In planar topology, a face represents the interior region of a planar graph enclosed by edges. Think of it as the area within a closed shape.
Understanding Faces in Planar Graphs
A face, specifically in the context of planar graphs, isn't quite the same as a "face" you might think of in 3D geometry. Here's a breakdown:
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Planar Graph Requirement: Faces exist only in planar graphs, meaning graphs that can be drawn on a plane without any edges crossing.
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Interior Region: The face is the area completely bounded by edges.
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Boundaries: The edges of a face form its boundary. A single face can have multiple boundaries. Imagine a donut - it has an outer boundary and an inner boundary creating a "hole".
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Exterior Face: A planar graph also has an exterior face, which is the unbounded region outside the graph.
Examples to Illustrate Faces
Let's consider a few examples:
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Simple Polygon: A simple polygon (like a triangle or square) has one face: the area enclosed by its sides.
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Polygon with a Hole: A square with a smaller square cut out of the middle has one face: the area between the outer square and the inner square. This single face is bounded by the outer and inner squares.
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A Graph with Disconnected Components: Consider a graph with two separate circles. There are two interior faces (one inside each circle) and one exterior face.
Formal Definition (Euler's Formula Implication)
Euler's formula for planar graphs is relevant: v - e + f = 2, where:
- v is the number of vertices (corners).
- e is the number of edges (lines).
- f is the number of faces (regions, including the exterior face).
This formula highlights the relationship between vertices, edges, and faces in planar graphs and how faces contribute to their overall structure.
Key Takeaways
- Faces are interior areas enclosed by edges in a planar graph.
- A face can have multiple boundary components.
- The exterior region outside the graph is also considered a face.
- Euler's formula establishes a mathematical relationship between vertices, edges, and faces.
