o b e
      \         /
       \       /
     (  \     /  )
         \   /
          \ /
           @ a c e

   ( ( a , ( ) ) , b c , ( d ) b e ) a c e

   Figure 2.  Cactus Graph and Cactus Expression

One traverses a cactus by beginning at the left hand side of the root node,
reading off the list of paints that one encounters at that point, climbing
up the left hand side of the leftmost lobe, marking that ascent by means
of a left parenthesis, traversing whatever cactus one happens to reach
at the first node above the root, that done, proceeding from left to
right along the top side of the lobe, marking each interlobal span
by means of a comma, traversing each cactus in turn that one meets
along the way, on completing the last of them climbing down the
right hand side of the lobe, marking that descent by means of
a right parenthesis, and then traversing each cactus in turn,
in left to right order, that is incident with the root node.

For the time being I will continue with the informal presentation
of the subject, and merely mention that additional discussion and
formal definitions of cactus graphs, cactus expressions, and the
relationships between them can be found at the following sites:

LOC.  http://stderr.org/pipermail/inquiry/2004-December/thread.html#2135
LOC.  http://forum.wolframscience.com/showthread.php?threadid=649

PERS.  http://stderr.org/pipermail/inquiry/2004-April/thread.html#1341
PERS.  http://stderr.org/pipermail/inquiry/2004-May/thread.html#1391
PERS.  http://forum.wolframscience.com/showthread.php?threadid=297

Jon Awbrey

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