Question

For every polygon P⊆R², the set R²\P has exactly two regions. Each of these has the entire polygon P as its frontier.

Let G be a plane graph and e an edge of G.
(i)If X is the frontier of a face of G, then either e ⊆ X or X ∩ e = ∅
(ii) If e lies on a cycle C⊆G, then e lies on the frontier of exactly two faces of G, and these are contained in distinct faces of C.
(iii) If e lies on no cycle, then e lies on the frontier of exactly one face of G.

Answer 1

The statement is discussing properties of plane graphs and their faces, edges, and cycles.

Step-by-step explanation:

Based on the given statements, let's break down the three parts:

(i) If X is the frontier of a face of G, if e ⊆ X or X ∩ e = ∅. This means that if X is the boundary of a face in the plane graph G, then either the edge e is completely contained within X or there is no intersection between X and e.

(ii) If e lies on a cycle C, then e lies on the frontier of exactly two faces of G, and these faces are contained in distinct faces of C. This means that if e is part of a cycle in G, then e forms the boundary of two faces in G, and these two faces are within different faces of the cycle C.

(iii) If e lies on no cycle, then e lies on the frontier of exactly one face of G. This means that if e does not belong to any cycle in G, then e forms the boundary of only one face in G.