(a) [BB] Show that the graph is planar by drawing an isomorphic plane graph with straight edges. (b) [BB] Label the regions defined by your plane graph and list the edges which form the boundary of each region. (c) [BB] Verify that V-E R=2, N ≤2 E, and E ≤3 V-6 (where, as defined in the proof of Corollary 14.1 .3, N is the sum of the numbers of edges on the boundaries of all regions). | Numerade

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SOLVED: A connected planar simple graph with 16 edges such that every vertex has degree 4. In how many regions does this planar graph split the plane

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SOLVED:Verify that a contraction of a planar graph is planar.

SOLVED: 9.6 Prove or disprove the following: (a) Every subgraph of a planar graph is planar. (b) Every subgraph of a nonplanar graph is nonplanar. (c) If G is a nonplanar graph, then G contains a proper nonplanar subgraph. (d) If G does not contain K5 or K3,3 as a subgraph, then G is planar. (e) If G is a graph of order n and size m with m ≤ 3n - 6, then G is planar. (f) If G is a graph with one or more triangles and contains no subdivision of K5 as a subgraph, then G is planar.

SOLVED:Give an example of each of the following: (i) a countable planar graph; (ii) a countable non-planar graph.