Computational Topology (Jeff Erickson)
Shortest Noncontractible Cycles
A tie is a noose, and inverted though it is,
it will hang a man nonetheless if he’s not careful.
— Yann Martel,
Life of Pi
(2001)
(written:) Yakka foob mog. Grug pubbawup zink wattoom gazork. Chumble ¨spuzz.
(spoken:) I love loopholes.
— Calvin, explaining Newton’s First Law of Motion ‘in his own words’
Bill Watterson,
Calvin and Hobbes
(January 9, 1995)
8 Shortest Noncontractible Cycles
In this lecture, I’ll describe several algorithm to compute the shortest noncontractible cycle in a combi
natorial 2manifold. The input is a cellularly embedded graph
G
on a 2manifold
Σ
(or equivalently, a
polygonal schema for
Σ
) with nonnegatively weighted edges, and our goal is to compute a cycle in
G
of
minimum length that is not contractible in
Σ
.
This problem has several natural motivations. the length of the shortest noncontractible cycle. In
topological graph theory, the length of the shortest noncontractible cycle in an embedded graph is called
the
handle girth
[
1
]
or
edgewidth
of the embedding
[
15
,
18
]
; graphs that have embeddings with large
edgewidth share many useful combinatorial properties with planar graphs. A closely related concept is
the
facewidth
or
representativity
of a graph embedding, which is the minimum number of
faces
of
G
intersecting a noncontractible cycle on the surface, or equivalently, half the edgewidth of the radial
graph
G
±
[
17
]
. The length of the shortest noncontractible cycle in a Riemannian manifold is called
the
systole
(to be more speciﬁc, the
homotopy 1systole
) of the manifold
[
2
]
. Shortest noncontractible
cycles are good indicators of
topological noise
of geometric surface models reconstructed from point
clouds or volume data.
The algorithms described below all require that edge weights are nonnegative. We treat the graph
as a continuous metric space, where the edge weights represent distance. I will assume throughout the
lecture that the shortest path between any two
vertices
of the graph is unique; this assumption can be
enforced if necessary using standard perturbation schemes. However, we do
not
assume that the edge
weights satisfy the triangle inequality.
1
8.1 Thomassen’s 3Path Property
The ﬁrst efﬁcient algorithm to compute shortest noncontractible cycles is an easy consequence of the
following observation of Thomassen
[
18
]
.
Lemma 8.1.
Let
x
and
y
be two points on a surface
Σ
, and let
α
,
β
,
γ
be paths in
Σ
from
x
to
y
. If the
loops
α
·
¯
β
and
β
·
¯
γ
are contractible, then the loop
α
·
¯
γ
is also contractible.
Proof:
The concatenation of any two contractible loops is contractible.
±
Corollary 8.2.
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 Fall '08
 Staff
 Life of Pi

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