AM8002
Fall 2014
Week 11 –
Cop number of
outerplanar graphs
Dr. Anthony Bonato
Ryerson University
Planar graphs
• a graph is planar if it can be drawn in the plane
without edge crossings
Theorem 11.1 (Aigner, Fromme,84): planar graphs
have cop number at most 3.
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Example: Buckeyball graphs
• girth 5, 3-regular, so have cop number ≥ 3
(and so = 3 by theorem).
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Ideas behind proof
• cop territory: induced subgraph so that if the
robber entered he would eventually be caught
(not necessarily immediately)
• cop territory starts as a maximum order
isometric path
• inductively grow cop territory, until it is entire
graph
• show that the cop territory is always one of three
kinds, and that we can always enlarge it so it
remains one of the three kinds
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Outerplanar graphs
• a graph is outerplanar if its vertices can be
arranged on a circle with the following
properties:
1. Every edge joins two consecutive vertices
on the circle, or forms a chord on the
circle.
2. If two chords intersect, then they do so at
a vertex.
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Examples
• cycles
• each of these are maximal outerplanar:
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Characterization
Theorem 11.2 (Kuratowski,1930)
A graph is planar if it does not contain a
subdivision of K5 or K3,3.
Theorem 11.3 (Harary, Chartrand,1967)
A graph is outerplanar if it does not contain
a subdivision of K4 or K2,3.
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Cop number of
outerplanar graphs
Theorem 11.4 (Clarke, 02) If G is
outerplanar, then it’s cop number is at
most 2.
• proof is simpler than planar case, but still
needs some care
• overall idea is similar: enlarge cop territory
• two cases: no cut vertices, or some cut
vertices
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