Undirected graphs and networks

Transcription

1 Gen. Maths h. 1(1) Page 1 Thursday, ecember 0, :10 PM Undirected graphs and networks 1 V co covverage rea of study Units 1 & Geometry In this chapter 1 Vertices and edges 1 Planar graphs 1 ulerian paths and circuits 1 Hamiltonian paths and circuits 1 Trees

2 1 General Mathematics Undirected graphs and networks Graphs are an efficient way of summarising data in many practical problems. The graphs we will be dealing with in this chapter differ from those that we have worked with in the past, as they consist of points connected by various lines. s there is no particular order or direction to these lines, the graphs are defined as undirected graphs or networks. Undirected graphing is an area of mathematics dealing with problems such as planning a delivery route to visit a number of shops while travelling the least distance, designing a communications network to link a number of towns, organising the flow of work in a factory, or allocating jobs for increased efficiency. elow are examples of undirected graphs or networks you may have come across: Route map for the Melbourne Metropolitan Tram Network 1 v, w Lamp H O H O= = O (H O) (O ) The chemical molecules for water (left) and carbon dioxide (right) Milliammeter 0 m + + Rectifier O91 n electrical circuit ohm Voltmeter + The Swiss mathematician Leonhard uler (pronounced oyler; ) developed much of the theory of undirected graphs in his work on topology and graph theory.

3 hapter 1 Undirected graphs and networks 1 Vertices and edges The map at right shows the main roads linking a number of country towns in Victoria. We can see that there are three main roads leading into Traralgon, but only one joining Leongatha and Yarram. The map is an example of an undirected graph or network since there are no arrows showing a particular direction. The network consists of vertices (the towns) and edges (the roads). The degree of a vertex is defined as the number of edges leading to or from that vertex. Therefore the degree of the Traralgon vertex is because there are roads leading to or from there. The degree of the Yarram vertex is because there are only roads leading to or from there. The graph is considered to be a connected graph since it is possible to reach each vertex from any other one. connected graph must have all vertices joined to at least one other vertex. There cannot be any isolated vertices. We use the term multiple edge if there is more than one edge linking two vertices. or example, in the figure at right there is a multiple edge between vertices and. The figure also contains a loop at vertex. n edge which connects a vertex to itself is defined as a loop. When calculating the degree of a vertex, a loop counts as. Thus the degree of vertex is ; 1 for the edge connecting to plus for the loop. Morwell Moe Traralgon Trafalgar Leongatha Yarram The figure is not a connected graph because not all vertices are connected to at least one other vertex. is therefore an isolated vertex as it is not able to be reached from the other vertices. WORK xample or the following graph, state: a the number of vertices b the number of edges c the degree of each vertex d whether the graph is connected. THINK WRIT a ount the number of vertices. Note: The vertices are the points labelled,,, and. b ount the number of edges. Note: The number of edges is found by counting the number of lines joining the vertices. Remember: loop is counted as. c ount the degree of each vertex, that is, the number of edges leading to or from each vertex. Note: 1. contains a loop, which is counted as when totalling edges.. The degree of is abbreviated to deg() etc. d nswer the question. 1 a The number of vertices is, that is,,,, and. b The number of edges is 9. c Vertex has connections to other vertices, so deg() =. Vertex has connections to other vertices, so deg() =. Vertex has connections to other vertices, so deg() =. Vertex has connections to other vertices, so deg() =. Vertex has 0 connections to other vertices, so deg() = 0. d The graph is not connected, as vertex has no edges leading to or from it; that is, it is isolated from each of the other vertices.

4 1 General Mathematics WORK xample raw a connected graph which has vertices, loops and multiple edges. etermine the number of edges and the degree of each vertex. Note: There are numerous ways of drawing this connected graph using the given information. THINK WRIT 1 raw points and label them,,,,,. raw edges connecting the points to. Note: s this diagram represents a connected graph, each vertex must be connected to at least one other vertex. Select two of the points and draw a loop on each of them. Select pairs of points and draw an extra edge connecting each specific pair. Note: or this example, each step has been drawn in a different colour to highlight each stage. ount the number of edges. Note: loop is counted as when totalling edges. There are 1 edges. etermine the degree of each of the vertices, deg() = ; that is, that is, count the number of edges leading to ig 1 or from each vertex. Note: Vertices and have loops; thus they will each have an extra added to their vertex sum. deg() = ; that is, deg() = ; that is, ig 1 deg() = ; that is, deg() = ; that is, deg() = ; that is, ig 17 remember remember 1. n undirected graph or network consists of vertices and edges.. The degree of a vertex equals the number of edges connected to it.. loop adds to the degree of a vertex.. connected graph has no isolated vertices.

5 hapter 1 Undirected graphs and networks 17 1 Vertices and edges WORK xample 1 1 or each of the following graphs, state: a b c d e f 1 i the number of vertices ii the number of edges iii the degree of each vertex. Which of the graphs in question 1 are connected? raw the graphs in question 1 that are not connected, and include extra edges to make the graphs connected. Which of the graphs in question 1 contain loops? WORK xample raw a connected graph which has 7 vertices, loops and multiple edges. etermine the number of edges and the degree of each vertex. raw a connected graph that has: a vertices and 8 edges b vertices and edges c vertices and 1 edges d vertices and edges e vertices and edges. 7 etermine the degree of each vertex in question. 8 If a graph has vertices, what is the least number of edges it could have so that it is connected? 9 raw the following graphs: a number of vertices is, deg() =, deg() =, deg() = and deg() = b number of vertices is, deg() =, deg() =, deg(g) = and deg(h) = or this undirected graph: a determine the number of vertices, V b determine the number of edges, c determine the degree of each vertex d determine whether or not the graph is connected e if the graph is not connected, suggest how it may become connected. G H

6 18 General Mathematics 11 multiple choice The degree of the vertex,, in this diagram is: 1 multiple choice If a connected graph has vertices, the least number of edges it could have is: multiple choice The graph at right consists of: 8 edges and vertices edges and 1 vertices 10 edges and vertices edges and 8 vertices 1 edges and vertices

7 hapter 1 Undirected graphs and networks 19 Planar graphs Undirected graphs that can be drawn with no crossing edges are called planar graphs or networks. lectronic circuits are examples of planar graphs. Sometimes it is possible to redraw undirected graphs that have crossing edges and remove the crossovers. or example, the graph on the left may be redrawn as the graph on the right, a planar graph. To show that the above graph was planar, we simply redrew each of the vertices and then added in the edges making sure that there were no crossovers. However, this is not always possible. The graph at right is not planar. To redraw the graph, we commence with the edges from, and then add in the edges from. When we try to join to, we find that it must cross over an edge. Regions planar graph divides the plane into a number of regions. region, R, is an area from which it is not possible to move unless an edge is crossed. Regions exist within the graph as well as outside the graph. The graph at right has vertices, 9 edges and regions (that is, regions from within the graph and 1 region outside the graph). 1

8 10 General Mathematics WORK xample Redraw the following networks as connected planar graphs if possible. a b c G THINK H I WRIT G a 1 Redraw each of the vertices. a dd in the edges making sure there are no crossovers. nswer the question. b 1 Redraw each of the vertices. b dd in the edges making sure there are no crossovers. G H I The network is a connected planar graph as there are no edges which cross over. nswer the question. c 1 Redraw each of the vertices. c dd in the edges making sure there are no crossovers. The network is not a connected planar graph. It is not possible to reach a particular vertex from all other vertices. nswer the question. G The network is a connected planar graph as there are no edges which cross over. rom the results obtained in worked example we find that: igure and type of graph Number of vertices (V) Number of regions (R) Number of edges () V + R a Planar b Not planar 1 c Planar 7 7 1

9 hapter 1 Undirected graphs and networks 11 a b In fact, if we analysed numerous planar graphs we would find, as uler did, that in each case the final column V + R would equal. Leonhard uler developed a number of important results in the theory of planar graphs, one of which is shown below. or any connected planar graph, the relationship between the number of vertices, V, regions, R, and edges,, is given by: V + R = This is known as uler s Law. WORK xample i etermine whether uler s formula holds for the given graph and state whether the graph is a connected planar graph. ii ompare the sum of the degrees of all the vertices and the number of edges. iii stablish how many odd-degree vertices there are. etermine whether uler s formula holds for the given information and state whether a connected planar graph would be produced. i V = 7, R =, = 8 ii V = 7, R =, = 10 THINK WRIT a i 1 Write down uler s formula. a i V + R = etermine the number of vertices. V = etermine the number of edges. = 8 etermine the number of regions. R = Note: Remember there is an outside region which must be included in the total. Substitute the V, R, and values into the LHS of uler s formula. LHS = V + R = + 8 valuate. = 10 8 = 7 ompare the answer obtained with the RHS of the equation. RHS = RHS = LHS 8 nswer the question. uler s formula holds for the given graph so the given graph is a connected planar graph. ii 1 etermine the degree of each ii deg() = deg() = vertex. deg() = deg() = deg() = deg() = etermine the degree of vertices sum = 1 etermine the number of edges of There are 8 edges. the graph. ompare the results obtained in The degree of all the vertices is twice steps and. iii etermine how many odd-degree vertices there are. the number of edges. iii There are odd-degree vertices, that is, an even number of odd-degree vertices. ontinued over page

10 1 General Mathematics THINK WRIT b i 1 Write down uler s formula. b i V + R = Substitute the V, R, and values into the LHS of uler s formula. LHS = V + R = valuate. = 11 8 = ompare the answer obtained with the RHS of the equation. RHS = RHS LHS nswer the question. uler s formula does not hold for the given information. This combination of values would not produce a connected planar graph. ii 1 Write down uler s formula. ii V + R = Substitute the V, R, and values into the LHS of uler s formula. LHS = V + R = valuate. = 1 10 = ompare the answer obtained with the RHS of the equation. RHS = RHS = LHS nswer the question. uler s formula holds for the given information. This combination of values would produce a connected planar graph. s we observed in worked example, and as Leonhard uler discovered: or any connected planar graph: the sum of the degree of all the vertices = number of edges there is always an even number of odd-degree vertices. We will work with Leonhard uler s results in the following exercise. remember remember 1. planar graph has no crossover edges.. planar graph divides the plane into a number of regions.. When counting regions, the region around the outside of the graph is counted as 1.. or any connected planar graph: (a) uler s Law states that V + R = (b) the sum of the degree of all the vertices = number of edges (c) there is always an even number of odd-degree vertices.

11 hapter 1 Undirected graphs and networks 1 1 Planar graphs WORK xample 1 Redraw the following networks as connected planar graphs, if possible. a b c d a opy and complete the table below for each of the following graphs. i ii iii iv v vi vii viii b i ii iii iv v vi vii viii Number of vertices (V) Number of regions (R) Number of edges () opy and complete the following sentence. or any planar graph, V + R =. This formula is known as Law. V + R WORK xample a a etermine whether uler s formula holds for the given graph and state whether the graph is a connected planar graph. b ompare the sum of the degrees of all the vertices and the number of edges. c stablish how many odd-degree vertices there are.

12 1 General Mathematics omplete the tasks listed below for each of the following graphs. a b c d e f G i ind the number of vertices. ii ind the number of regions. iii Write down the degree of each vertex. iv ind the total of all these degrees. v Write down the number of edges. vi ompare the results obtained in parts iv and v and then copy and complete the following sentence: or any connected planar graph, the sum of the degrees of all the vertices = number of edges. or each of the graphs in question, write down how many vertices there are whose degree is an odd number. Is the following statement true or false? In any connected graph, there is always an even number of odd-degree vertices. WORK xample b 7 etermine whether uler s formula holds for the given information and state whether a connected planar graph would be produced. a V = 8, R =, = 10 b V =, R =, = c V = 7, R =, = 10 d V = 10, R = 8, = 1 e V =, R =, = f V =, R = 8, = g V =, R =, = 8 h V = 7, R = 8, = 1 i V = 9, R =, = 11 j V = 1, R =, = 1 8 Use uler s formula for each of the connected planar graphs below to determine the value of the unknown. a V =, R = 7, =? b V =, R =, =? c V =, R =?, = d V =, R =?, = e V =?, R = 8, = 11 f V =?, R =, = g V = 8, R =, =? h V =, R =?, = 10 i V =?, R = 8, = 1 j V =?, R =, = 7 9 multiple choice The number of vertices a connected planar network with edges and regions has is: 1

13 hapter 1 Undirected graphs and networks multiple choice The number of regions a connected planar network with 8 edges and 7 vertices has is: multiple choice or the given graph, V = 8, R =, = 1 V = 7, R =, = V =, R =, = 1 V = 7, R =, = V = 8, R =, = 1 Schlegel diagrams ig 0 tetrahedron (one of the platonic solids) may be represented as a planar graph. (It may help to imagine a squashed version of the tetrahedron.) H G tetrahedron Such a representation is called a Schlegel diagram. raw the corresponding Schlegel diagram for each of the following platonic solids. cube octahedron dodecahedron icosahedron Map colouring How many different colours are needed to colour a map so that no two adjoining regions are the same colour? Regions that meet at a point only may be the same colour. 1 opy and colour the map at right using the minimum of colours.

14 1 General Mathematics The simplified map of frica below shows borders between countries. Print it from your Maths Quest and colour it using the minimum number of colours. frica Political N km zimuthal qual rea Projection Suggest the minimum number of colours needed to colour any map.

15 hapter 1 Undirected graphs and networks 17 ulerian paths and circuits path is a series of vertices connected by edges: it displays how to travel from one vertex to another via an edge. Just as in everyday ig 1 life a path is something we can walk along, we may consider a path in a graph as something that can be traversed. In a path the vertices are used only once and the edges are traversed (passed over) only once; that is, edges and vertices listed in a path may not be repeated. Some examples of paths in the graph above are:, and. The sequence is not a path since there is no edge which connects vertex to vertex. circuit (or cycle) is a path which starts and finishes at the same vertex and no edge is traversed (passed over) more than once. n example of a circuit from the above graph is. 1. path is a series of vertices connected by edges.. circuit (or cycle) is a path which starts and finishes at the same vertex and no edge is traversed more than once. n ulerian path is a path which uses each edge in a graph only once, however a vertex may be repeated. ig or the graph at right, an ulerian path could start at, travel to then finish at. nother ulerian path is. o you remember a children s game in which you must draw a picture of an envelope (or a house) without lifting your pencil off the paper and without going over any line twice? This is an example of an ulerian path. graph may have more than one ulerian path or it may not be possible to draw any ulerian paths. n example of a graph for which an ulerian path is not possible is shown at right. If an ulerian path starts and finishes at the same vertex, it is called an ulerian circuit. Therefore, an ulerian circuit is a 1 path which starts and finishes at the same vertex, uses all the edges and does not go over any edge twice. n ulerian circuit may be drawn for the graph at right. The ulerian circuit is 1 1. ulerian paths and circuits have many real-life applications in areas such as planning delivery routes and communication networks. onsider the following example of a practical application of an ulerian circuit. Postal workers collect their mail from the distribution centre, deliver mail along their particular route where each street (edge) is crossed once. t the end of their delivery route, the postal workers return to the distribution centre. Other examples include recycling and garbage collection, newspaper and junk mail deliveries and so on.

16 18 General Mathematics 1. n ulerian path is a path which uses each edge in a graph only once.. n ulerian circuit is an ulerian path which starts and finishes at the same vertex. or each of the following graphs determine whether it is possible to draw an: i ulerian path and ii ulerian circuit. a WORK xample ig b ig 7 THINK a i 1 Specify a path in which each edge is only used once. Note: vertex may be used more than once. nswer the question. ii 1 Specify a path which begins and ends at the same vertex but uses each edge only once. WRIT a i egin at vertex ; travel to. It is possible to specify an ulerian path from the given graph. The ulerian path is. ii egin at vertex ; travel to. annot get back to without going over edges already covered. ttempt another path. Note: In this case any path attempted will lead to the same result. nswer the question. egin at vertex, travel to. annot get back to without going over edges already covered. It is not possible to specify an ulerian circuit from the given graph without going over some of the edges twice. b i 1 Specify a path in which each edge is only used once. Note: vertex may be used more than once. nswer the question. b i egin at vertex ; travel to. It is possible to specify an ulerian path from the given graph. This is actually a special case of an ulerian path as it is also an ulerian circuit.

17 hapter 1 Undirected graphs and networks 19 THINK ii 1 Specify a path which begins and ends at the same vertex but uses each edge only once. nswer the question. Note: In this case it is possible to obtain an ulerian circuit from any vertex. WRIT ii egin at vertex ; travel to. It is possible to specify an ulerian circuit from the given graph. One possible ulerian circuit is. fter studying many undirected graphs dealing with ulerian paths and circuits, it was found that for an ulerian path and an ulerian circuit to be obtained from a given network, the following characteristics had to be satisfied: 1. n ulerian path is possible if the number of odd-degree vertices is 0 or.. n ulerian circuit is possible if each of the vertices has even degrees. Let us see if these characteristics have been satisfied in worked example. Recall that in worked example a, only an ulerian path was satisfied. However, in worked example b, both an ulerian path and an ulerian circuit were specified. If we examine the degree of each vertex in worked example a, we find that there are two odd number degree vertices; that is, and. Therefore the characteristics for an ulerian path have been satisfied. However, we were unable to specify an ulerian circuit as each vertex did not have an even degree. If we examine the degree of each vertex in worked example b, we find that there are 0 odd number degree vertices. Vertex egree 1 Odd Odd Vertex egree This implies that each vertex has an even degree. Therefore the characteristics for both an ulerian path and an ulerian circuit have been satisfied. remember remember 1. path is a series of vertices connected by edges.. circuit (or cycle) is a path which starts and finishes at the same vertex and no edge is traversed more than once.. n ulerian path is a path which uses each edge in a graph only once.. n ulerian circuit is an ulerian path which starts and finishes at the same vertex.. n ulerian path is possible if the number of odd vertices is 0 or.. n ulerian circuit is possible if each of the vertices has even degrees.

18 170 General Mathematics 1 ulerian paths and circuits WORK xample 1 or which of the following graphs is it possible to draw an: a ulerian path b ulerian circuit? i ii iii iv v vi opy and complete this table for the graphs in question 1. Vertex Number of odd vertices ulerian path (Yes/No) ulerian circuit (Yes/No) egree of vertex in graph i egree of vertex in graph ii egree of vertex in graph iii Note: The shaded regions indicate that the particular vertex does not exist for the given graph. Using the results from question, copy and complete these rules to check if it is possible to draw an ulerian path or an ulerian circuit. a n ulerian path is possible if the number of odd degree vertices is or. b n ulerian circuit is possible if the vertices all have degrees. or the following networks: i state the degree of each vertex ii specify whether an ulerian path is possible egree of vertex in graph iv egree of vertex in graph v egree of vertex in graph vi

19 hapter 1 Undirected graphs and networks 171 iii specify whether an ulerian circuit is possible. a b c d G H The graph at right shows a number of roads in a town. The G council is organising a street sweeping route so that the same road is not used twice. The street sweeping truck is to start and finish at the council depot,. H I K a xplain why it is not possible to do this without going down one street twice. J b If one street were to be left out, it would be possible to complete the task without going down the same street twice. Which street has to be left out? The network at right shows a number of paths in a park. a Is it possible to start at and walk along each pathway once H and return to? I G b If it is, draw such a path. If not, which path needs to be walked along twice? J 7 multiple choice or this graph, a suitable ulerian path would be: Questions 8 and 9 refer to the following graphs. i ii iii 8 multiple choice The graphs which have an ulerian path are: i only i and ii only i, ii and iii ii and iii only i and iii only 9 multiple choice The graphs which have an ulerian circuit are: i only ii only iii only i and ii only ii and iii only 10 multiple choice If a graph has only even degree vertices then: it is possible to draw an ulerian circuit but not an ulerian path it is not possible to draw either an ulerian circuit or an ulerian path it is possible to draw an ulerian circuit but it depends on the graph whether an ulerian path can be drawn it is possible to draw an ulerian path but not an ulerian circuit it is possible to draw an ulerian path and an ulerian circuit. H G WorkSHT 1.1

20 17 General Mathematics Hamiltonian paths and circuits ulerian paths are used when we need to find a way to travel along each edge only once. This is useful in areas such as postal or delivery routes and garbage collections. However, there are occasions when we are interested in travelling to each vertex only once, but it is not important that we travel along each edge. or example, if the vertices are five tourist areas to be visited in a town, we may be interested in visiting each area (vertex) but not in travelling along each road. path that passes through each vertex once is called a Hamiltonian path (named after Sir William Hamilton [180 ], a Scottish mathematician). Hamiltonian path must pass through each vertex once, but does not have to use each edge. Usually all of the edges are not required to draw a Hamiltonian path. In the graph shown (above right), one Hamiltonian path is. nother is. It is possible to specify a number of Hamiltonian paths from a given graph. Hamiltonian path that starts and finishes at the same vertex is called a Hamiltonian circuit, in this case, one vertex is used twice: for starting and finishing. In the network at right, is an example of a Hamiltonian circuit. 1. Hamiltonian path passes through each vertex only once. It may not use all of the edges.. Hamiltonian circuit is a Hamiltonian path which starts and finishes at the same vertex. WORK xample etermine which of the following have a: i Hamiltonian path ii Hamiltonian circuit. a b c THINK a i 1 Specify a Hamiltonian path for the given graph. Note: ach vertex must be used only once. However, each edge does not need to be used. There may be more than one Hamiltonian path. nswer the question. WRIT a i egin at vertex ; travel to. Or begin at vertex ; travel to. It is possible to specify a Hamiltonian path from the given graph. Possible Hamiltonian paths include: or.

21 hapter 1 Undirected graphs and networks 17 THINK ii 1 Specify a Hamiltonian circuit for the given graph. Note: We must begin and end at the same vertex and pass each of the other vertices once only. nswer the question. WRIT ii egin at vertex ; travel to. It is possible to specify a Hamiltonian circuit from the given graph. possible Hamiltonian circuit is. b i 1 Specify a Hamiltonian path for the given graph. nswer the question. b i egin at vertex ; travel to. It is possible to specify a Hamiltonian path from the given graph. The Hamiltonian path is. ii 1 Specify a Hamiltonian circuit for the given graph. Note: To get back to vertex, we will have to go through vertices and. nswer the question. ii egin at vertex ; travel to. It is not possible to get back to without passing through and again. It is not possible to specify a Hamiltonian circuit as we cannot get back to the start, that is, to vertex, without passing through and again. c i 1 Specify a Hamiltonian path for the given graph. nswer the question. c i egin at vertex ; travel to. We cannot go any further as we are unable to get to vertex and. It is not possible to specify a Hamiltonian path as the graph is not connected. ii 1 Specify a Hamiltonian circuit for the given graph. nswer the question. ii egin at vertex ; travel to. gain, it is not possible to reach vertices and and then get back to. It is not possible to specify a Hamiltonian circuit as the graph is not connected.

22 17 General Mathematics The graph at right shows a delivery network with O being the central office. O Gaetano must deliver items to each of the six places labelled 1,, 1...,. The edges represent roads between the places. Plan a delivery route for Gaetano to deliver the items to each of the six places, without going to the same place twice. Gaetano must return to the central office after the items have been delivered. THINK WRIT 1 Specify a Hamiltonian path for the given egin at vertex O; travel to 1 O. graph. Note: Gaetano must visit each vertex only once and start and finish at O. Therefore, we need a Hamiltonian circuit. heck that each vertex (except O) has ach vertex (except O) has only been used WORK xample 7 only been used once. nswer the question. Note: In this example there is more than one possible solution. once. One possible solution for Gaetano s route is O 1 O. nother possible solution is O 1 O. Shortest path The shortest path in a graph has a number of applications in business: minimising the distance travelled, minimising the time for a journey or a series of jobs and minimising transport costs. In shortest path problems, we usually have a starting point (for example a depot, main office or a warehouse) and require a Hamiltonian circuit with the least distance. One method for finding the shortest path is to move along the network choosing the edge with the shortest distance or least cost until a Hamiltonian circuit is formed. WORK xample distributor supplies retail outlets labelled,,, and from a warehouse, W. The distances along the roads are shown in the network at right. ind the shortest delivery route that goes to each retail outlet and returns to the warehouse. THINK WRIT 1 Obtain a Hamiltonian circuit starting and finishing at W. Look at the possible routes W, W and W. hoose the route of the least distance, that is, W. ontinue from, until a Hamiltonian circuit is formed choosing the least distance. Highlight the selected route. nswer the question. 8 9 km km W 8 km km 7 km 10 km 1 km 11 km 9 km 18 km egin at vertex W, travel to W. 9 km km W 8 km km 7 km 10 km 1 km 11 km 9 km 18 km The shortest delivery route, W W, is highlighted in the above diagram. The distance for the shortest route is km, that is, = km.

23 hapter 1 Undirected graphs and networks 17 In other problems we may not require a Hamiltonian circuit as many shortest path situations require simply travelling in one direction from the starting point to the destination. method similar to that used in worked example 8 may be used; choosing the shortest distance edge and moving along the network until the destination is reached. ind the shortest distance along the roads between towns and G in the given diagram. THINK 1 WORK xample 9 Obtain the route which gives the shortest distance from to G. Note: We do not require a Hamiltonian circuit as we are moving directly from to G and do not need to visit all of the towns. Start at vertex. Look at the possible routes, and. hoose the route of the least distance, that is,. rom, look at G, and. gain choose the route of the least distance, that is,. Note: The distance from towns to G via G was shorter than the more direct route of to G (that is, 7 km compared to 1 km). 10 km 17 km 1 km WRIT egin at vertex ; travel to G. 1 km 11 km 18 km 9 km 9 km 17 km Highlight the selected route. nswer the question. The shortest delivery route from and G, G, is highlighted in the above diagram. The distance for the shortest route is 8 km, that is, = 8 km. km 11 km 18 km 9 km 9 km 1 km 1 km 10 km 1 km km 17 km 1 km G km 1 km 1 km 10 km km 1 km G There is a lot of trial and error involved in calculating the shortest distance; it is advisable to use a pencil, eraser and scrap paper for calculations to decide the shortest path. remember remember 1. Hamiltonian path passes through each vertex only once. It may not use all of the edges.. Hamiltonian circuit is a Hamiltonian path which starts and finishes at the same vertex.. To find the shortest path in a network, choose the edge of least distance and move along the network until the destination is reached. This usually requires a trial and error approach.

24 17 General Mathematics 1 Hamiltonian paths and circuits WORK xample 1 etermine which of the following have a: a Hamiltonian path b Hamiltonian circuit. i ii iii iv v vi WORK xample 7 The graph at right represents a delivery route with O being the central office. courier must deliver items to each of the eight 1 O places labelled 1,,..., 8. The edges represent roads between the places. Plan a delivery route for the courier to deliver the 7 8 items to each of the eight places without going to the same place twice. The courier must return to the central office after the items have been delivered.

25 hapter 1 Undirected graphs and networks 177 WORK xample 8 WORK xample 9 onstruct a graph for which a Hamiltonian circuit is 1. raw a graph that does not have a Hamiltonian path. Which graphs in question 1 also have an ulerian path? distributor located at O supplies shops labelled to. The distances are in kilometres. ind the shortest delivery route from the depot at O that visits each shop and returns to the depot. 7 delivery firm has to collect goods at storage places,, G, H and I and deliver them to the depot at. ind the shortest route from the depot visiting each storage place and returning to the depot O km 0 km 8 ind the length of the shortest Hamiltonian circuit in each of the following. a b c 11 d (Hint: You may find it easiest to start in the top left-hand corner of each network.) 9 The table shows the distance, in km, between storage points,,,, and G and a depot. G Note: The shaded region indicates that there is no road connecting the two towns. a Transfer the information from the above table onto a copy of the graph at right. G b ind the shortest path from the depot, visiting each storage point and finishing at the depot. 10 ind the shortest distance along the roads, between the towns S and in each of the folowing diagrams. (Note that distances are in kilometres.) a 8 b c S 10 S 18 G 7 S G G 11 km 8 km 1 km 8 km 7 km 10 km I km H 1 km G 9 km

26 178 General Mathematics 11 or the network shown in the diagram at right, find the shortest distance between: a and H b and G. 1 tour of six country towns is to start and finish at endigo. ind the shortest distance tour. 1 The distance, in km, between five towns is shown in the table below Note: The shaded region indicates that there is no road connecting the two towns. a Transfer the information from the above table onto a copy of the graph at right. b tour is planned to start and finish at and visit all of the towns. ind the shortest distance for the tour. c If the tour were to start and finish at, would the shortest distance be the same? d new tour, starting and finishing at, is planned to visit all the towns and a further town,. The distances from to, to and to are 7, 9 and 19 km respectively. ind the shortest distance tour now possible G H 9 1 Mildura km 10 km Swan Hill Ouyen 10 km 0 km 1 km Kerang 11 km Horsham 19 km endigo 7 km km 1 multiple choice 1 Hamiltonian circuit for the graph at right is: G G G G G multiple choice G WorkSHT 1. The length of the shortest Hamiltonian circuit is: km 7 km km 0 km 1 km 1 km km 1 km km km 17 km 9 km 1 km 1 km 1 km

27 hapter 1 Undirected graphs and networks 179 Trees tree is a connected graph without any circuits, loops or multiple edges. xamples of trees include: 1 and Since a tree cannot have any circuits or loops, it contains only one region. xamples of graphs which are not trees include: and 1 This is not a tree This is not a tree This is not a tree because it is because the graph because the graph a circuit. contains a loop. contains a multiple edge. Trees are used in transportation and communication networks, in computer programming, in planning projects to represent independent activities or structures, and in road and railway planning. WORK xample 10 Which of the following are trees? a 1 b 1 c 1 THINK WRIT a etermine whether the graph is connected. a This graph does not represent a tree since it is not connected. b 1 etermine whether the graph is b The graph is connected. connected. etermine whether the graph contains any circuits, loops or multiple edges. This graph does not represent a tree since it contains a circuit, that is, 1 1. c 1 etermine whether the graph is connected. c The graph is connected. etermine whether the graph contains any circuits, loops or multiple edges. ount the number of regions. This graph does represent a tree as it meets each of the criteria, that is, this is a connected graph without any circuits, loops or multiple edges. There is only 1 region. In order for any network to be defined as a tree, it must satisfy the following criteria: 1. the graph must be connected. the graph must not contain any circuits, loops or multiple edges. the graph must contain only one region.

28 180 General Mathematics a b WORK xample 11 Remove the edges from this graph to produce a tree. omment on the relationship between the number of edges and the number of vertices. THINK a 1 Look at the given graph and identify any circuits. Remove one of the edges from circuit, say. WRIT a There are circuits:,, and. Remove the edge from circuit Remove the edge from circuit. b 1 ount the number of vertices, V. b V = ount the number of edges,. = nswer the question. The difference between the vertices and edges in a tree is 1; that is, V = 1. The tree obtained in worked example 11 is one possible spanning tree for the graph. spanning tree is a tree that includes all the vertices in the graph. ind other spanning trees in worked example 11 by removing different edges. Often it is necessary to find a minimal spanning tree; that is, a spanning tree with the minimum length (or cost, or time). To find a minimal spanning tree: 1. select the edge with the minimum value. If there is more than one such edge, choose any one of them.. select the next smallest edge, provided it does not create a cycle.. repeat step until all the vertices have been included.

29 hapter 1 Undirected graphs and networks 181 a b WORK xample 1 ind the minimal spanning tree of the network shown at right. omment on the relationship between the number of edges and the number of vertices. THINK a 1 Select the edge with the smallest value and highlight it. WRIT a The smallest edge is Select the next smallest edge and highlight it. Select the next smallest edge and highlight it. The next smallest edge is. The next smallest edge is. ontinue this process until vertices and are connected and highlighted. Note: and respectively are the next smallest edges. nswer the question. b 1 ount the number of vertices, V. b V = ount the number of edges,. = nswer the question. 7 The minimal spanning tree is shown above. Its value is =. The difference between the vertices and edges in a tree is 1; that is, V = 1. remember remember 1. tree is a connected graph without any circuits, loops or multiple edges which contains only one region.. spanning tree is a tree that includes all the vertices in the graph.. minimal spanning tree is a spanning tree with the minimum length (or cost, or time).. To find a minimal spanning tree: (a) select the edge with the minimum value. If there is more than one such edge, choose any one of them. (b) select the next smallest edge, provided it does not create a cycle. (c) repeat step b until all the vertices have been included.. The vertices and edges in a tree are related by the equation V = 1.

30 18 General Mathematics 1 Trees WORK xample 10 1 Which of the following are trees? a b c d e f g h or those networks which are not trees, give reasons why they are not. WORK xample 11 hange those graphs in question 1 that are not trees into trees by removing or adding edges. a i raw a tree with vertices. ii How many edges are there? b i raw a tree with 11 vertices. ii How many edges are there? c opy and complete: If a tree has V vertices, the number of edges is given by =. Use the result from question to find how many vertices there are in a tree with: a 8 edges b 9 edges c 1 edges d 17 edges e 0 edges onstruct spanning trees for the following graphs: a b c d WORK xample ind the value of the minimal spanning tree for each of the following graphs. a b c d e 9 f communication network is to be developed linking six towns. The distances (in km) between the towns are shown in the graph at right. alculate the minimal spanning tree so that the length of cabling used to connect the towns is a minimum

31 hapter 1 Undirected graphs and networks 18 8 multiple choice The following graphs which represent trees are: i ii iii iv 9 i and iv only i and ii only ii and iii only iii and iv only all of them multiple choice The network at right, when changed to a tree, will resemble which of the following?

32 18 General Mathematics summary Vertices and edges n undirected graph or network consists of vertices and edges. The degree of a vertex is the number of edges leading to or from that vertex. loop counts as edges. In a connected graph it is possible to reach each vertex from any other vertex. connected graph must not have any isolated vertices. Planar graphs planar graph has no crossover edges. planar graph divides the plane into a number of regions. When counting regions, the region around the outside of the graph is counted as 1. planar graph Multiple edges or any connected planar graph: V = uler s Law states that: V + R =. = 8 The sum of the degree of all the vertices = R = number of edges. V + R There is always an even number of odd-degree = + 8 vertices. = ulerian paths and circuits path is a series of vertices connected by edges. circuit (or cycle) is a path which starts and finishes at the same vertex and no edge is traversed more than once. n ulerian path is a path which uses each edge in a graph only once, however a vertex may be repeated. n ulerian circuit is an ulerian path which starts and finishes at the same vertex. n ulerian path is possible if the number of odd vertices is 0 or. n ulerian circuit is possible if each of the vertices has even degrees. Hamiltonian paths and circuits Hamiltonian path passes through each vertex only once. It is not necessary to use all of the edges. Hamiltonian circuit is a Hamiltonian path that starts and finishes at the same vertex. To find the shortest path in a network, choose the edge of least distance and move along the network until the destination is reached. This usually requires a trial and error approach. Trees tree is a connected graph without any circuits, loops or multiple edges and contains only one region. spanning tree is a tree that includes all the vertices in the graph. minimal spanning tree is a spanning tree with the minimum length (or cost, or time). To find a minimal spanning tree: 1. select the edge with the minimum value. If there is more than one such edge, choose any one of them.. select the next smallest edge, provided it does not create a cycle.. repeat step until all the vertices have been included. The vertices and edges in a tree are related by the equation V = 1. Loop dge Vertex Not a planar graph

33 hapter 1 Undirected graphs and networks 18 HPTR review Multiple choice 1 Which of these graphs is not a connected graph? 1 Questions and refer to the diagram at right. The vertices with degree are: and only and all of them none of them The number of edges in total is: or the graph at right: V =, =, R = V =, =, R = V =, =, R = V =, =, R = V =, =, R = The number of regions that a connected planar network with 1 edges and 8 vertices has is: 7 The description that does not satisfy uler s formula for a planar network is: V =, = 7, R = V = 9, = 1, R = 8 V = 1, = 1, R = 1 V = 9, = 1, R = V =, = 11, R = 8 Questions 7 and 8 refer to the following graphs. i ii iii iv v G

34 18 General Mathematics The graph that has an ulerian path is: i only ii only iii only iv only v only 8 The graph that has an ulerian circuit is: i only ii only iii only iv only v only 9 If a graph has vertices of odd degree and all the other vertices of even degree, then: it is possible to draw an ulerian path and an ulerian circuit it is possible to draw an ulerian path but not an ulerian circuit it is possible to draw an ulerian circuit but not an ulerian path it is not possible to draw either an ulerian circuit or an ulerian path it is possible to draw an ulerian circuit but it depends on the graph whether an ulerian path can be drawn Hamiltonian circuit for the graph at right is: The length of the shortest Hamiltonian circuit is: ,, 1 distributor supplies four shops from a warehouse. The distances of the shops from the warehouse range from 10 km to 0 km. The shortest delivery route from the warehouse to all four shops and back to the warehouse could be found by using: an ulerian path an ulerian circuit a Hamiltonian path a Hamiltonian circuit a minimal spanning tree. 1 1 i ii iii iv The graphs that are trees are: all of them i only i and ii only i and iv only iii only 1 1 The minimal spanning tree of the graph at right has a length of:

35 hapter 1 Undirected graphs and networks When converted to a spanning tree, the network at right will resemble which of the following? 1 Short answer 1 or the graph at right: a write down the number of vertices b write down the number of edges c write down the degree of each vertex d establish if the graph is connected e if it is not connected, suggest how it may become connected. G H I 1 raw a connected graph with 10 vertices, 18 edges and loops. 1 Re-draw the graph at right to show clearly that it is a planar graph. 1 a Using uler s Law, determine whether the following would produce a connected planar graph. i V =, R =, = ii V = 7, R =, = 8 iii V = 10, R = 1, = 0 iv V =, R =, = 9 b Use uler s Law for each of the connected planar graphs below to determine the value of the unknown. i V = 10, R = 9, =? ii V = 8, R = 7, =? iii V =, R =?, = iv V =?, R =, = 1 or the graph at right: a how many vertices and edges are there in the graph? b write down the degree of each vertex c explain how your answer to b means that the graph does not have an ulerian circuit d write down an ulerian path for the graph. 1

36 188 General Mathematics Trish delivers the local paper after school each Tuesday afternoon. H On most days she collects the papers from the printers and M I L distributes them along the route illustrated at right. J a i If Trish is able to start from any point, is she able to deliver the papers going down each of the streets only once? K ii If she is, write down one possible route she may take. iii Is this the only possible route? If not, write down an alternative route. b i If the papers are dropped off at Trish s house, (vertex L), is she able to complete her round and return home going down each of the streets only once? ii If so, which type of circuit has been completed? iii Write down two possible circuits that Trish could have completed. 7 a Write down two Hamiltonian circuits starting at vertex 1 for the network at right. b alculate the least value of the Hamiltonian circuit for the graph. c Give one real-life situation for which the least value Hamiltonian circuit found above would be appropriate. d Is a Hamiltonian circuit possible for the graph in question? 8 The distance, in km, between five towns is shown in the table below Note: The shaded region indicates that there is no road connecting the two towns. a Transfer the information from the above table onto a copy of the graph at right. b tour is planned to start and finish at and visit all of the towns. ind the shortest distance for the tour. c If the tour were to start and finish at, would the shortest distance be the same? d new tour, starting and finishing at, is planned to visit all the towns and a further town,. The distances from to, to and to are 80, 9 and 9 km respectively. ind the shortest distance tour now possible. 9 a ind a spanning tree for the graph at right. b How many edges are there in a tree with: i vertices? ii 11 vertices? iii 1 vertices? c raw an example of each of the trees in part b. 10 ind the shortest distance from S to in the following network S G 1 1

37 hapter 1 Undirected graphs and networks 189 nalysis The map shows six towns, labelled to, on an island. The lines represent roads linking the towns, with distances given in kilometres. The map may be treated as a planar network with the towns as vertices and the roads as edges. 1 a What is a planar network? 1 b xplain what the statement the degree of = means. 0 c Write down the degree of each vertex. 19 d alculate S, the sum of the degrees of all the vertices. e How many edges are there? f Write down a formula linking S and for this network ( = number of edges). ll the roads on the island are to be re-surfaced, with the work starting at. g Give a route to be used by the workers so that each road is re-surfaced and no road is travelled twice. h Is the route above an example of: i an ulerian circuit? ii an ulerian path? iii a Hamiltonian circuit? iv a Hamiltonian path? i alculate the shortest distance from to. communication network is to be established on the island. j xplain why a minimal spanning tree for the network would be used in planning the communication network. k alculate the minimal spanning tree for the network. supermarket chain has stores at each of the towns. ach store is to be visited by the regional manager, starting and finishing at the main office in. l Outline the shortest route (that is, the least distance to travel). m Is the route above an example of: i an ulerian circuit? ii an ulerian path? iii a Hamiltonian circuit? iv a Hamiltonian path? test yourself HPTR 1