Let G be a connected planar graph of order n having e=3 n-6 edges. Provr. that, in any planar re

step 1 We're asked to show that if G is a connected simple graph with V vertices and E edges, or the nu

Let G be a connected planar graph of order n having e=3 n-6...

Key Concepts

Triangulation

Triangulation in the context of planar graphs refers to the division of the plane into triangular regions by the edges of the graph. This concept is important because it shows that in a maximal planar graph, where additional edges would disrupt planarity, every face is a triangle, reflecting the graph’s maximality.

Face Boundary (Degree of a Face)

The degree of a face in a planar graph is the number of edges that form the boundary of the face. In a triangulated or maximal planar graph, each face has a degree of three, meaning each face is bounded by exactly three edges. This concept is tied to the structural properties of planar graphs and their maximal variants.

Euler's Formula

Euler’s formula is a foundational result for planar graphs, stating that for any connected planar graph, the number of vertices minus the number of edges plus the number of faces equals two. This relation is crucial in linking the structural elements of the graph and is often used to establish bounds on the number of edges and faces.

Planar Graph

A planar graph is a graph that can be drawn on the plane without any edges crossing. This property is fundamental to the study of graphs embedded in two-dimensional spaces and is closely related to topological and combinatorial properties such as face structure and Euler’s formula.

Maximal Planar Graph

A maximal planar graph is a planar graph to which no additional edges can be added without violating planarity. Equivalently, each face in a maximal planar graph is a triangle, which provides a criterion for maximality and is essential for understanding edge count limits in planar graphs.

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