Using mathematical induction on the number of edges prove Euler's formula: r = e -- v + 2 for connected planar simple (CPS) graphs, where r, e, and v are the number of regions, edges, and vertices, respectively. (Hint: Any planar representation of CPS graph can be constructed starting with a single vertex and then successively adding an edge and a vertex or adding an edge between existing vertices as long as it doesn't cross another edge.)

Using mathematical induction on the number of edges prove Euler's formula: r = e -- v + 2 for connected planar simple (CPS) graphs, where r, e, and v are the number of regions, edges, and vertices, respectively. (Hint: Any planar representation of CPS graph can be constructed starting with a single vertex and then successively adding an edge and a vertex or adding an edge between existing vertices as long as it doesn't cross another edge.)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Using mathematical induction on the number of edges prove Euler's formula: r = e -- v + 2 for
connected planar simple (CPS) graphs, where r, e, and v are the number of regions, edges, and vertices,
respectively.
(Hint: Any planar representation of CPS graph can be constructed starting with a single vertex and then
successively adding an edge and a vertex or adding an edge between existing vertices as long as it
doesn't cross another edge.)

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