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Let G be a simple graph. G is said to be maximal planar if it is...

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Advance Math

Let G be a simple graph. G is said to be maximal planar if itis planar and the addition of any new edge to G results in a(simple) non-planar graph. Examples of maximal non-planar graphsare K4 , K5 minus an edge, and K3,3 minus an edge.

(a) Show that a maximal planar graph is connected.

(b) Show that a maximal planar graph of order ?3 has nobridges.

(c) Show that every face of a maximal planar graph of order ?3is bounded by a triangle.

Answer & Explanation Solved by verified expert
4.4 Ratings (680 Votes)
aIf an edge e of a maximal planar graph is in exactlytwo triangles then Ge is also maximal planar Since Ge hasexactly three fewer edgesIf G is a maximal planar graph with n4 vertices then there areat least n such edges Induction on nLet G be a maximal planar graph with at least four verticesAssume that there are vertices uv such that Guv isdisconnected Let X be one component of Guv and    See Answer
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