Planar Graph and Trees

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1 Dr. Nahid Sultana December 16, 2012

2 Tree Spanning Trees Minimum Spanning Trees Maps and Regions Eulers Formula Nonplanar graph Dual Maps and the Four Color Theorem

3 Tree Spanning Trees Minimum Spanning Trees A graph T is called a tree if T is connected and T has no cycles. Example: Tree Forest 5 leaves 4 leaves

4 Tree Spanning Trees Minimum Spanning Trees A graph T is called a tree if T is connected and T has no cycles. Example: Tree Forest 5 leaves 4 leaves If every connected component of a graph G is a tree, then G is called a forest.

5 Tree Spanning Trees Minimum Spanning Trees A graph T is called a tree if T is connected and T has no cycles. Example: Tree Forest 5 leaves 4 leaves If every connected component of a graph G is a tree, then G is called a forest. A graph without cycles is called cycle-free. The tree consisting of a single vertex with no edges is called the degenerate tree.

6 Tree Spanning Trees Minimum Spanning Trees A graph T is called a tree if T is connected and T has no cycles. Example: Tree Forest 5 leaves 4 leaves If every connected component of a graph G is a tree, then G is called a forest. A graph without cycles is called cycle-free. The tree consisting of a single vertex with no edges is called the degenerate tree. A leaf is a vertex with degree 1 in a tree (or forest).

7 Tree Spanning Trees Minimum Spanning Trees Every tree has the following properties: 1. There is a unique simple path between every pair of vertices. 2. Adding an edge between nonadjacent vertices in a tree creates a graph with a cycle. 3. Removing any edge disconnects the graph. 4. If the tree has at least two vertices, then it has at least two leaves. 5. The number of vertices in a tree is one larger than the number of edges.

8 Tree Spanning Trees Minimum Spanning Trees A subgraph T of a connected graph G is called a spanning tree of G if T is a tree and T includes all the vertices of G.

9 Tree Spanning Trees Minimum Spanning Trees A subgraph T of a connected graph G is called a spanning tree of G if T is a tree and T includes all the vertices of G. Example: A connected graph with a spanning tree highlighted.

10 Tree Spanning Trees Minimum Spanning Trees A subgraph T of a connected graph G is called a spanning tree of G if T is a tree and T includes all the vertices of G. Example: A connected graph with a spanning tree highlighted. Theorem: Every connected graph contains a spanning tree.

11 Tree Spanning Trees Minimum Spanning Trees A minimal spanning tree (MST) of a connected weighted graph G is the spanning tree of G with the smallest possible sum of edge weights.

12 Tree Spanning Trees Minimum Spanning Trees A minimal spanning tree (MST) of a connected weighted graph G is the spanning tree of G with the smallest possible sum of edge weights. Find MST T of a connected graph G with n vertices: notice that T must have n 1 edges. There are two algorithms: Algorithm 1: 1. Arrange the edges in the order of decreasing weights 2. Proceeding sequentially, delete each edge that does not disconnect the graph until n 1 edges remain. Example: In class

13 Tree Spanning Trees Minimum Spanning Trees Algorithm 2: 1. Arrange the edges in the order of increasing weights 2. Starting with only with the vertices and proceeding sequentially, add each edge which does not result in a cycle until n 1 edges are added. Example: In class

14 Tree Spanning Trees Minimum Spanning Trees Algorithm 2: 1. Arrange the edges in the order of increasing weights 2. Starting with only with the vertices and proceeding sequentially, add each edge which does not result in a cycle until n 1 edges are added. Example: In class The weight of a MST is unique, but the MST itself is not. Different MST can occur when two or more edges have the same weight.

15 Maps and Regions Eulers Formula Nonplanar graph A graph/multigraph which can be drawn in the plane so that its edges do not cross is said to be planar.

16 Maps and Regions Eulers Formula Nonplanar graph A graph/multigraph which can be drawn in the plane so that its edges do not cross is said to be planar. Example:

17 Maps and Regions Eulers Formula Nonplanar graph A particular planar representation of a planar graph is called a map. A map divides the plane into a number of regions or faces.

18 Maps and Regions Eulers Formula Nonplanar graph A particular planar representation of a planar graph is called a map. A map divides the plane into a number of regions or faces. Example: the planar graph of K 4 has 4 regions, one of which is exterior to the graph.

19 Maps and Regions Eulers Formula Nonplanar graph A particular planar representation of a planar graph is called a map. A map divides the plane into a number of regions or faces. Example: the planar graph of K 4 has 4 regions, one of which is exterior to the graph. Degree of a region r, written deg(r), is the length of the cycle or closed path which borders r.

20 Maps and Regions Eulers Formula Nonplanar graph A particular planar representation of a planar graph is called a map. A map divides the plane into a number of regions or faces. Example: the planar graph of K 4 has 4 regions, one of which is exterior to the graph. Degree of a region r, written deg(r), is the length of the cycle or closed path which borders r. Theorem: The sum of the degrees of the regions of a map is equal to twice the number of edges.

21 Maps and Regions Eulers Formula Nonplanar graph Example:

22 Maps and Regions Eulers Formula Nonplanar graph Example: The borders of all regions are cycles except for r 3. But if we start at the vertex C and move counter-clockwise around r 3, then we obtain the closed path: (C, D, E, F, E, C) where the edge {E, F } occurs twice.

23 Maps and Regions Eulers Formula Nonplanar graph Example: The borders of all regions are cycles except for r 3. But if we start at the vertex C and move counter-clockwise around r 3, then we obtain the closed path: (C, D, E, F, E, C) where the edge {E, F } occurs twice. deg(r 1 ) = 3,deg(r 2 ) = 3,deg(r 3 ) = 5,deg(r 4 ) = 4,deg(r 5 ) = 3. The sum of the degrees is 18, which is twice the number of edges.

24 Maps and Regions Eulers Formula Nonplanar graph Theorem: Let G = (V, E) be a connected planar graph, and let v = V, e = E, and r = number of regions. Then: v e + r = 2.

25 Maps and Regions Eulers Formula Nonplanar graph Theorem: Let G = (V, E) be a connected planar graph, and let v = V, e = E, and r = number of regions. Then: Example: v e + r = 2.

26 Maps and Regions Eulers Formula Nonplanar graph Theorem: Let G = (V, E) be a connected planar graph, and let v = V, e = E, and r = number of regions. Then: Example: v e + r = 2. Here v = 6, e = 9, and r = 5. And v e + r = = 2, as expected.

27 Maps and Regions Eulers Formula Nonplanar graph Theorem: Suppose a connected planar graph has v 3 vertices and e edges, then e 3v 6.

28 Maps and Regions Eulers Formula Nonplanar graph Theorem: Suppose a connected planar graph has v 3 vertices and e edges, then e 3v 6. Is K 5 planar?

29 Maps and Regions Eulers Formula Nonplanar graph Theorem: Suppose a connected planar graph has v 3 vertices and e edges, then e 3v 6. Is K 5 planar? A graph is called complete bipartite graph if its vertices V can be partitioned into two subsets M and N such that each vertex of M is connected to each vertex of N, denoted by K m,n, where m = M and n = N. Example:

30 Maps and Regions Eulers Formula Nonplanar graph Theorem: The following properties of a graph are equivalent (that is, if the graph has any one of the properties, then it has all of the properties): 1. The graph is bipartite. 2. The graph does not contain any cycles with odd length. 3. The graph does not contain any closed paths with odd length.

31 Maps and Regions Eulers Formula Nonplanar graph Theorem: The following properties of a graph are equivalent (that is, if the graph has any one of the properties, then it has all of the properties): 1. The graph is bipartite. 2. The graph does not contain any cycles with odd length. 3. The graph does not contain any closed paths with odd length. K 3,3 is not planar.

32 Maps and Regions Eulers Formula Nonplanar graph Theorem: The following properties of a graph are equivalent (that is, if the graph has any one of the properties, then it has all of the properties): 1. The graph is bipartite. 2. The graph does not contain any cycles with odd length. 3. The graph does not contain any closed paths with odd length. K 3,3 is not planar. Theorem: (Kuratowski) A graph is nonplanar iff it contains a subgraph homeomorphic to K 3,3 or K 5.

33 Dual Maps and the Four Color Theorem A Vertex coloring or simply coloring of a given graph G is an assignment of colors to the vertices of G such that adjacent vertices have different colors.

34 Dual Maps and the Four Color Theorem A Vertex coloring or simply coloring of a given graph G is an assignment of colors to the vertices of G such that adjacent vertices have different colors. A graph G is n-colorable if there exists a coloring of G which uses n colors.

35 Dual Maps and the Four Color Theorem A Vertex coloring or simply coloring of a given graph G is an assignment of colors to the vertices of G such that adjacent vertices have different colors. A graph G is n-colorable if there exists a coloring of G which uses n colors. The minimum number of colors needed to paint G is called the chromatic number of G and is denoted by χ(g). Example in Class.

36 Dual Maps and the Four Color Theorem Example: Find the chromatic number of the bipartite graph K 2,4.

37 Dual Maps and the Four Color Theorem Example: Find the chromatic number of the bipartite graph K 2,4. Theorem: The following are equivalent for a graph G: 1. G is 2-colorable. 2. G is bipartite. 3. Every cycle of G has even length.

38 Dual Maps and the Four Color Theorem Consider a map M (planar representation of a planar graph/multigraph), Two regions of M are said to be adjacent if they have an edge in common.

39 Dual Maps and the Four Color Theorem Consider a map M (planar representation of a planar graph/multigraph), Two regions of M are said to be adjacent if they have an edge in common. Example:

40 Dual Maps and the Four Color Theorem Consider a map M (planar representation of a planar graph/multigraph), Two regions of M are said to be adjacent if they have an edge in common. Example: By a coloring of M we mean an assignment of a color to each region of M such that adjacent regions have different colors. A map M is n-colorable if there exists a coloring of M which uses n colors.

41 Dual Maps and the Four Color Theorem Consider a map M. In each region of M we choose a point, and if two regions have an edge in common then we connect the corresponding points with a curve through the common edge. These curves can be drawn so that they are noncrossing. Thus we obtain a new map M, called the dual of M, such that each vertex of M corresponds to exactly one region of M.

42 Dual Maps and the Four Color Theorem Example: Find the map which is dual to the following maps:

43 Dual Maps and the Four Color Theorem Example: Find the map which is dual to the following maps: Four Color Theorem (Appel and Haken): If the regions of any map M are colored so that adjacent regions have different colors, then no more than four colors are required.

44 in class.

45 The weight of a cycle in a given weighted graph G is defined as the sum of the weights of the edges in the cycle.

46 The weight of a cycle in a given weighted graph G is defined as the sum of the weights of the edges in the cycle. Now suppose we want to find a Hamiltonian cycle that has least possible weight. This is a very famous optimization problem known as the Traveling Salesperson Problem.

47 The weight of a cycle in a given weighted graph G is defined as the sum of the weights of the edges in the cycle. Now suppose we want to find a Hamiltonian cycle that has least possible weight. This is a very famous optimization problem known as the Traveling Salesperson Problem. Theorem: The complete graph K n contains (n 1)! 2 Hamiltonian cycles.

48 The weight of a cycle in a given weighted graph G is defined as the sum of the weights of the edges in the cycle. Now suppose we want to find a Hamiltonian cycle that has least possible weight. This is a very famous optimization problem known as the Traveling Salesperson Problem. Theorem: The complete graph K n contains (n 1)! 2 Hamiltonian cycles. Example:

49 In class.

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