Planar Graph Theory We say that a graph is planar if it can be drawn in the plane without edges crossing. We use the term plane graph to refer to a planar depiction of a planar graph. e.g. K4 is a planar graph Q1: The following is also planar. Find a plane graph version of the graph. A B F E D C A Method that sometimes works for drawing the plane graph for a planar graph: 1. Find the largest cycle in the graph. 2. The remaining edges must be drawn inside/outside the cycle so that they do not cross each other. Q2: Using the method above, find a plane graph version of the graph below. A B C D E F G H non e.g. K3,3: K5 Here are three (plane graph) depictions of the same planar graph: J N M J K J N I M K K I N M I O O L O L L A face of a plane graph is a region enclosed by the edges of the graph. There is also an unbounded face, which is the outside of the graph. Q3: For each of the plane graphs we have drawn, find: V = # of vertices of the graph E = # of edges of the graph F = # of faces of the graph Q4: Do you have a conjecture for an equation relating V, E and F for any plane graph G? Q5: Can you name the 5 Platonic Solids (i.e. regular polyhedra)? (This is a geometry question.) Q6: Find the # of vertices, # of edges and # of faces for each Platonic Solid. Q7: Find an equation relating V, E and F for the 5 Platonic Solids. Connected Graphs and Trees Formal Definition: We say a graph G is connected if there is a path between every pair of vertices. e.g. EVERY graph we have done so far. Non e.g. The following graph G, where V = {A,B,C,D,E,F,U,W} B E C U F W A D A graph G is called a tree if it is (1) connected and (2) acyclic (i.e. has no cycles). e.g. Graph G below: I R P S T J U O K M Q L N Q8: Draw a tree on 6 vertices, one on 8 vertices and one on 9 vertices. Q9: Can you draw a tree without at least two vertices of degree 1? Q10: In a tree T, take any two vertices u,v in T. How many paths are there between u and v? Q11: How many edges are there in a tree on n vertices? Q12: The cast of Lost has decided to stay on the island. They have five cities on the island. Below is a graph showing the cost (in coconuts) to build a road between each city. If there is not an edge between two cities, then there is a mountain in the way and the road cannot be built. Determine the least cost of making all the cities reachable from each other. NorthTown 9 3 5 11 8 Palm Leaf New Easy 9 7 3 10 Plane City Cabo San Lost Def’n: A spanning tree of a connected graph G is a connected acyclic subgraph T of G. e.g. Above, you found a spanning tree of the island of Lost. Note: A graph can have its edges weighted, as above. We call this a weighted graph. Def’n: A minimum weight spanning tree is (of course) a spanning tree of min. wt. Q13: Find a minimum weight spanning tree of the following graph G: 15 13 13 16 14 14 12 12 16 14 14 15 .