[Blindmath] Drawing Graphs in College Level Math Course

[Maria Kristic]

In addition to Sina's suggestions (I've also used those methods quite a bit
with these types of problems, more so the adjacency matrix but have used
both), I've also sometimes written out verbal descriptions, describing the
layout of the vertices and stating the edges in a systematic way, so that
the graph could easily be drawn/visualized by someone. I form these
descriptions after taking some time to think about the graphs and visualize
a possible solution diagram in my head; I'm totally blind, but from having
to visualize things a lot over the years in not often having tactile access
to diagrams or materials to communicate them in a tactile form, I've gotten
pretty efficient with it. For instance, here are possible solutions (since
there are several variations for some) for the below problem, written in the
way I might describe them in an assignment/exam, then I have some additional
comments following that:

1. A K_{5} graph. Five vertices arranged around the corners of a pentagon,
labeled, going clockwise from the single-vertex point at the top, V1 through
V5. Edges V1 to V2, V2 to V3, V3 to V4, V4 to V5, V5 to V1, V1 to V3, V1 to
V4, V2 to V4, V2 to V5, V3 to V5.
2. Five vertices arranged around the corners of a pentagon, labeled, going
clockwise from the single-vertex point at the top, V1 through V5. Edges V1
to V2, V2 to V3, V3 to V4, V4 to V5, V5 to V1, loop edge at V1, loop edge at
V2, loop edge at V3, loop edge at V4, loop edge at V5.
3. Five vertices arranged around the corners of a pentagon, labeled, going
clockwise from the single-vertex point at the top, V1 through V5. Two,
parallel edges from V1 to V2, two, parallel edges from V2 to V3, two,
parallel edges from V3 to V4, two, parallel edges from V4 to V5, two,
parallel edges from V5  to V1.
4. Five vertices arranged around the corners of a pentagon, labeled, going
clockwise from the single-vertex point at the top, V1 through V5. Edges V1
to V2, V2 to V3, three, parallel edges from V3 to V4, single edge from V4 to
V5, single edge from V5 to V1, loop edge at V1, loop edge at V2, loop edge
at V5.

If a Draftsman board is what I think it is--a rubber board with which one
uses a special pen and plastic sheets to create non-erasable, raised
diagrams--then she might want to visualize it in her head a bit before
drawing to prevent using several of the sheets of paper on which to work it
out if paper's an issue, or, if this is a homework assignment she's doing on
her own time, she might want to have something manipulative like Wikki Stix
with her to use to work out a diagram before drawing it (I've used coins
placed on top of Wikki Stix diagrams in appropriate places to represent
vertices when I've used them like this), if she wants to use the Draftsman.

If she's dictating the diagrams to a scribe, she'd want to dictate in a way
similar to my descriptions above. I hope you understand my logic in how I
listed the edges: for all of them, I start from a certain vertex (the top
point of the pentagon in this case), then walk around the pentagon clockwise
listing the edges encountered around the exterior of the pentagon, and then:
1. For the complete graph, since the definition of a complete graph is that
it's connected and each vertex is connected to every other in the graph by
an edge, I then went through, starting from that top point again, to fill in
the remaining edges. So, V1 is already connected to V2 and V5, but not yet
to V3 or V4, so there are edges from V1 to V3 and V1 to V4. V2 is already
connected to V1 and V3, but not yet to V4 and V5, so there are edges V2 to
V4 and V2 to V5. V3 is already now connected to V2 and V4 and V1, so it's
just not connected to V5, so there's an edge V3 to V5. V4 is now connected
to all of the other four vertices, and same is true for V5, so that's all
the edges of that graph. 2. I just listed the loop edges in the order that
they'd be encountered as one went clockwise around the pentagon in the same
order as the initial edges listed for all graphs (i.e., starting at V1). 3.
Instead of single edges, since this one has two, parallel edges between each
of the vertices, I listed the edges in the same order I would have for the
initial listed edges of the other graphs but indicated that they were two
and parallel in each case. 4. As I was initially walking around the exterior
of the pentagon, I felt it was most convenient to point out that there were
three edges instead of one from V3 to V4 right there, then I went back after
going around the exterior and filled in the loop edges as they would be
encountered in a clockwise order.

The descriptions, of course, vary a bit depending on the situation. For
instance, I might describe vertices in terms of rows/columns (vertices
arranged in three rows of two columns each; left column contains vertices V1
through V3 from top to bottom; right column is V4 through V6 from top to
bottom), indicate presence/location of connected as opposed to isolated
vertices (four vertices arranged around the corners of a square, edges
connecting them to form the exterior of a square, isolated vertex in the
centr of the square), etc.

Just my own $0.02...hope it helps...I'd be happy to help privately, too.

Regards,
Maria

-----Original Message-----
From: blindmath-bounces at nfbnet.org [mailto:blindmath-bounces at nfbnet.org] On
Behalf Of Missy Garber
Sent: Sunday, June 07, 2009 8:57 PM
To: 'Blind Math list for those interested in mathematics'
Subject: [Blindmath] Drawing Graphs in College Level Math Course

A mom on another email list posted the following question which pertains to
an assignment her daughter was given. Her daughter is taking a math course
in her first year of college. Can anyone offer some advice for her? How
would you approach this task? I'll paste her question below.

Thanks,
Missy


Is it possible for a totally blind person to do a math problem
that requires them to draw the following:

A connected graph with five vertices each of degree 4 with:
1. no loops and no multiple edges
2. loops but no multiple edges
3. multiple edges but no loops
4. both multiple edges and loops

This particular blind person has a draftsman
drawing board but has only drawn basic shapes for previous math
classes in public school. She is able to interpret (these types
of) diagrams fairly well if they are drawn and brailled/labeled
accurately which is usually not the case. Is there a way to
accommodate this?

I have heard that some or a least a few of the totally blind
college students in this area have the math classes
waived--eventually. This is not the preferred route. An
advisor put her in this class.

Ugh!!! and thank you!