SAMPLE. Networks. A view of Königsberg as it was in Euler s day.

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1 ack to Menu >>> How are graphs used to represent networks? H P T R 10 Networks How do we analyse the information contained in graphs? How do we use graphs to represent everyday situations? 10.1 Graph theory basics The ridges of Königsberg problem The problem that began the scientific study of networks is known as the Königsberg ridges problem. The problem began as follows: The centre of the old city of Königsberg in Germany was on an island in the middle of the Pregel River. The island was connected to the banks of the river and to another island by five bridges. Two other bridges connected the second island to the banks of the river, as shown below. SMPL view of Königsberg as it was in uler s day. 40 ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

2 ack to Menu >>> 408 ssential Standard General Mathematics simplified view of the situation is shown in the diagram opposite. The question was: an a continuous walk be planned so that all bridges are crossed only once? Whenever someone tried it, they either ended up missing a bridge or crossing one of the bridges more than once, as in the two routes shown below. nter the mathematician The Königsberg ridges problem was a well-known problem in 18th century urope and attracted the attention of the Swiss mathematician uler (pronounced Oil-er ). He started analysing the problem by drawing a simplified diagram to represent the situation, as shown below on the right. SMPL uler's diagram ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

3 This diagram is now what we call a network or a graph. The dots are called vertices (plural of vertex). The lines are called edges. uler found that the answer to the question depended on the degree of each vertex. The degree of a vertex is given by the number of edges attached to the vertex. The number (degree) may be odd or even. or uler s graph: vertex is of degree 5: we write deg() = 5; vertices, and are all of degree : we write, for example, deg() =. ll four vertices are of an odd degree. 1 5 hapter 10 Networks dge deg() = 4 deg() = 5 Vertex uler was able to prove that a graph with all odd vertices cannot be traced or drawn without lifting the pencil or going over the same edge more than once. The problem was solved. The seven bridges of Königsberg could not be crossed in a single walk without either missing a bridge or crossing one bridge more than once. With this analysis, a new area of mathematics was developed, which has many practical applications in today s world. These include: analysing friendship networks, scheduling airline flights, designing electrical circuits, planning large-scale building projects, and many more. This relatively new area of mathematics is now called graph theory, some aspects of which we will explore in this chapter. xercise 10 1 a With a pencil, or the tip of your finger, see whether you can trace out a continuous walk that crosses each of the bridges only once. ack to Menu >>> SMPL ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

4 ack to Menu >>> 410 ssential Standard General Mathematics b n eighth bridge has been added to the diagram as shown. With a pencil, or the tip of your finger, see whether you can trace out a continuous walk that crosses each of the bridges only once. Such a walk exists. or each of the graphs shown, complete the associated statements by filling in the boxes. a i The graph has vertices. ii The graph has edges. iii deg() = iv deg( ) = v The graph has odd vertices. b i The graph has vertices. ii The graph has edges. iii deg() = iv deg( ) = v The graph has odd vertices. c i The graph has vertices. ii The graph has edges. iii deg() = iv deg( ) = v The graph has odd vertices. The game of Sprouts is played between two people and involves drawing a network. Rules for the game of Sprouts Twoormore points are drawn on a piece of paper. These are network vertices. Players then take turns adding edges according to the following rules: 1 ach edge must join two vertices or itself. very time a new edge is drawn, a new vertex must be added somewhere on the edge. dges cannot cross nor pass through a vertex. 4 No vertex may have a degree greater than. 5 The last player able to add a new edge wins. SMPL or more information see: sample game of Sprouts is played on the following page. ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

5 ack to Menu >>> hapter 10 Networks 411 Sample game of Sprouts The starting vertices The starting vertices Step 1: Player 1 Step : Player Step : Player 1 Step 4: Player Step 5: Player 1 wins Player 1 wins because Player cannot draw in a new edge without creating a vertex of degree greater than. Will a game starting with two points always end in five steps? an you find a way of ending the game more quickly? oes the first player always win? What happens if you start with three, four or more points? xplore. 10. Isomorphic and connected graphs Isomorphic graphs One of the things that you will find when working with networks is that quite different-looking graphs actually contain the same information. When this happens, we say that these graphs are equivalent or isomorphic. orexample, the three graphs below appear to be different, but, in graphical terms they are isomorphic. Graph 1 Graph Graph SMPL This is because they contain the same information. ach graph has the same number of edges (5) and vertices (4), corresponding vertices have the same degree and the edges join up the vertices in the same way ( to, to, to, to, and to ). ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

6 ack to Menu >>> 41 ssential Standard General Mathematics However, the three graphs below, although having the same numbers of edges and vertices, are not isomorphic. This is because corresponding vertices do not have the same degree and the edges do not connect the same vertices. Graph 1 Graph Graph Isomorphic graphs Twographs are said to be isomorphic (equivalent) if: they have the same numbers of edges and vertices corresponding vertices have the same degree and the edges connect the same vertices. onnected graphs So far, all the graphs we have met have been connected. That is, every vertex in the graph can be reached from every other vertex in the graph. orexample, the three graphs shown below are all connected. Graph 1 Graph Graph The graphs are connected because, starting at any vertex, say, you can always find a path along the edges of the graph to take you to every other vertex. or example: In Graph 1 you can get from vertex to vertex by travelling along edge. similar statement can be made about all the other vertices. In Graph you can get from vertex to vertex by travelling along edge to vertex, then edge to vertex and finally edge to vertex. ll other vertices are accessible from vertex in a similar manner. In Graph you can get from vertex to vertex by travelling along edge to vertex, then edge to vertex. ll other vertices are directly accessible from vertex. However, the three graphs below are not connected, because there is not a path along the edges that connects vertex (for example) to every other vertex in the graph. SMPL Graph 1 Graph Graph ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

7 onnected graphs have application in a range of problems such as planning airline routes, communication systems and computer networks, where a missing connection can lead to an inoperable system. hapter 10 Networks 41 onnected graphs graph is connected if every vertex in the graph is accessible from every other vertex in the graph along a path formed by the edges of the graph. xercise 10 1 In each of the following sets of three graphs, two of the graphs are isomorphic (equivalent). In each case, identify the isomorphic graphs. a b c Graph 1 Graph 1 Graph Graph Graph Graph ack to Menu >>> SMPL Graph 1 Graph Graph ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

8 ack to Menu >>> 414 ssential Standard General Mathematics d e Graph 1 Graph 1 Graph Graph Which of the following graphs are connected? raw a connected graph with: a three vertices and three edges c four vertices and six edges 4 raw a graph that is not connected with: a three vertices and two edges c four vertices and four edges Graph b three vertices and five edges d five vertices and five edges. b four vertices and three edges d five vertices and three edges. Graph SMPL 5 What is the smallest number of edges that can form a connected graph with four vertices? ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

9 10. Planar graphs and uler s formula Planar graphs planar graph can be drawn on a plane (page surface) so that no edges intersect (cross), except at the vertices. hapter 10 Networks 415 planar graph: no intersecting edges Some graphs do not initially appear to be planar; for example, Graph 1 shown below left. However, Graph (drawn below right) is equivalent (isomorphic) to Graph 1. Graph is clearly planar. Graph 1: non-planar graph as drawn Not all graphs are planar. or example, the graph opposite cannot be redrawn in an equivalent planar form, no matter how hard you try. xample 1 Redrawing a graph in planar form Graph : planar form of Graph 1 Non-planar graph Redraw the graph shown opposite in a planar form. ack to Menu >>> SMPL ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

10 ack to Menu >>> 41 ssential Standard General Mathematics Solution (there are others) 1 Redraw the graph with edge removed. Note: We have removed edge because it intersects edge. Replace edge as a loop that avoids intersecting with the other three edges. The graph is now in an equivalent planar form: no edges intersect, except at vertices. aces of a graph The graph opposite can be regarded as dividing the paper it is drawn on into two regions. In the language of graphs, these regions are called the faces of the graph. One face, f 1,isbounded by the graph. The other face, f,isthe region surrounding the graph. Note that this outside face is infinite in extent. The graph opposite divides the paper into four regions, so we say that it has four faces: f 1, f, f and f 4. Here f 4 is an infinite face. uler s formula f 4 f 1 f f 1 f f uler discovered that, for connected planar graphs, there is a relationship between the number of vertices, v, the number of edges, e, and the number of faces, f. This relationship can be expressed as v e + f = This is known as uler s formula. or the graph opposite: f 1 f f f 4 v =, e = 8 and f = 4. So v e + f = = confirming uler s formula. SMPL ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

11 ack to Menu >>> hapter 10 Networks 41 uler s formula oraconnected planar graph, v e + f = where v = number vertices, e = number of edges and f = number of faces. xample Verifying uler s formula onsider the connected planar graph shown opposite. a Write the number of vertices, v, the number of edges, e, and the number of faces, f. b Verify uler s formula. Solution a 1 There are four vertices:,,,,sov = 4. Number of vertices: v = 4 There are six edges:,,, ( ) and, Number of edges: e = so e =. There are four faces, so f = 4. Tip: Mark the faces in on the diagram. o not forget the f 1 infinite face f 4 that surrounds the graph. f f f 4 Number of faces: f = 4 b 1 Write down uler s formula. Substitute the values of v, e, and f.valuate. Write your conclusion. xample uler s formula: v e + f = v e + f = = uler s formula is verified. Using uler s formula to determine the number of faces of a graph SMPL connected planar graph has four vertices and five edges. etermine the number of faces. Solution 1 Write v and e. v = 4, e = 5 Write uler s formula. uler's formula: v e + f = Substitute the values of v and e f = 4 Solve for f. 1 + f = f = 5 Write your answer. Thegraph has three faces. ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

12 418 ssential Standard General Mathematics xercise 10 1 Which of the following graphs are drawn in planar form? Show that each of the following graphs is planar by redrawing it in an equivalent planar form. a b c d e f H G or each of the following graphs: i state the values of v, e and f ii verify uler s formula. a b ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard SMPL ack to Menu >>>

13 ack to Menu >>> hapter 10 Networks 419 c d 4 oraplanar connected graph, find: a f given v = 4 and e = 4 b v given e = and f = c e given v = and f = d v given e = and f = 4 e f given v = 4 and e = f v given e = and f = 4 g f given v = and e = 11 h e given v = 10 and f = Traversable networks The Königsberg ridges problem of planning a walk that crosses each of the seven bridges only once is an example of what is called traversing a network. Traversing a network means finding a route Start through the network, along the edges, that uses all the edges,butonce only. n example is shown opposite. Start at vertex and follow the arrows to vertex. This network is traversable. Not all networks can be traversed. or, example, no route can be found through the network opposite that uses all the edges only once. Try itand see. This network is not traversable. inish Network 1: traversable Network : non-traversable One way of finding out whether a network is traversable is to try a number of routes through the networks and see whether they work. This is known as trial and error. However, for all but the simplest networks, this can become very tedious after a while. ortunately, we do not have to use trial and error to see whether a network can be traversed. The solution has to do with the order of the vertices and whether they are of odd or even degree. SMPL Rules for determining whether a network is traversable oranetwork to be traversable, it must first be connected. connected network is traversable if: all vertices are of even degree; or exactly two vertices are of odd degree and the rest are of even degree. If a network has more than two vertices of odd degree, it is not traversable. ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

14 ack to Menu >>> 40 ssential Standard General Mathematics Using these rules, we can see why Network 1 on the previous page is traversable: it has two odd vertices ( and ) and the remaining vertex ()iseven. Likewise, Network is not traversable because it has more than two odd vertices (,, and ). xample 4 Traversing a network or each of the following networks: etermine whether the network can be traversed, and state why. If traversable, check by identifying a path that traverses the network. a b c Solution a Traversable: all even vertices Start/inish Note: In each case, more than one path is possible. xercise 10 b Traversable: two odd vertices, the rest even Start inish or each of the following networks: etermine whether the network can be traversed, and state why. If traversable, check by identifying a path that traverses the network. 1 c Not traversable: more than two odd vertices SMPL ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

15 4 5 ack to Menu >>> hapter 10 Networks uler paths and circuits Paths and circuits path is a sequence of edges, linking successive vertices, that connects two different vertices in a network. orexample, in the graph opposite: -- is a path. It starts at vertex, passes through vertex and ends at vertex. --- is another path. It starts at vertex, passes through vertices and, and ends at vertex. Paths in networks can take many forms. They can involve just one edge, or up to all the edges. SMPL circuit is like a path but starts and finishes at the same vertex. orexample, in the graph opposite: -- is a circuit. It starts at, passes through vertices and, and returns to vertex. circuit can involve just two edges, or up to all the edges in a network. ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

16 ack to Menu >>> 4 ssential Standard General Mathematics uler paths n uler path passes along every edge in a connected network, but uses each edge only once. Itmay pass through any vertex more than once. If a network is traversable, it has uler paths. real-life example of an uler path is the route used to collect the garbage in a housing estate. Ideally, it can be planned so that the garbage truck has to travel down each street only once. orexample, the graph above is traversable. It has several uler paths. Two of these paths, and , are shown below. They connect vertices and. (ollow the arrows.) Start inish uler circuits inish Start n uler circuit is an uler path that starts and finishes at the same vertex. real-life example of an uler circuit is the route followed by a country road inspector. Ideally, a route can be planned so that the inspector can start and finish in their hometown, but travel along each road only once. orexample, in the graph opposite, Start/inish ---- is an uler circuit. It starts and finishes at vertex (follow the arrows). n uler circuit has as many start/finish points as there are vertices in the network. SMPL onditions for uler paths and uler circuits To have an uler path or uler circuit, a network must first be connected. To have an uler path but not an uler circuit, the connected network must have exactly two vertices of odd degree, with the remaining vertices having even degree. n uler path will start at one of the odd vertices and finish at the other. To have an uler circuit, the connected network must have all vertices of even degree. n uler circuit starts and finishes at the same vertex. It can be any vertex in the network. If there are more than two vertices of odd degree, the network is not traversable, soitis not possible to have either an uler path or an uler circuit. ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

17 ack to Menu >>> hapter 10 Networks 4 xample 5 Identifying uler circuits and paths or each of the following networks: etermine whether the network has an uler path only, an uler circuit, or neither, and state why. If the network has an uler path only, or an uler circuit, show one example. a b c Solution a uler circuit: all even vertices Start/inish b uler path only: two odd vertices, the rest even Start Note: In each case, more than one path is possible. inish pplication of uler paths and circuits c Neither: more than two odd vertices uler paths and circuits have many practical applications. You have already met one application of an uler path: the Königsberg ridges problem. In everyday life, uler paths relate to situations such as delivering mail, inspecting roads, picking up garbage and cleaning streets in a city. In all these situations there is a need to travel along each road or street in an area, but to do so no more than once. uler circuits apply to the same situations, but with the added condition of wanting to return to the starting point. SMPL xercise 10 1 or each of the following networks: etermine whether the network has an uler path only, an uler circuit, or neither, and state why. ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

18 ack to Menu >>> 44 ssential Standard General Mathematics If the network has an uler path only, or an uler circuit, show one example. a b c d e f g h i road inspector lives in Town and is Town required to do an inspection of the roads connecting the neighbouring towns,, and. The network of roads is shown on the right. Town Town Town Town a an the inspector set out from Town, carry out his road inspection by travelling over every road linking the five towns only once, and return to Town? Why? b If he can, show a possible route. SMPL postman has to deliver letters to the houses located on the network of streets shown on the right. a Is it possible for the postman to H start and finish his deliveries at the same point in the network without retracing his steps at some stage? If not, why not? b It is possible for the postman to G start and finish his deliveries at different points in the network without retracing his steps at some stage. Why is this so? Identify one such route. ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

19 hapter 10 Networks 45 4 Two islands in a river are connected to the banks of the river and to each other by bridges as shown. There is also another bridge connecting the two banks of the river, as shown. a raw a graph to represent this situation. Label the vertices,, and to represent the banks of the river and the two islands. Use the edges of the graph to represent the bridges. b It is not possible to plan a walk that passes over each bridge only once. Why not? c Show where another bridge can be added to make such a walk possible. xplain why it can now be done by drawing a graph that represents the new situation. 10. Hamilton paths and circuits uler paths and circuits focus on edges. Hamilton paths and circuits focus on vertices. Hamilton path passes through every vertex in a connected network once and once only.it may or may not involve all the edges. Hamilton circuit is a Hamilton path that starts and finishes at the same vertex. Hamilton path involves all the vertices but not necessarily all the edges. orexample, in the graph opposite, --- is a Hamilton path. It starts at vertex and ends at vertex.(ollow the arrows.) Note that it does not involve all edges. Hamilton circuit is a Hamilton path that starts and finishes at the same vertex. orexample, in the graph opposite, is a Hamilton circuit. It starts and finishes at vertex.(ollow the arrows.) Note that it does not involve all edges. Unfortunately, unlike uler paths and circuits, there are no simple rules for determining whether a network contains a Hamilton path or circuit. It is just a matter of trial and error. Start Start/ inish ack to Menu >>> inish SMPL ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

20 a ct k o M e n> u > > 4 ssential Standard General Mathematics pplications of Hamilton paths and circuits Hamilton paths and circuits have many practical applications. In everyday life, Hamilton circuits relate to situations like the following: courier leaves her depot to make a succession of deliveries to a variety of locations before returning to her depot. She does not like to go past each location more than once. tourist plans to visit all the historic sites in a city without visiting each site more than once. You are planning a trip from Melbourne to visit Shepparton, Wodonga, endigo, Swan Hill, Natimuk, Warrnambool and Geelong before returning to Melbourne. Hamilton path would apply to situations like the following: You plan a trip from Melbourne to Mildura, with visits to endigo, Halls Gap, Horsham, Stawell and Ouyen on the way. In all these situations, there would be several suitable paths, and other factors, such as time taken or distance travelled, would need to be taken into account in order to determine the best route. This is an issue addressed in the next section: weighted graphs. xercise 10 1 List a Hamilton path for the network shown: a starting at and finishing at b starting at and finishing at G. SMPL Identify a Hamilton circuit in each of the following networks (if possible), starting at each time. a b c ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard G H G

21 ack to Menu >>> hapter 10 Networks 4 d e f G H I H G 10. Weighted graphs and the shortest path problem We plan a trip starting at Town and Town travelling through all the other towns Town Town shown on the network below, before Town returning to Town.Wedonot want to pass through any town more than Town Town once. There are many routes we could choose. ny Hamilton circuit is appropriate. Two possible routes, Route 1 and Route, are shown below. Town Town Town Town Town Route 1 Town Town Town Town Town Town Route Town ut which is the best route? There are several ways in which a route could be regarded as best. It could be the route with the shortest distance, the route that takes the shortest time, the route that costs the least amount of money, or even the most scenic route. In this instance, we will take the best route to mean the shortest distance route. To see which is the shortest distance Town route, we need to add the distances Town 5 Town between the towns to the graphs. We do 8 this by writing the distances (in km) Town 4 5 beside the relevant edges on the graph. When we do this we say we are adding weights to the graph, and we call the resulting graph a weighted graph. Town 9 Town SMPL ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

22 ack to Menu >>> 48 ssential Standard General Mathematics Now we add up the distances along Route 1 and Route. Town 8 Town Town Town Town Route 1 (------): Length = = 40 km Town Town 8 Town Town Town Town Route (------): Length = = km Town Thus Route is the shortest path through the network that will enable you to start and finish at Town, visit every other town on the way and not pass through any town more than once. Shortest path The shortest path that passes through each vertex once only is the shortest Hamilton circuit. While there are systematic ways of determining the shortest path in networks, they are complicated to apply and beyond the scope of this book. onsequently, finding shortest paths in networks is a matter of trial and error. xample inding the shortest path ind the shortest path through the network shown that: starts and finishes at the same point, passes through each vertex once only. State its length. Strategy: The path we want is the shortest Hamilton circuit in the network. SMPL Solution 1 Identify the Hamilton circuits in the network. etermine the length of each circuit. hoose the shortest path and write your answer and its length. 5 TheHamilton circuits are: and or Length = = 1 units or -----: Length = = 1 units Theshortest path is and its length is 1 units. 4 9 ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

23 ack to Menu >>> hapter 10 Networks 49 xercise 10G 1 The following graphs represent the roads between a number of towns labelled,,, and. The length of each road is given in kilometres. Identify the shortest route, starting and finishing at, that would enable each town to be visited once only, and determine its distance. The graphs are not drawn to scale. a The graph below shows a mountain bike rally course. ompetitors must pass through each of the checkpoints,,,,, and. The average times (in minutes) taken to ride between the checkpoints are shown on the edges of the graph. ompetitors must start and finish at 10 checkpoint. They can pass through the 1 other checkpoints in any order they wish What is the route that would have the 8 Start/inish shortest average completion time? ind the shortest path from vertex to vertex in this network. b SMPL 4 ind the shortest Hamilton path for the graph on the right, starting at vertex G ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

24 ack to Menu >>> 40 ssential Standard General Mathematics 10.8 Minimum spanning trees Trees tree is a connected graph that contains no circuits, multiple edges or loops. It may be part of a large graph. orexample, Graphs 1 and below are trees. Graph is not a tree. Graph 1: a tree Graph : a tree Loop Multiple edges Graph : not a tree ircuit tree with n vertices has (n 1) edges. Graph 1 above (a tree) has eight vertices and seven edges, but Graph (not a tree) has 10 vertices and 1 edges. very connected graph contains one or more trees. Two trees are shown on the graph opposite. Spanning trees Tree 1 Tree spanning tree connects all the vertices in a connected graph but has no circuits, multiple edges or loops. SMPL n example of a spanning tree is shown on the graph opposite. There are several other possibilities. Spanning tree oranetwork of n vertices, a spanning tree will have n vertices and (n 1) edges. orexample, the network above has eight vertices. spanning tree for this network will have eight vertices and seven edges. ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

25 ack to Menu >>> hapter 10 Networks 41 xample ind two spanning trees for the network shown opposite. inding a spanning tree in a network Solution 1 The network has five vertices and seven edges. spanning tree will have five (n)vertices and four (n 1) edges. To form a spanning tree, remove any three edges, making sure that: all the vertices remain connected. there are no multiple edges or loops. Spanning tree 1 is formed by removing edges, and. Spanning tree is formed by removing edges, and. Note: Several other possibilities exist. Minimum spanning trees and Prim s algorithm orweighted graphs, it is possible to determine the length of each spanning tree, by adding up the weights of the edges in the tree. or the spanning tree shown opposite: Required vertices = 5 Required edges = 4 Spanning tree 1 Spanning tree SMPL Length = = units Spanning tree ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

26 ack to Menu >>> 4 ssential Standard General Mathematics The minimum spanning tree is the spanning tree of minimum length (may be minimum distance, minimum time, minimum cost, etc.). There may be more than one minimum spanning tree in a graph. Minimum spanning trees have applications such as planning the layout of a computer network, or a water supply system to a new housing estate. In these situations, we might want to minimise the amount of cable or water pipe needed to do the job. lternatively, we might want to minimise the time needed to complete the job, or the cost. Prim s algorithm is a set of rules to determine a minimum spanning tree for a graph. Prim s algorithm for finding a minimum spanning tree 1 hoose a starting vertex (any will do). Inspect the edges starting from this vertex and choose the one with the lowest weight. (If two edges have the same weight, the choice can be arbitrary.) You now have two vertices and one edge. Next, inspect the edges starting from the two vertices. hoose the edge with the lowest weight, provided it does not form a cycle. (If two edges have the same weight, the choice can be arbitrary.) You now have three vertices and two edges. Repeat until all the vertices are connected, then stop. The result is a minimum spanning tree. xample 8 pplying Prim s algorithm pply Prim s algorithm to obtain a minimum spanning tree for the network shown, and calculate its length. Solution 1 Start at : is the lowest-weighted edge (). raw it in SMPL rom or : is the lowest-weighted edge (5). raw it in. 5 ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

27 rom, or : is the lowest-weighted edge (). raw it in. 4 rom,, or : is the lowest-weighted edge (5). raw it in. 5 rom,,, or : is the lowest-weighted edge (). raw it in. ll vertices have now been joined. The minimum spanning tree is determined. ind the length of the minimum spanning tree by adding the weights of the edges. xercise 10H hapter 10 Networks Minimum spanning tree Length = = 1 units 1 connected graph has eight vertices and ten edges. Its spanning tree has vertices and edges. Which of the following graphs are trees? ack to Menu >>> SMPL 5 ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

28 ack to Menu >>> 44 ssential Standard General Mathematics or each of the following graphs, draw three different spanning trees. a b c 4 or each of the following connected graphs, use Prim s algorithm to determine the minimum spanning tree and its length. a b c d e f G SMPL 5 Water is to be piped from a water tank to seven outlets on a property. The distances (in metres) of the outlets from the tank and from each other are shown in the network below. Starting at the tank, determine the minimum length of pipe needed. Outlet Outlet Outlet 1 Outlet G Tank Outlet 5 Outlet Outlet 11 ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

29 ack to Menu >>> hapter 10 Networks 45 Power is to be connected by cable from apower station to eight substations ( to H). The distances (in kilometres) of the substations from the power station and from each other are shown in the network opposite. etermine the minimum length of cable needed. 1 Power station G H SMPL 19 ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

30 ack to Menu >>> 4 ssential Standard General Mathematics Review Key ideas and chapter summary Graph or network Vertices and edges egree of a vertex Isomorphic graphs Multiple edges and loops graph or network consists of a set of points called vertices and a set of lines called edges. ach edge joins two vertices. In the graph above,,,,, and are the vertices and the lines,,,,,,, and are the edges. The degree of vertex, written deg(), is given by the number of edges attached to the vertex. orexample, in the graph above: deg() = and deg() =. Twographs are said to be isomorphic (equivalent) if: they have the same number of edges and vertices corresponding vertices have the same degree and the edges connect the same vertices orexample, the two graphs below are isomorphic or equivalent. SMPL multiple edges loop The graph above is said to have multiple edges, asthere are two edges joining and. has one edge, which links to itself. This edge is called a loop. ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

31 ack to Menu >>> hapter 10 Networks 4 Path onnected graph ircuit Planar graph path can be thought of as a sequence of edges. or example, in the graph below: --, G--- and -G--- are all examples of paths. G graph is connected if there is a path between each pair of vertices. This graph is connected. This graph is not connected. circuit is a sequence of edges linking successive vertices that starts and finishes at the same vertex. orexample, in the graph below: and --- are both circuits graph that can be drawn in such a way that no two edges intersect, except at the vertices, is called a planar graph. This graph is planar. SMPL This graph is not planar. Review ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

32 ack to Menu >>> 48 ssential Standard General Mathematics Review uler s formula uler path ondition for an uler path uler circuit ondition for an uler circuit Hamilton path or any connected planar graph, uler s formula states: v e + f = v = the number of vertices. e = the number of edges. f = the number of faces. orexample: f 1 f v =, e = 4, f = v e + f = 4 + = path that includes every edge just once (but does not start and finish at the same vertex) is called an uler path. To have an uler path (but not an uler circuit), a network must be connected and have exactly two vertices of odd degree, with the remaining vertices having even degree. The uler path will start at one of the odd vertices and finish at the other. Start orexample, the graph opposite is connected. It has two odd vertices and three even vertices. It has an uler path that starts at one of the odd vertices and finishes at the other odd vertex. inish n uler path that starts and finishes at the same vertex is called an uler circuit. To have an uler circuit, anetwork must be connected and all vertices must be even. Start orexample, in the network inish opposite, all vertices are even. It has an uler circuit. The circuit starts and finishes at the same vertex. SMPL Hamilton path is a path through agraph that passes through each vertex exactly once, but does not necessarily start and finish at the same vertex. f Start inish ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

33 ack to Menu >>> hapter 10 Networks 49 Hamilton circuit Weighted graphs Tree Spanning tree Minimum spanning tree Hamilton circuit is a Hamilton path that starts and finishes at the same vertex. Start inish weighted graph is one where a number is associated with each edge. These numbers are called weights. 5 4 tree is a connected graph that contains no circuits, multiple edges or loops. tree with n vertices has n 1 edges. The tree above has 8 vertices and edges. spanning tree is a graph that contains all the vertices of a connected graph, without multiple edges, circuits or loops. Spanning tree SMPL Prim s algorithm minimum spanning tree is a spanning tree for which the sum of the weights of the edges is as small as possible. Prim s algorithm is a systematic method for determining a minimal spanning tree in a connected graph. 9 Review ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

34 ack to Menu >>> 440 ssential Standard General Mathematics Review Skills check Having completed this chapter you should be able to: recognise and determine the number of vertices, edges and faces of a graph recognise isomorphic graphs, connected graphs, planar graphs, weighted graphs, trees and spanning trees use uler s formula for connected planar graphs know and apply the definitions of uler paths and circuits know the conditions for a graph to have an uler path or uler circuit locate an uler path or circuit in a graph know and apply the definitions of Hamilton paths and circuits locate a Hamilton path or circuit in a graph determine the shortest path in a weighted graph know and apply the definition of a spanning tree use Prim s algorithm to determine a minimum spanning tree and its length. Multiple-choice questions The following graph relates to Questions 1 to 1 The number of vertices in the graph above is: 5 9 The number of edges in the graph above is: 5 9 The number of faces in the graph above is: The degree of vertex in the graph above is: The number of even vertices in the graph above is: The graph above has: an uler path but not an uler circuit several uler paths but no uler circuits an uler circuit several uler circuits neither an uler path nor an uler circuit. SMPL ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

35 ack to Menu >>> hapter 10 Networks 441 orwhich one of the following graphs is the sum of the degrees of the vertices equal to 14? 8 or the graph on the right, the sum of the degrees of the vertices is: or the graph in Question 8: v = 9, e = 1, f = 5 v = 9, e = 1, f = v = 10, e = 1, f = 5 v = 9, e = 14, f = v = 10, e = 1, f = 5 10 uler s formula for a planar graph is: v e = f + v e + f = v + e + f = e v + f = v e = f 11 connected graph with 10 vertices divides the plane into five faces. The number of edges connecting the vertices in this graph will be: oraconnected graph to have an uler path but not an uler circuit: all vertices must be odd all vertices must be even there must be exactly two odd vertices and the rest even there must be exactly two even vertices and the rest odd an odd vertex must be followed by an even vertex 1 oraconnected graph to have an uler circuit: all vertices must be odd all vertices must be even there must be exactly two odd vertices and the rest even there must be exactly two even vertices and the rest odd an odd vertex must be followed by an even vertex SMPL Review ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

36 ack to Menu >>> 44 ssential Standard General Mathematics Review 14 Which one of the following graphs has an uler path but not an uler circuit? 15 Which one of the graphs in Question 14 has an uler circuit? 1 Which one of the following graphs has an uler circuit? 1 or the graph shown, which additional edge could be added to the network so that the graph formed would contain an uler path? 18 The length of the shortest path from to in the network shown is: SMPL ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

37 ack to Menu >>> hapter 10 Networks Which one of the following paths is a Hamilton circuit for the graph shown here? Of the following graphs, which one has both uler and Hamilton circuits? 1 Which one of the following graphs is a tree? SMPL Which one of the following graphs is a spanning tree for the network shown on the right? Review ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

38 ack to Menu >>> 444 ssential Standard General Mathematics Review park ranger wants to check all of the trails in a national park, starting at and returning to the Park Office. She would like to do it without having to travel over the same trail more than once. If possible, the route she should follow is: an uler path an uler circuit a Hamilton path a Hamilton circuit a minimum spanning tree 4 park ranger wants to check all of the camp sites in a national park, starting at and returning to the Park Office. The campsites are all interconnected with trails. She would like to check the campsites without having to visit each camp site more than once. If possible, the route she should follow is: an uler path an uler circuit a Hamilton path a Hamilton circuit a minimum spanning tree 5 The park authorities plan to pipe water to each of the camp sites from a spring located in the park. They want to use the least amount of water pipe possible. If possible, the water pipes should follow: an uler path an uler circuit a Hamilton path a Hamilton circuit a minimum spanning tree SMPL ach week, a garbage run starts at the Park Office and collects the rubbish left at each of the camp sites and dumps the rubbish at a tip outside the park. The plan is to visit each camp site only once. If possible, the garbage run should follow: an uler path an uler circuit a Hamilton path a Hamilton circuit a minimum spanning tree ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

39 ack to Menu >>> hapter 10 Networks 445 Short-answer questions 1 raw a connected graph with: a four vertices, four edges and two faces b four vertices, five edges and three faces c five vertices, eight edges and five faces. Redraw each of the following graphs in a planar form. a b or the network shown, write down: a the degree of vertex b the numbers of odd and even vertices c the route followed by an uler path starting at vertex. 4 or the network shown, write down: a the degree of vertex b the numbers of odd and even vertices c the route followed by an uler circuit starting at vertex. 5 or the network shown, determine: a the length of the shortest Hamilton path connecting vertex to vertex b the length of the minimum spanning tree. SMPL or the network shown, determine: a the length of the shortest Hamilton circuit starting and finishing at vertex b the length of the minimum spanning tree. 5 c Review ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard

40 ack to Menu >>> 44 ssential Standard General Mathematics Review xtended-response question 1 The diagram opposite shows the network of walking tracks in a small national park. These tracks connect the camp sites to each other and to the Park Office. The lengths of the tracks (in metres) are also shown Park office a The network of tracks is planar. xplain what this means. b Verify uler s rule for this network. c ranger at camp site 8 plans to visit camp sites 1,,, 4 and 5 on her way back to the Park Office. What is the shortest distance she will have to walk? d How many even and how many odd vertices are there in the network? e ach day, the ranger on duty has to inspect each of the tracks to make sure that they are all passable. i Is it possible for her to do this starting and finishing at the Park Office? xplain why. ii Identify one route that she could take. f ollowing a track inspection after wet weather, the Head Ranger decides that it is necessary to put gravel on some walking tracks to make them weatherproof. i What is the minimum length of track that will need to be gravelled to ensure that all camp sites and the Park Office are accessible along a gravelled track? ii Show the tracks to be gravelled on a diagram. g ranger wants to inspect each of the camp sites but not pass through any camp site more than once on his inspection tour. He wants to start and finish his inspection tour at the Park Office. i What is the technical name for the route he wants to take? ii With the present layout of tracks, he cannot inspect all the tracks without passing through at least one camp site twice. Suggest where an additional track could be added to solve this problem. iii With this new track, write down a route he could follow. SMPL 80 ambridge University Press Uncorrected Sample Pages vans, Lipson, Jones, very, TI-Nspire & asio lasspad material prepared in collaboration with Jan Honnens & avid Hibbard