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From: eklhad@ihnet.UUCP (K. A. Dahlke)
Newsgroups: net.puzzle
Subject: Re: Don't cross my path !
Message-ID: <150@ihnet.UUCP>
Date: Fri, 17-Aug-84 15:06:07 EDT
Article-I.D.: ihnet.150
Posted: Fri Aug 17 15:06:07 1984
Date-Received: Sun, 19-Aug-84 01:18:03 EDT
References: <684@sbcs.UUCP> <785@abnjh.UUCP>
Organization: AT&T Bell Labs, Naperville, IL
Lines: 22

The three houses / three utilities problem has been around for a while.
The three angry workers going to work is isomorphic, and more interesting.
Planar graph theory is sufficient.
You are looking for Kerotowski's theorem (name probably badly miss-spelled).
It states: a graph can be placed in a plane with no crossings
if and only if the graph does not contain a K3,3 or a K5.
A K3,3 is the houses/utilities construct.
A K5 is five points all interconnected.
Another impossible puzzle: five homes have telephones with wires
directly connecting each house with the other four houses
(no crossings of ccourse).
This (a K5) can also never be planar.

I find the theorem absolutely amazing.
A program could take an arbitrary scrawly graph and tell you if it could
be put in a plane, just by checking for K3,3 and K5.
I don't know if the program could actually rearrange the graph 
to make it planar.  That would be interesting.
Someone in the CAD world has probably solved this problem.
-- 

Karl Dahlke    ihnp4!ihnet!eklhad