GATE CS 2015 Set 1 — Question 44
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Engineering Mathematics → Graph Theory (Math) → Planarity
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Question
Let be a connected planar graph with 10 vertices. If the number of edges on each face is three, then the number of edges in is ___________.
Correct answer
24 to 24
Solution
For a connected planar graph, Euler's formula states:where is the number of vertices, is the number of edges, and is the number of faces.Given:
Every face is bounded by 3 edges (triangulation). The sum of degrees of the faces is equal to twice the number of edges:Since each face has 3 edges:Substitute and into Euler's formula:The number of edges in is 24.
Every face is bounded by 3 edges (triangulation). The sum of degrees of the faces is equal to twice the number of edges:Since each face has 3 edges:Substitute and into Euler's formula:The number of edges in is 24.
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