Prove Lemma (Switch Edges). Suppose that, starting from some embeddings of planar graphs with di…

Problem 12.9. (a) Prove Lemma (Switch Edges). Suppose that, starting from some embeddings of planar graphs with disjoint sets of vertices, it is possible by two successive applications of constructor operations to add edges e and then f to obtain a planar embedding, F. Then starting from the same embeddings, it is also possible to obtain F by adding f and then e with two successive applications of constructor operations. Hint There are four cases to analyze, depending on which two constructor opera tions are applied to add e and the f. Structural induction is not needed (b) Prove Corollary Permute Edges). Suppose that, starting from some embeddings of pla nar graphs with disjoint sets of vertices, it is possible to add a sequence of edges eo, e1,...,en by successive applications of constructor operations to obtain a pla nar embedding, Then starting from the same embeddings, it is also possible to obtain F by applications of constructor operations that successively add any permutation of the edges eo, e1,...,en. Hint: By induction on the number of switches of adjacent elements needed to con- vert the sequence 0,1 ,n into a permutation 2 (0, (1) (n) (c) Prove Corollary (Delete Edge). Deleting an edge from a planar graph leaves a planar graph. (d) Conclude that any subgraph of a planar graph is planar.

Prove Lemma (Switch Edges). Suppose that, starting from some embeddings of planar graphs with disjoint sets of vertices, it is possible by two successive applications of constructor operations to add edges e and then f to obtain a planar embedding, F. Then starting from the same embeddings, it is also possible to obtain F by adding f and then e with two successive applications of constructor operations. Prove Corollary (Permute Edges). Suppose that, starting from some embeddings of planar graphs with disjoint sets of vertices, it is possible to add a sequence of edges e_0, e_1, ellipsis, e_n by successive applications of constructor operations to obtain a planar embedding, F. Then starting from the same embeddings, it is also possible to obtain F by applications of constructor operations that successively add any permutation of the edges e_0, e_1, ellipsis, e_n. Prove Corollary (Delete Edge). Deleting an edge from a planar graph leaves a planar graph. Conclude that any subgraph of a planar graph is planar.

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