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1
P
01 INCOMPLETENESS: finite graph theory 1
by
Harvey M. Friedman
January 30, 2006
In this abstract, all digraphs are simple; i.e., have no
loops or multiple edges. The results remain unaffected if
loops are allowed (but not if multiple edges are allowed).
For digraphs G, we write V(G) for the set of all vertices
of G, and E(G) for the set of all edges of G. A digraph on
a set E is a digraph G where V(G) = E.
A dag is a directed acyclic graph; i.e., a digraph with no
cycles.
Let G be a digraph, and A V(G). We write GA for the set
of all tails of edges in G whose heads lie in A. I.e.,
GA = {y V(G): (
$
x A)((x,y) E(G))}.
We begin by quoting a well known theorem about dags. We
call it the complementation theorem.
COMPLEMENTATION THEOREM (finite dags). For all finite dags
G there is a unique A V(G) such that GA = V(G)\A.
We can look at the Complementation Theorem in terms of a
large independent set. We say that A V(G) is independent
in G if and only if there is no edge connecting elements of
A.
COMPLEMENTATION THEOREM (finite dags). Every finite dag has
a unique independent set A such that V(G)\A GA.
We will focus on digraphs on sets of the form [1,n]
k
. Here
k,n
≥
1 and [1,n] = {1,2,.
..,n}.
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 Fall '08
 JOSHUA
 Math, Graph Theory

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