Introduction to Graph Theory. A problem to solve. A problem to solve. A problem to solve. Notes. Notes. Notes. Notes. Dr. James Burk.

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1 Course: Math 400: {Graph Theory} Office: Science Building Fall 2016 A problem to solve A map of Searcy... A problem to solve A UPS driver has ten stops to make... A problem to solve 15 possible roads to take between stops...

2 A problem to solve Is it possible to visit each stop exactly once?? A problem to solve Can we determine the cheapest route for the driver to travel, if we knew the distances (or costs) of each of these roads? Graph Theory - An Introduction Graph Theory is a branch of mathematics with applications to: computer science optimization theory linguistics chemistry biology and many many others... Graph Theory - An Introduction Definition A graph is a collection of points called vertices and lines called edges that connect some of the vertices.

3 Graph Theory - An Introduction Definition A graph is a collection of points called vertices and lines called edges that connect some of the vertices. Graph Theory - An Introduction Definition A graph is a collection of points called vertices and lines called edges that connect some of the vertices. Graph Theory - An Introduction Definition A graph is a collection of points called vertices and lines called edges that connect some of the vertices. Graph Theory - Some Properties Each vertex has a number associated to it called its degree: Definition The degree of a vertex is the number of edges attached to that vertex. Example: What is the degree of each vertex in these graphs?

4 Graph Theory - Some Properties Something to note about graphs and the degrees of vertices: Each edge has two ends. And each of these two ends contributes to the degree of a vertex. So the total of the degrees of all the vertices must be twice the number of edges in the graph. Theorem (Sum of Degrees Theorem) In any graph, the sum of the degrees of the vertices equals twice the number of edges. Graph Theory - Some Properties Any graph is either connected or disconnected. Definition A graph is connected if any two vertices are connected by a series of edges. If not, the graph is disconnected. We ll see some examples a little later today. Graph Theory - Some Properties Some graphs are the same even though they aren t drawn in exactly the same way. Consider these graphs: We say these graphs are isomorphic. Graph Theory - Some Properties Some things to note here: The length of each edge doesn t matter. Edges don t have to be straight. The arrangement of vertices often doesn t matter. There could be more than one edge between vertices this depends on the context of the graph. Just because edges are drawn so that they intersect doesn t mean that there is a vertex at their intersection.

5 Graph Theory - An example Suppose you have several computers and printers that need to be connected to a server and form a network: Graph Theory - An example Suppose you have several computers and printers that need to be connected to a server and form a network: Graph Theory - An example Suppose you have several computers and printers that need to be connected to a server and form a network: Graph Theory - An example Suppose you have several computers and printers that need to be connected to a server and form a network:

6 Graph Theory - Another Example There are three student organizations on a very small campus: Math club, Science club and Track. Amanda is a member of the Science club and Track, Brent is a member of Math club, and Carla is a member of Math club and Track. A R B S C T Graph Theory - Another Example There are three student organizations on a very small campus: Math club, Science club and Track. Amanda is a member of the Science club and Track, Brent is a member of Math club, and Carla is a member of Math club and Track. A R B S C T Graph Theory - Another Example There are three student organizations on a very small campus: Math club, Science club and Track. Amanda is a member of the Science club and Track, Brent is a member of Math club, and Carla is a member of Math club and Track. A R B S C T Graph Theory - Another Example There are three student organizations on a very small campus: Math club, Science club and Track. Amanda is a member of the Science club and Track, Brent is a member of Math club, and Carla is a member of Math club and Track. A R B S C T

7 Graph Theory - Another Example Any set of information that can be presented in a table can be represented by a graph. For example: # flights Houston Chicago Miami Conroe LA Willis Houston Chicago Miami Conroe LA Willis H LA Con Chi M W Graph Theory - Some History The 18th century, in Konigsburg, Prussia. There were 7 bridges in the city. During a walk through the city, a question was asked: Is it possible to leave home, cross each bridge exactly once, and then return home? Graph Theory - An Introduction The bridge problem can be rephrased as: Is it possible to start at one vertex, and walk along the edges so that: 1 all edges are crossed exactly once, and 2 the walk ends at the same vertex it started. If this is possible, the graph is called an Eulerian graph, after Leonard Euler (pronounced Oiler ) Exercises page 1 of 2 1 For each graph below, find the number of edges, the number of vertices, the degree of each vertex. Verify that the Sum of Degrees Theorem is true for each graph. (a.) The so-called House-X graph (b.) The Petersen graph (c.) A Dorogovtsev-Goltsev-Mendes graph

8 Exercises page 2 of 2 2 Draw a graph with 6 vertices and 12 edges. 3 State a few instances where drawing or using a graph might be necessary or useful. 4 Are these graphs isomorphic? Why or why not? 5 Are these graphs isomorphic? Why or why not? Course: Math 400: {Graph Theory} Office: Science Building jlburk@harding.edu Fall 2016 Eulerian graphs Definition A graph is said to be Eulerian if there is a path starting and ending at the same vertex that uses each edge exactly once. This path is then called an Euler circuit. This has many practical applications: snow plow routes computer chip architecture any application where complete coverage & efficiency are needed Eulerian graphs So, how do we know if a given graph is Eulerian? First, a reminder of two definitions: Definition For any vertex, its degree is the number of edges at that vertex. Example:

9 Eulerian graphs Definition A graph is connected if any vertex can be reached from any other vertex through some edges. Example: Eulerian graphs Definition A graph is connected if any vertex can be reached from any other vertex through some edges. Example: Eulerian graphs Examples: Do any of these three graphs possess an Euler circuit? So, we are now ready for the first theorem in Graph Theory: Theorem (Euler Circuit Theorem) A connected graph is Eulerian when every vertex has even degree. Eulerian graphs But this last one should be familiar: it s the graph representing the Konigsburg bridges! So, since this graph is not Eulerian, there is no way to traverse all seven bridges exactly once, ending where you start.

10 Eulerian graphs Theorem (Euler Circuit Theorem) A connected graph is Eulerian when every vertex has even degree. This theorem only says an Euler circuit exists. It doesn t tell us how to find it. If we use this theorem to determine there is an Euler circuit, how do we find it? Finding Euler Circuits There is an algorithm (a list of steps) that will find an Euler circuit within a graph. But first, a definition: Definition If a connected graph contains an edge that, when removed, causes the graph to be disconnected, that edge is called a cut edge. How to find Euler circuits Fleury s Algorithm 1 Start at any vertex. Walk along any edge to another vertex, (temporarily) removing this edge. 2 From this vertex, walk along any edge to another vertex, trying to avoid a cut edge. Remove this edge as well. 3 Repeat step 2 until all edges have been removed (walked on) and the walk ends at the starting vertex. Fleury s Algorithm an example Here s a graph: it s Eulerean, so there must be an Euler circuit

11 Fleury s Algorithm an example Let s label the vertices so that it s easier to find an Euler circuit: A F B E C Let s find an Euler circuit starting at vertex E. D Fleury s Algorithm an example Remove the edge from E to B (not a cut edge): A F B E C D E B Fleury s Algorithm an example Remove the edge from B to F (a cut edge, but no other choice): A F B E C D E B F Fleury s Algorithm an example Remove the edge from F to A (not a cut edge): A F B E C D E B F A

12 Fleury s Algorithm an example Remove the edge from A to C (a cut edge, but no other choice): A F B E C D E B F A C Fleury s Algorithm an example Remove the edge from C to F (not a cut edge): A F B E C D E B F A C F Fleury s Algorithm an example Remove the edge from F to E (a cut edge, but no other choice): A F B E C D E B F A C F E Fleury s Algorithm an example Remove the edge from E to C (not a cut edge): A F B E C E B F A C F E C D

13 Fleury s Algorithm an example Remove the edge from C to D (a cut edge, but no other choice): A F B E C E B F A C F E C D D Fleury s Algorithm an example Remove the edge from D to E (not a cut edge): A F B E C E B F A C F E C D E D Fleury s Algorithm Things to notice about Fleury s algorithm: We found an Euler circuit starting at vertex E. E B F A C F E C D E We could use this to find one starting at any other vertex, for example: A C F E C D E B F A This algorithm could be used to find many other Euler circuits! They are not unique! Fleury s Algorithm Another example: use Fleury s algorithm to find the Euler circuit in the graph below A B F G C E D

14 Eulerian Walks What if, given a graph, we are interested in traversing all edges, but we don t care if the starting and ending points are the same? Definition An Euler walk is a path that uses each edge exactly once, but ends at a point different from its starting point. Example: Eulerian Walks Theorem (Euler Walk Theorem) A connected graph contains an Euler Walk when all but two vertices have even degree. that is, two with odd degree, and the rest with even degree. The Euler walk will begin and end at the vertices with odd degree. Eulerian Walks Example: A campground has 7 campsites, with 11 bike trails. Is it possible to start at one campground, bike each trail once, and end at the same campground? NO Is it possible to start at one campground, bike each trail once, and end at another campground? YES Homework page 1 of 2 1 Is it possible to leave one city, travel all the roads exactly once, and return to the starting city? Why or why not? Caldwell Alameda Burley Grange Fairmont Dover Evanston 2 Does the graph below contain an Euler walk? Why or why not? If it does, find one. Caldwell Alameda Burley Grange Fairmont Dover Evanston

15 Homework page 2 of 2 3 Does the graph below contain an Euler circuit? Why or why not? If it does, find one. B A C F E G D 4 Use fleury s algorithm to find an Euler circuit (starting at vertex A) in the graph below: Course: Math 400: {Graph Theory} Office: Science Building jlburk@harding.edu Fall 2016 Another problem When a graph represents locations (cities, intersections, houses, etc.), and routes between these locations, we usually let: vertices represent the locations, and edges represent the roads between these locations. If the graph contains an Euler circuit or an Euler walk, all edges can be traversed exactly once. That is, each road could be traveled exactly one time. What if we want to visit each city exactly once, regardless of how many times each road is traveled? Hamiltonian graphs Definition A graph is Hamiltonian if there is a circuit that passes through each vertex exactly once. This path is called a Hamiltonian circuit Warning: Just like Eulerian graphs, the Hamiltonian circuit (if it exists) is usually NOT unique!! Examples:

16 Another example: here s a map of the regions of Ireland Is it possible to leave one region, visit each region exactly once, and return to the starting point? Is it possible to leave one region, visit each region exactly once, and return to the starting point? Is it possible to leave one region, visit each region exactly once, and return to the starting point?

17 Is this graph Hamiltonian? YES! Why could this be important? UPS scheduling, a special case of the Traveling Salesman Problem. So, how do we know whether or not a given graph is Hamiltonian? It looks like a graph will contain a Hamiltonian circuit if there are enough edges between vertices. More specifically,... Theorem (Hamilton Circuit Theorem) Suppose we have a connected graph that has no multiple edges, and has n vertices. If every vertex has degree at least n 2, then the graph is Hamiltonian. Examples: Theorem (Hamilton Circuit Theorem) Suppose we have a connected graph that has no multiple edges, and has n vertices. If every vertex has degree at least n 2, then the graph is Hamiltonian. Definition (Complete graphs) A graph is complete if there is exactly one edge between every pair of vertices. Examples:

18 A question to think about: How many edges are in the complete graph with n vertices? 4 vertices: # of edges: 6 degree of each vertex: 3 5 vertices: # of edges: 10 degree of each vertex: 4 6 vertices: # of edges: 15 degree of each vertex: 5 7 vertices: # of edges: 21 degree of each vertex: 6 Why talk about complete graphs right now? Because there are always more than enough edges to satisfy the Hamilton Graph theorem: Theorem (Hamilton Circuit Theorem) Suppose we have a connected graph that has no multiple edges, and has n vertices. If every vertex has degree at least n 2, then the graph is Hamiltonian. So every complete graph is definitely Hamiltonian!! WARNING: This theorem only says a Hamiltonian circuit exists. It doesn t tell us how to find it. So, how can we find them? Finding Efficient Routes We ll actually do something much better. In many contexts, the edges of a graph are defined in such a way that some edges are better (or worse) than others. For example, a UPS driver might prefer to drive a shorter road than a longer road between cities. We could keep track of these differences with a labeled graph. Finding Efficient Routes Example: Many graphs will carry more information...

19 Finding Efficient Routes Example: Some Hamiltonian circuits are better than others. How do we find the best Hamiltonian circuit? Finding Efficient Routes First, we ll find the best Hamilton circuits within a complete graph (one in which all vertices are connected by exactly one edge.) [Because of the Hamilton Graph Theorem, all complete graphs are Hamiltonian!] Wait: Isn t this impractical? Not all graphs are complete! We can make any graph complete, by adding edges, and labeling these new edges with numbers that make sense. More on this later. Finding Efficient Routes We ll learn two algorithms: the Greedy Algorithm and the Edge-Picking Algorithm. They will both give us efficient Hamiltonian circuits in complete labeled graphs. Then, we choose the best one. We could win $1 million! Homework 1 Which of the graphs below contain an Euler circuit? An Euler walk? Are any of them Hamiltonian? Why or why not? 2 Is the graph below Hamiltonian? Why or why not? Can you find the Hamilton circuit? Caldwell Alameda Burley Grange Fairmont Dover Evanston

20 Homework 1 How many Hamilton circuits does the complete graph K 4 below contain? 2 Draw the complete graph with 5 vertices. Find at least 2 Hamilton circuits. Course: Math 400: {Graph Theory} Office: Science Building jlburk@harding.edu Fall 2016 finding efficient Hamiltonian circuits in complete graphs 1 Choose a vertex as a starting point, and travel along the edge that has the smallest weight. 2 After arriving at the next vertex, travel along the edge of smallest weight that connects to a vertex that has not yet been visited. 3 Continue until all vertices have been visited. 4 Return to the starting vertex. Example:

21 Example: Example: Example: Example:

22 Example: Example: Total cost or weight : = 26 This Greedy Algorithm has this name because it has us choose the cheapest option at each chance we get. The next algorithm uses a different strategy: choose the routes first, the vertices second. Course: Math 400: {Graph Theory} Office: Science Building jlburk@harding.edu Fall 2016

23 finding efficient Hamiltonian circuits in complete graphs 1 Choose a vertex as a starting point, and travel along the edge that has the smallest weight. 2 After arriving at the next vertex, travel along the edge of smallest weight that connects to a vertex that has not yet been visited. 3 Continue until all vertices have been visited. 4 Return to the starting vertex. WARNING!! Be careful in the Greedy Algorithm, choosing a different starting point might give a different circuit!! So, for a complete graph with 5 vertices, the Greedy Algorithm could give 5 different efficient Hamilton circuits!! Monday s worksheet gave an example of this!! The Edge-Picking Algorithm The Edge-Picking Algorithm finding efficient Hamiltonian circuits in complete graphs 1 Mark the edge that has the smallest weight. 2 Mark the edge of next-smallest weight, as long as it does not complete a circuit, and does not add a third marked edge to a single vertex. 3 Continue until no more edges can be marked. That is, until all vertices have been used. 4 Return to the starting vertex. The Edge-Picking Algorithm Example:

24 The Edge-Picking Algorithm Example: The Edge-Picking Algorithm Example: The Edge-Picking Algorithm Example: The Edge-Picking Algorithm Example:

25 The Edge-Picking Algorithm Example: The Edge-Picking Algorithm Example: Total cost or weight : = 26 The Edge-Picking Algorithm For this labelled graph, both algorithms chose the same Hamilton circuit! This will not always happen! Comparing algorithms An example: miles driven running errands Errand Home Post Office Grocery Video store Bank Home Post Office Grocery Video store Bank

26 Total cost or weight : = 73

27 The Edge-Picking Algorithm The Edge-Picking Algorithm The Edge-Picking Algorithm The Edge-Picking Algorithm

28 The Edge-Picking Algorithm The Edge-Picking Algorithm Total cost or weight : = 74 Comparing Algorithms Greedy Algorithm: Edge-Picking Algorithm: Total cost = 73 Total cost = 74 Non-complete labelled graphs Now, what about finding efficient routes in non-complete graphs such as: We have a couple of options for making this graph complete:

29 Non-complete labelled graphs Depending on the context, we could: Non-complete labelled graphs Or, we could: Course: Math 400: {Graph Theory} Office: Science Building Fall 2016 More with Graph Theory One of these graphs is not like the others:

30 More with Graph Theory Definition A tree is a connected graph that contains no circuits. Examples: Theorem If a tree has n vertices, then it will have n 1 edges. More with Graph Theory There is one very important fact about trees which we will take advantage of: Theorem In a tree, there is always exactly one way to get from each vertex to any other vertex without repeating any of the edges. Finding trees inside a graph Given any graph, can we find a tree inside that graph? Here s a graph: Definition It s not a tree! A spanning tree for a graph is a tree inside the graph that includes every vertex. Finding trees inside a graph Definition A spanning tree for a graph is a tree inside the graph that includes every vertex. Here s a graph that is not a tree: Two different spanning trees:

31 Finding trees inside a graph Definition A spanning tree for a graph is a tree inside the graph that includes every vertex. Here s another graph that is not a tree: Two different spanning trees: Finding trees inside a graph So, given any graph, there may be many spanning trees inside that graph. Before we go any further, let s talk about why these might be important. communication networks, air conditioning ducts, and cancer research: Finding trees inside a graph Finding trees inside a graph

32 In [16] a methodology based on Game Theory for the analysis of gene expression data is studied. Roughly speaking, the starting point is the observation of a picture of gene expressions in a sample of cells under a biological condition of interest, for example a tumor. Then, Game Theory plays a primary role to quantitatively evaluate the relevance of each gene in regulating or provoking the condition of interest, taking into account the observed relationships in all subgroups of genes. In this paper, an alternative model based on minimum cost spanning tree representation of gene expression data has been introduced. One of the main characteristics of this model is the possibility to avoid the dichotomization technique required for microarray games introduced in [17]. minimum costs spanning tree situation, cooperative game theory, gene expression analysis, microarray Gene expression occurs when genetic information contained within DNA is transcripted into messenger ribonucleic acid (mrna) molecules and then translated into the proteins. So, estimates of the amount of mrna for a given gene yields a quantitative evaluation of the amount of proteins coded by that gene, that is its gene expression level. Nowadays, DNA microarray technology is available for taking pictures of gene expressions. Within a single experiment of this sophisticated technology, the ratio of expression levels of thousands of genes under two different experimental conditions can be estimated. The expression level of each gene collected from a cell under a biological condition of interest (e.g. atumor),canbecomparedtotheexpression level of the same gene collected from a control cell (usually an healthy one), and the ratio of expression levels between Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for pro t or commercial advantage and that copies bear this notice and the full citation on the rst page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior speci c permission and/or a fee. GameNets 06, October 14, 2006, Pisa, Italy. Copyright 2006 ACM X/06/10...$5.00 the two conditions can be computed for each gene. If k>0 experiments are performed, each gene i in the set of studied genes N can be associated to a k-dimensional vector of real-valued ratio of expression levels, that we will refer as the expression vector of gene i, for each i N. In [17] the class of microarray games has been introduced. Via a dichotomization technique applied to gene expression data, it is constructed a game whose characteristic function takes values on the interval [0, 1]. The objective of such agameistostresstherelevance( sufficiency )ofgroupsof genes in relation to a specific biological condition or response of interest (e.g. a disease of interest). In that paper, it has been discussed the possibility of applying game-theoretical tools that can take into account the relationships which exist among genes, like the Shapley value. The highest Shapley values of the game should point to the most influential genes, so that it could be useful as a hint for pointing at the genes that mostly deserve further investigation. Microarray games may be useful to describe the magnitude of the association of a given biological condition of the original cell (e.g. atumor)withagivengeneexpressioncategory (e.g., abnormal expression, down regulation, strong variability etc.) of genes realized in each coalition of genes, i.e. each subset of N. Aswealreadymentioned,inorderto set up a microarray game, the real-valued ratio of expression levels must firstly be turned into expression categories using appropriate cutoffs on the continuous domain. Akeyissueinthedefinitionofmicroarraygamesistheoperational definition of association between a gene expression category and a biological condition realized in a nonempty coalition S N. GivenasetofgenesN collected from a cell, the authors in [17] claimed that a sufficient condition to realize in a nonempty coalition S N the association between the expression category and a biological condition of the cell is that all the genes which present the expression category belong to coalition S (sufficiency principle for groups of genes). For example, if the expression category is over expression and the biological condition of the cell is tumor, then the association between the tumor and the over expression category is realized in a coalition S N, S, ifalltheoverexpressedgenesinthetumorcellbelong to S. Subsequently, given a set of observed cells, the magnitude of the association is measured, in each coalition S, astheobservedfrequencyofassociationsrealizedins according to the sufficiency principle. Following the example above, if a nonempty coalition of genes S N contains Finding trees inside a graph Minimum cost spanning tree situations and gene expression data analysis Stefano Moretti Unit of Molecular Epidemiology, National Cancer Research Institute (IST) Largo R. Benzi 10, Genoa, Italy Telephone number:+39-(0) stefano.moretti@istge.it ABSTRACT Keywords 1. INTRODUCTION Finding trees inside a graph So, how do we find a spanning tree inside a given graph? Just like with Hamilton circuits, we ll do something better: We ll find the best spanning tree in a graph with labeled edges. This will be called the minimal spanning tree for the graph. Minimal spanning trees Example: Find a spanning tree that has the minimal cost. Minimal spanning trees Example: Cost: = 28.5

33 Minimal spanning trees Example: Cost: = 19.5 Minimal spanning trees Luckily, we have an algorithm for finding mininal spanning trees. Kruskal s Algorithm finding minimal spanning trees in a graph: 1 Choose any edge with smallest weight. 2 Choose the edge with the next smallest weight. 3 Continue choosing new edges with smallest weight, making sure not to create a circuit. 4 Repeat until all vertices have been visited. Minimal spanning trees Example: use Kruskal s Algorithm to find a minimal spanning tree in the following graph: Total cost: Minimal spanning trees Example: use Kruskal s Algorithm to find a minimal spanning tree in the following graph: Total cost: 1 +

34 Minimal spanning trees Example: use Kruskal s Algorithm to find a minimal spanning tree in the following graph: Total cost: Minimal spanning trees Example: use Kruskal s Algorithm to find a minimal spanning tree in the following graph: Total cost: Minimal spanning trees Example: use Kruskal s Algorithm to find a minimal spanning tree in the following graph: Total cost: Minimal spanning trees Example: use Kruskal s Algorithm to find a minimal spanning tree in the following graph: Total cost: = 16.5 Kruskal s algorithm will ALWAYS find the minimal spanning tree!!

35 Homework - page 1 of 2 1 How many edges are in a tree that has 15 vertices? 2 How many vertices are in a tree that has 15 edges? 3 Find all spanning trees in the following graph: 4 Find all spanning trees in the following graph: Homework - page 2 of 2 4 Draw two different spanning trees from the following graph: 5 Use Kruskal s Algorithm to find the minimal spanning tree inside the following labeled graph. What is the total cost? Homework - page 2 of 2 6 Use Kruskal s Algorithm to find the minimal spanning tree in the following labeled graph. What is the total cost? Course: Math 400: {Graph Theory} Office: Science Building jlburk@harding.edu Fall 2016

36 Here s a map of the counties of Ireland, before the Irish Civil War: Note that adjacent counties have different colors. How many colors are used? Some applications of planar graphs Here s a map of the US (without Alaska or Hawaii): Note that adjacent states have different colors. How many colors are used? Map coloring In both cases, only four colors were needed to color the regions of the maps. Is there a map in which more than four colors would be needed? Map coloring Here s a map of the counties of Texas:

37 Map coloring Here s the same map with many colors used: Map coloring The question we have is this: What is the least number of colors needed to color a map so that adjacent regions are different colors? Definition A map is n-colorable if it can be colored with n colors, with adjacent regions different colors. Map coloring Let s represent each map as a graph. Example: Draw each region of the map as a vertex. Map coloring Draw an edge for each pair of regions that share a border (are adjacent). Is this map is 3-colorable!

38 Map coloring Is this map is 3-colorable! YES! Is it 2-colorable? NO! Map coloring What can we say about 2-colorable graphs? Theorem A graph is 2-colorable exactly when there are no circuits with an odd number of vertices. Not 2-colorable Yes, it is 2-colorable The Utilities Problem A question was asked some time ago: Suppose there are three houses A, B, and C. And three utilites X, Y, and Z. Is it possible to connect all three utilities to all three houses without any of the connections crossing over each other? The Utilities Problem It s easy to see how this question can be posed in the realm of graphs: Is it possible to connect each of A, B, and C to each of X, Y, and Z without crossing edges? [This will be an example of a complete bipartite graph.]

39 The Utilities Problem The Utilities Problem The Utilities Problem The Utilities Problem

40 The Utilities Problem The Utilities Problem It doesn t look like we can do it! Those graphs in which we can are very special... Planarity Definition A graph is said to be planar if it can be drawn so that none if its edges intersect. Warning: Just because a graph is drawn with intersecting edges, doesn t mean that it s not a planar graph!! Planarity Example: Is this graph planar?

41 Planarity Example: Is this graph planar? Planarity Example: Is this graph planar? Planarity Example: Is this graph planar? Yes! The graph is planar. Planarity Example: Is this graph planar?

42 Planarity Example: Is this graph planar? Planarity Example: Is this graph planar? Planarity Example: Is this graph planar? No, it s not planar! Planarity Some fun with planar graphs:

43 Planarity For any graph, how do we know if it s planar or not? This is not an easy question to answer. But, there is a nice property of a graph when it is planar Euler s formula. First, another definition. Euler s Formula Take any planar graph drawn with no intersecting edges. The edges divide the plane into different regions called faces. The region surrounding the graph is called the infinite face. Examples: Number of faces = 4 Number of faces = 6 Euler s Formula Theorem (Euler s Formula) Take any planar graph drawn with no intersecting edges. Let v be the number of vertices in the graph. Let e be the number of edges in the graph. Let f be the number of faces in the graph. Then v e + f = 2. Euler s Formula Verify Euler s formula v e + f = 2 for the following graph: v e + f = = 2 v = 4 e = 6 f = 4

44 Euler s Formula Verify Euler s formula v e + f = 2 for the following graph: v = 6 e = 10 f = 6 v e + f = = 2 Back to Map Coloring In the 1850 s, several mathematicians asked the following question: Is it possible to color every map with only 4 colors, so that adjacent regions have different colors? In other words: Is every planar graph 4-colorable? This question wasn t answered until 1976, when two mathematicians (Haken & Appel) proved: Theorem (The Four-Color Theorem) Every planar graph is 4-colorable. Map coloring Theorem (The Four-Color Theorem) Every planar graph is 4-colorable. And every map can be represented as a planar graph. So, no matter how complicated the map, four colors will be all that is needed to color each region with adjacent regions having different colors. Map coloring Here s a map of Texas with four colors used:

45 Homework 1 Is the following graph planar? 2 Is the following graph planar? 3 Draw a complete bipartite graph with 8 vertices. Is it planar? Homework 4 Is the graph in the previous problem 2-colorable? 3-colorable? 5 Is the graph in problem #1 2-colorable? 3-colorable? 4-colorable? 6 Is the graph in problem #2 2-colorable? 3-colorable? 4-colorable? More Map Coloring Course: Math 400: {Graph Theory} Office: Science Building jlburk@harding.edu Fall 2016 More Map Coloring Another application of map coloring scheduling Suppose every student club is required to have a float in this year s homecoming parade. We have one day to decide what each club s float will be. So, each club must meet for an hour to decide on their float. Problem: some people are in more than one club, so all clubs can t meet at the same time. How many different meeting times do we need so that all clubs can meet to discuss their parade float? More Map Coloring

46 Another application of map coloring scheduling In this table, an X means that there is one student in both clubs... Skiing SGA Debate NHS Paper Service Dem. Rep. X X Skiing SGA X X X X Debate X X X X NHS X X X X X Paper X X X X Service X X X X X Dem. X X X Rep. X X X More Map Coloring Scheduling Let s organize this in a graph: More Map Coloring Scheduling Let s organize this in a graph: More Map Coloring Scheduling Let s organize this in a graph: An edge between vertices means that these clubs can t meet at the same time. More Map Coloring

47 Scheduling Let s organize this in a graph: Will 2 meeting times be enough? Is the graph 2-colorable? NO! More Map Coloring Scheduling Will 3 meeting times be enough? Is the graph 3-colorable? I don t think so! More Map Coloring Scheduling Will 4 meeting times be enough? Is the graph 4-colorable? YES! More Map Coloring Another application traffic management Consider an intersection: There are 6 lines of traffic: A, B, C, D, E, F Some of the traffic lines can t move without risk of collision. For example, A and D. How many different cycles of traffic lights will be needed? More Map Coloring

48 Another application traffic management One vertex for each line of traffic. More Map Coloring Another application traffic management One edge for each possibility for collision. More Map Coloring Another application traffic management Are 2 light cycles enough? Is the graph 2-colorable? NO!! More Map Coloring Another application traffic management Are 3 light cycles enough? Is the graph 3-colorable? YES!! More Map Coloring

49 Homework Eight political committees must meet on the same day, but some members are on more than one committee. So those committees that have members in common cannot meet at the same time. An X in the table below indicates that the two corresponding committees share a common member. Use graph coloring to determine the least number of meeting times that will be necessary so that all members can attend their committee meetings. A B F J Ed. He FA Ho Approp. X X X Budget X X Finance X X X X Judiciary X X X X Ed. X X Health X X X For. Aff. X X X Housing X X X More Map Coloring