Algorithmic Graph Theory: Planar Graphs

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• Let F be the set of faces of a plane graph with m edges. Then d(f) = 2m. ∑ f∈F • If a plane graph G has two faces with the same boundary, then G is a cycle. Paul Bonsma (HU Berlin) AGT: <strong>Planar</strong> graphs January 31, 2012 6 / 15
Proposition 10.1 Let G, φ be a 2-connected plane graph without loops. Then every facial walk is a cycle. Proof: Suppose there exists a facial walk W = v 0 ,e 0 ,v 1 ,e 1 ,...,e k−1 ,v k that is not a cycle, so w.l.o.g. v 0 = v i for some 2 ≤ i ≤ k −2 (as G contains no loops). Let f be the face bounded by W. Then a simple closed curve can be drawn in f ∪φ(v 0 ), which separates v 1 from v k−1 . So v 0 is a cut vertex, a contradiction. □ Paul Bonsma (HU Berlin) AGT: <strong>Planar</strong> graphs January 31, 2012 7 / 15
Page 1 and 2: Algorithmic Graph Theory Section 10
Page 3 and 4: • Let X ⊆ R 2 be the union of a
Page 5: Facial walks • Let f be a face of
Page 9 and 10: • A plane graph is a triangulatio
Page 11 and 12: Proposition 10.6 Let G be a simple
Page 13 and 14: Lemma 10.9 Let G be a 3-connected g
Page 15: Proof, continued: Let x 0 ,...,x k

Algorithmic Graph Theory: Planar Graphs

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