SAMPLE. MODULE 5 Undirected graphs

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1 H P T R MOUL Undirected graphs How do we represent a graph by a diagram and by a matrix representation? How do we define each of the following: graph subgraph vertex edge (node) loop isolated vertex bipartite graph degenerate graph connected graph circuit tree spanning tree complete graph simple graph uler path uler circuit Hamilton path Hamilton circuit adjacency matrix planar graph degree of a vertex How do we apply uler s formula to planar graphs? How do we determine the shortest path between two given vertices on a graph? How do we find the minimum spanning tree of a graph? How do we apply Prim s algorithm to find the minimum spanning trees of a connected graph? How do we apply the Hungarian algorithm?.1 Introduction and definitions hockey team belongs to a competition in which there are six teams:,,,, and. few weeks into the season, some of the teams have played each other: has played, and has played, and has played, and has played and has played and has played, and SMPL igure.1 ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown 1 and McMenamin

2 hapter Undirected graphs 1 This situation can be represented graphically. The teams can be represented by points and two such points are joined by a line whenever the teams they represent have played each other. figure such as igure.1 is called a graph. The points,,,, and are called the vertices or nodes of the graph and the lines connecting the vertices are called edges. graph consists of a set of elements called vertices and a set of elements called edges. ach edge joins two vertices. The vertices and of a graph are adjacent vertices if they are joined by an edge. The graph can be represented by a table or a matrix. 1 is used to denote that there is one edge connecting the two vertices and a 0 indicates that there is no edge. The matrix shown below is called an adjacency matrix Table for figure.1 Matrix for figure.1 The adjacency matrix of the graph is an n n matrix in which for example the entry in row and column is the number of edges joining vertices and. nother graph, igure., is shown opposite. Note that three edges leave from. The graph is said to have multiple edges as there are two edges joining and. has one edge, which links to itself. This edge is called a loop. The table and matrix appear as shown below. Note: loop is recorded as one edge in an adjacency matrix igure. SMPL ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin

3 1 ssential urther Mathematics Module Networks and decision mathematics xample 1 Representing a graph by a table or matrix onstruct the table and matrix corresponding to the graph shown, which represents three houses,, and, connected to three utility outlets, gas (G), water (W) and electricity (). Solution G W G W G W igure. G W G W igure. is called a bipartite graph as the set of vertices is partitioned into two disjoint sets (,, ) and (gas, water, electricity) and each edge has a vertex in each set. bipartite graph is a graph whose set of vertices can be split into two subsets X and Y in such a way that each edge of the graph joins a vertex in X and a vertex in Y. t each non-isolated vertex,,inagraph there will be some edges joined to (incident with vertex ). The degree of vertex, written as deg() isthe number of edges incident with it. origure.: deg = deg = deg = Note: deg = Loops add two to the degree of avertex. SMPL simple graph is a graph with no loops or multiple edges. igure. is a simple graph. or a simple graph, the sum of the degrees of the graph is equal to twice the number of edges of the graph. the sum of the degrees of a simple graph is even. igure. ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin igure.

4 hapter Undirected graphs 17 origure.: Sum = deg() + deg() + deg() + deg() + deg() = = 1 orasimple graph, the entries for the corresponding table are either 1s or 0s. Note that the sum of the 1s = 1 = the number of edges. If is an isolated vertex (i.e. there are no edges incident to ) then deg() = 0. See igure.. graph is said to be degenerate if all its vertices are isolated. See igure igure. igure.7 Many applications of graphs involve getting from one vertex to another. In order to discuss this, the idea of a path is introduced. path in the graph in igure.8 from to could be, G, G,,,, G, G (shown in red). Not all G the edges or vertices in a path are required to be different. path can be thought of as a sequence of edges. igure.8 graph is said to be connected if there is a path between each pair of vertices. SMPL The graph in igure.9 is connected. The graph in igure. is not connected. circuit is a sequence of edges linking successive vertices that starts and finishes at the same vertex. In igure.9, one circuit is. nother circuit is. igure.9 ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin

5 18 ssential urther Mathematics Module Networks and decision mathematics subgraph of a graph consists of selected edges and vertices of the graph with the same links as the original graph and with the selected vertices including all the endpoints of the selected edges. igures.10a and.10b show two subgraphs of igure.9. igure.10a igure.10b igure.10c The graph shown in red in igure.10c is not a subgraph as it has an edge that does not exist in the original graph. xercise 1 This section of a road map can be considered as a graph, with towns as vertices and the roads connecting the towns as edges. a Give the degree of: i Town ii Town iii Town H b onstruct the table (matrix) for this graph. c Is this graph simple? Why? Town or each of the following graphs give the associated table (matrix). a b c e Which of the graphs in Question are: a simple? b connected? f g Town h d Town Town H Town SMPL ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin

6 hapter Undirected graphs 19 or each of the following, draw the graph for the given table (matrix). a b d e a raw three connected subgraphs, with four vertices, of the graph shown. (Note that there are many such subgraphs.) b What is the degree of vertex? c Give the sum of the degrees of the vertices of the graph. d Give a circuit from passing through vertices (including ).. Planar graphs and uler s formula c f Leonard uler (pronounced oiler ) was one of the most prolific mathematicians of all time. He contributed to mathematics in an amazing array of topics. His proof of the formula named after him is considered to be the beginning of the branch of mathematics called topology. The application of uler s formula is developed in this section. lectrical circuits can be represented by a graph, as in igure.11. SMPL igure.11 or certain electrical circuits it is advantageous to not have connections and crossing. igure.11 can be redrawn as shown in igure.1 so that these edges do not cross. igure.1 ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin

7 0 ssential urther Mathematics Module Networks and decision mathematics graph that can be drawn in such a way that no two edges meet (or have common points), except at the vertices where they are both incident, is called a planar graph. Not all graphs are planar. igure.1 cannot be redrawn so that the edges have no intersection points except at the vertices. It is true that all simple graphs with four or fewer vertices are planar. igure.1 onsider a connected planar graph shown in igure.1. This graph has vertices and 8 edges. f onnected planar graphs also have faces or regions. f f 1 f igure.1 has four faces: f 1, f, f and f,where f is an infinite face. There is a relationship between the number of vertices, edges and faces in a connected planar graph. igure.1 Let v denote the number of vertices. Let e denote the number of edges. Let f denote the number of faces. Note that in igure.1: v e + f = 8 + = This result holds for any connected planar graph and is known as uler s formula. xample uler s formula states: v e + f = SMPL Verifying uler s formula Verify uler s formula for the graph shown. f 1 f f Solution 1 The vertices are,,,,. There are vertices so v =. v = There are faces as shown on the diagram. f = on t forget the infinite face f 1 :So f =. The edges are,,,,, and. e = 7 There are 7 of them, so e = 7. v e + f = 7 + = ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin f

8 hapter Undirected graphs 1 xample Verifying uler s formula tetrahedron may be represented graphically by a connected planar graph. Verify uler s formula for the graph shown. Solution v =, f =, e = v e + f = + = xample Using uler s formula connected planar graph has vertices and 8 edges. How many faces does the graph have? raw a connected planar graph with vertices and 8 edges. Solution uler's formula: v e + f = v = ande= f = + f = f = xercise 1 or each of the following graphs: i state the values of v, e and f f f 1 f f ii verify uler s formula. a b c d Show that the following graphs are planar by redrawing them in a suitable form: a b c SMPL f ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin

9 ssential urther Mathematics Module Networks and decision mathematics oraplanar connected graph, find: a f, ifv = 8 and e = 10 b v, ife = 1 and f = c f, ifv = and e = 1 d e, ifv = 10 and f = 11 a Represent this cube as a planar graph. b Verify uler s formula for this graph.. omplete graphs complete graph is a graph with edges connecting all pairs of vertices. The complete graph with n vertices is denoted by K n. K K K Note: K and K are planar while K is not. n(n 1) The complete graph with n vertices, K n, has edges. orexample: ( 1) K has = edges ( 1) K has = 10 edges complete graph could be used to represent a round robin competition (a competition in which each side plays every other side once). orexample, for five teams,,,, and are shown. ach edge represents a match that has been played. The adjacency matrix has 1 s in all positions except the main diagonal. This graph could be drawn for six people at a party. ach edge indicates that a conversation took place. The adjacency matrix has 1s in all positions except the main diagonal SMPL ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin

10 hapter Undirected graphs xercise 1 competition consists of six teams. In the competition each team plays every other team. a How many matches are there? b Represent the competition with the complete graph K. c Represent the competition with an adjacency matrix. a How many edges does K 7 have? b raw K 7. How many handshakes could take place among 8 people. Represent this with a graph.. uler and Hamilton paths uler paths In a paper published in 17, Leonard uler solved the problem that had intrigued the citizens of Königsberg, which lay near the mouth of the Pregel River. The river divided the city into four parts, as shown in igure.1. There were seven bridges. Pregel River igure.1 The problem was posed as follows: ould we walk over each of the seven bridges once only and return to our starting point? uler represented this situation as shown in igure.1. uler showed that this graph cannot be traversed completely in a single cyclical trail. That is, no matter which vertex is chosen as the starting point it is impossible to cover the graph and come back to the chosen starting vertex while using each edge only once. igure.1 SMPL We recall that for a graph, a path is a sequence of edges. In igure.17, one path from to is e 1, e, e (red). nother path from to is e (green). e 1 e e e e igure.17 ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin

11 ssential urther Mathematics Module Networks and decision mathematics Recalling the definition of circuit in section.1: a circuit is a path that begins and finishes at a particular vertex. Inigure.18, one circuit is e 1, e, e, e, e, e (red). This circuit begins and finishes at. path that includes every edge just once is called an uler path. n uler circuit is an uler path that starts and finishes at the same vertex. Identifying uler paths and circuits e e 1 e e e e igure.18 useful way of identifying uler circuits is to look for a connected graph where all vertices have an even degree. The converse result also holds: if a graph has an uler circuit, then it is connected and each vertex has an even degree. Using this result, it can be seen that the graph for the Königsberg bridge problem does not contain an uler circuit as vertex has an odd degree. connected graph has an uler path starting at vertex and finishing at vertex if and are the only odd vertices of the graph. xample Identifying an uler circuit a xplain why the graph shown opposite has an uler circuit. b List an uler circuit for this graph. Solution a The graph has only even vertices. b nuler circuit is. xample Identifying an uler path a xplain why the graph opposite has an uler path between and. b List one possible uler path (there are several) between and. SMPL Solution a and are odd vertices. The remaining vertices are even. b. Hamilton paths Hamilton path is a path through a graph that passes through each vertex exactly once. Hamilton circuit is a Hamilton path which starts and finishes at the same vertex. ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin

12 hapter Undirected graphs It is easy to remember the difference between Hamilton paths (circuits) and uler paths (circuits). Hamilton graphs are defined in terms of vertices and uler graphs are defined in terms of edges. Unfortunately, unlike the condition for an uler circuit, there is no nice condition to identify when a graph is a Hamilton circuit. It is just a matter of trial and error. xample 7 Identifying a Hamilton circuit List a Hamilton circuit for the graph shown. Solution H G Hamilton circuit is G H. Not every graph that has a Hamilton circuit has an uler circuit, and also not every graph that has an uler circuit has a Hamilton circuit. The graph in xample 7 has a Hamilton circuit but not an uler circuit. The graph in igure.19 has an uler circuit but not a Hamilton circuit. xercise igure.19 1 a Which of the following graphs has: i an uler circuit? ii an uler path but not an uler circuit? b Name the uler circuits or paths found. a b c G H SMPL d e ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin

13 ssential urther Mathematics Module Networks and decision mathematics List a Hamilton circuit for each of the following. a b d G H G I e List a Hamilton path for this graph starting at and finishing at G.. Weighted graphs weighted graph is one where a number is associated with each edge. These numbers are called weights. xamples of weighted graphs arise when the vertices of a graph are towns on a map and the edges are the roads between the towns. The number assigned to each edge is the distance between the towns represented by the vertices of that edge. igure.0 is a weighted graph representing towns and the roads connecting them. Snugvill 7 1 ppleville 0 udstop Heavytown learview SMPL Melville igure.0 11 The numbers in a weighted graph may also represent times, fuel consumption, cost, and so on. The number attached to an edge is called the weight of that edge. c G H ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin

14 Trees tree is a connected graph that contains no circuits. igure.1 is a tree. tree has no multiple edges. The simplest tree is a single vertex. tree with n vertices has n 1 edges. Minimum spanning trees cable is needed to connect the communication systems of the four towns in figure.. It is convenient to lay the cable alongside existing roads but it is not necessary to lay the cable next to all roads. subgraph connecting all four towns can be chosen. It is sensible for such a subgraph to be a tree (in red in figure.) shown in figure.. hapter Undirected graphs 7 igure.1 igure. igure. spanning tree is a subgraph that contains all the vertices of the original graph and is a tree. minimum spanning tree for a weighted graph is a spanning tree for which the sum of the weights of the edges is as small as possible. igure. is a minimum spanning tree for figure.. SMPL Prim s algorithm Prim s algorithm is a set of rules to determine a minimum spanning tree for a graph. xamples of where Prim s algorithm applies include such problems as the following or a certain number of cities, a railroad network is to be developed and the cost of connecting any two given cities is known. ind how all the cities should be connected by rail to minimise total cost. Similar problems involve utility connections, e.g. water, gas, electricity. G ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin

15 8 ssential urther Mathematics Module Networks and decision mathematics Prim s algorithm: 1 hoose a vertex and connect it to a second vertex chosen so that the weight of the edge is as small as possible. In each step thereafter, take the edge with the lowest weight, producing a tree with the edges already selected. (If two edges have the same weight the choice can be arbitrary.) Repeat until all the vertices are connected and then stop. xample 8 pplying Prim s algorithm pply Prim s algorithm to obtain a minimum spanning tree for the graph shown. Write down its weight, and compare it to the weight of the original graph. Solution Step 1 Step Step Step The total weight is 17. The total weight of the original graph is 0. xercise 8 Step SMPL 1 ind a minimum spanning tree for each of the following graphs and give the total weight. a b G 17 7 ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin

16 c y trial and error find the shortest path from to. hapter Undirected graphs 9 d H G ind the shortest Hamilton path for the following graph, starting at. 7 SMPL G 8 ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin

17 0 ssential urther Mathematics Module Networks and decision mathematics Review Key ideas and chapter summary Graph Vertices (nodes) and edges djacent vertices djacency matrix Multiple edges and loops ipartite graph egree of a vertex Simple graph egenerate graph Path graph consists of a set of elements called vertices and a set of elements called edges. ach edge joins two vertices figure such as the one drawn opposite is called agraph. The points,,,, and are called the vertices or nodes of the graph and the lines connecting the vertices are called edges of the graph. The vertices and of a graph are adjacent vertices if they are joined by an edge The adjacency matrix of the graph is an n n matrix in which for example the entry in row and column is the number of edges joining vertices and.or the graph above the adjacency matrix is as shown The graph is said to have multiple edges as there are two edges joining and. has one edge, which links to itself. This edge is called a loop. bipartite graph is a graph whose set of vertices can be split into two subsets X and Y in such a way that each edge of the graph joins a vertex in X and a vertex in Y. t each non-isolated vertex,, inagraph there will be some edges joined to (adjacent with vertex ). The degree of vertex, written as deg()isthe number of edges incident with it. Loops are counted twice. SMPL onnected graph simple graph is a graph with no loops or multiple edges. graph is said to be degenerate if all its vertices are isolated. path can be thought of as a sequence of edges of the form, G, G,,,, G, G G graph is said to be connected if there is a path between each pair of vertices. ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin

18 hapter Undirected graphs 1 ircuit Subgraph Planar graph uler s formula omplete graph uler path uler circuit Hamilton path Hamilton circuit Weighted graphs Tree Spanning tree Minimum spanning tree Prim s algorithm circuit is a sequence of edges linking successive vertices that starts and finishes at the same vertex. subgraph of a graph consists of selected edges and vertices of the graph with the same links as the original graph and with the selected vertices including all the endpoints of the selected edges. graph that can be drawn in such a way that no two edges meet (or have common points), except at the vertices where they are both incident, is called a planar graph. uler s formula states that for any connected planar graph, v e + f =, where v = the number of vertices, e = the number of edges, and f = the number of faces. complete graph is a graph with edges connecting all pairs of vertices. path that includes every edge just once is called an uler path. n uler circuit is an uler path that starts and finishes at the same vertex. Hamilton path is a path through a graph that passes through each vertex exactly once. Hamilton circuit is a Hamilton path that starts and finishes at the same vertex. weighted graph is a graph where a number is associated with each edge. These numbers are called weights. tree is a connected graph that contains no circuits. tree has no multiple edges. The simplest tree is a single vertex. tree with n vertices has n 1 edges. spanning tree is a subgraph that contains all the vertices of the original graph and is a tree. minimum spanning tree for a weighted graph is a spanning tree for which the sum of the weights of the edges is as small as possible. Prim s algorithm is a set of rules to determine a minimum spanning tree for a graph. SMPL Review ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin

19 ssential urther Mathematics Module Networks and decision mathematics Review Skills check Having completed this chapter you should be able to: construct an adjacency matrix from a graph and draw the graph from an adjacency matrix determine whether a graph is simple, connected, planar or complete determine whether a graph has an uler circuit or path by trial and error, determine whether a graph has a Hamilton path or circuit find a minimum spanning tree of a graph Multiple-choice questions 1 The minimum number of edges for a graph with seven vertices to be connected is: 7 1 Which of the following graphs is a spanning tree for the 1 network shown? SMPL or the graph shown, which vertex has degree? Q S Q T S R U P U T R connected graph on 1 vertices divides the plane into 1 regions. The number of edges connecting the vertices in this graph will be: ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin

20 hapter Undirected graphs Which of the following graphs does not have an uler circuit? connected planar graph divides the plane into a number of regions. If the graph has eight vertices and these are linked by 1 edges, then the number of regions is: or the graph shown, which of the following paths is a Hamilton circuit? 8 orwhich one of the following graphs is the sum of the degrees of the vertices equal to 1? 9 The sum of the degrees of the vertices on the graph shown here is: 0 1 SMPL 10 or the graph shown, which additional arc could be added to the network so that the graph formed would contain an uler path? ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin Review

21 ssential urther Mathematics Module Networks and decision mathematics Review 11 or the graph shown here, the minimum length spanning tree has length: Of the following graphs, which one has both uler and Hamilton circuits? 1 Which one of the following graphs has an uler circuit? 7 SMPL 1 Which one of the following is a spanning tree for the graph shown here? ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin

22 hapter Undirected graphs Which one of the following graphs has an uler circuit? 1 1 Which one of the following graphs provides a counter-example to the statement: or a graph with seven vertices, if the degree of each vertex is greater than then the graph contains a Hamilton circuit? 17 orwhich one of the following graphs is the sum of the degrees of the vertices equal to 0? SMPL Review ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin

23 ssential urther Mathematics Module Networks and decision mathematics Review 18 Which one of the following paths is a Hamilton circuit for the graph shown here? PQRSTP PQRSTUVP PQUVRSTP PQRSTUVUTP PQRSTUVRVQUTP 19 our towns,,, and are linked by roads as shown. Which of the following graphs could be used to represent the network of roads? ach edge represents a route between two towns xtended-response questions 1 This question is about the vertices of a graph and the degree of a vertex. In Graph below, there are four vertices (the dots). Graph Graph Graph SMPL 1 a omplete the table for Graph. egree b Study Graphs, and and then consider the statement: Number of vertices In any graph the total number of vertices of odd degree is an even number. Is this statement true for Graphs, and? How many vertices of odd degree does each graph have? P U Q T V S R ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin

24 hapter Undirected graphs 7 c To Graph an extra edge is added joining two vertices of even degree. oes the above statement still hold true for this graph? xplain the reasons for your answer. d rom Graph an edge joining an odd degree vertex to an even degree vertex is removed. oes the above statement still hold true for this graph? xplain the reasons for your answer. onsider the graph shown here. a xplain why this is a planar graph. b or this graph, write down: i the number of vertices ii the number of edges iii the number of faces Note: the region outside the graph is counted as a face. c raw a spanning tree for this graph. d or the spanning tree drawn in c, write down i the number of vertices ii the number of edges iii the number of faces Let H denote a planar graph with n vertices. a If T denotes a spanning tree of H, specify: i the number of vertices in T ii the number of edges in T iii the number of faces in T b Hence verify that, for the graph T: number of vertices number of edges + number of faces =. The map shows six campsites,,,,, 9 and which are joined by paths. The numbers by the paths show lengths in kilometres of lake sections of the paths. 1 a i omplete the graph opposite which shows the shortest direct distances between 9 campsites. (The campsites are represented by vertices and paths are represented by 8 edges.) 7 ii telephone cable is to be laid to enable each 8 campsite to phone each other campsite. or environmental reasons, it is necessary to lay the cable along the existing paths. What is the minimum length of cable necessary to complete this task? (cont d.) SMPL Review ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin

25 8 ssential urther Mathematics Module Networks and decision mathematics Review iii ill in the missing entries for the matrix shown for the completed graph formed above b walker follows the route. i How far does this person walk? ii Why is the route not a Hamilton circuit? iii Write down a route that a walker could follow which is a Hamilton circuit. iv ind the distance walked in following this Hamilton circuit. c It is impossible to start at and return to by going along each path exactly once. n extra path joining two campsites can be constructed so that this is possible. Which two campsites need to be joined by a path to make this possible? SMPL ambridge University Press Uncorrected Sample pages Jones, vans, Lipson TI-Nspire & asio lasspad material in collaboration with rown and McMenamin