Further Maths AS Discrete Year 12

11 Hamiltonian A Hamiltonian graph is a graph for which Graphs and Networks a Hamiltonian exists. Minimum Spanning Trees 12 Complete A is a simple graph in which every Make sure that you learn Kruskal’s 1 Vertex/ A vertex is a point on a graph that is connected Graph pair of vertices is connected by an edge. 1 properly You are expected to Node to other vertices by edges. A complete graph with n vertices is denoted as Kn. remember this algorithm and be fluent in 2 Arc/ Edge its use. If you do the suggested questions, An edge is a line connecting two vertices on a 13 Bipartite A bipartite graph is one in which the vertices fall this shouldn’t be a problem. graph. Graph into two sets, and each edge has a vertex from 3 Weight A numerical value (often a distance) associated one set at one end, and a vertex from the other set 2 Make sure that you learn Prim’s algorithm with an arc on a network. at the other end. properly Again, you are expected to A bipartite graph with r vertices in one set remember this algorithm and be fluent in 4 Trail A trail is a sequence of edges in which the end of and s vertices in the other, in which every vertex in its use. Also note that you need to be able one edge (except the last) is the beginning of the each set is connected to each vertex in the other (a to apply Prim’s algorithm both directly to a next, and no edge is repeated. complete bipartite graph) is denoted as Kr,s network, and to an incidence matrix 5 Cycle representing a network. A cycle is a closed path (i.e. the end of the 14 Adjacency An adjacency matrix (incidence matrix) is a way of last edge is the start of the first, and Matrix representing a graph on a matrix. Rows and Remember which algorithm is which! no vertices are repeated except that the columns correspond to vertices, and the entries in 3 Sometimes you may be told which final vertex is the same as the first). the matrix correspond to the number algorithm to use, in which case it is 6 Connected A graph is connected if a path exists between of edges from one vertex to another. essential that you use the right one! Graph Otherwise, you may be asked to use a every pair of vertices. 15 Simple A graph in which there are no loops and in which Graph suitable algorithm and state its name – if 7 there is no more than one edge connecting any The degree (or order, or valency) of a vertex is you carry out one of the pair of vertices. the number of ends of edges connecting into it. correctly but give the wrong name you will 8 Subgraph A subgraph of a graph is a subset of 16 Tree A tree is a graph in which all vertices are lose an easy mark. connected by edges, but there are no cycles (a the vertices together with a subset of the edges. Remember that more than one minimal cycle is a circuit of edges). 4 9 Subdivision A subdivision of a graph occurs when a spanning tree may exist for a network. E.g. new vertex is added on an edge so that the edge 17 Euler’s If v is the number of vertices in a this network has four possible MSTs, shown Formula becomes two edges. So if there is an edge AB, connected planar graph, e is the number of edges, below. and you add a vertex P so that there is no longer and f is the number of faces (i.e. distinct regions an edge AB, but instead there are two edges AP defined by edges, including the infinite outer one) then the following formula is satisfied:v−e+f=2. and BP, then you have subdivided the graph. This is Euler's formula for connected planar graphs. 10 Loop An edge whose beginning and end both link into 18 Connected A graph is planar if it can be drawn in two the same vertex. Planar dimensions without any of the edges crossing. Graphs Further Maths AS Discrete Year 12

Travelling Salesperson Problem The Route Inspection Problem Network Flows Remember that the nearest neighbour Make sure you pair up odd vertices Flow A flow along an arc in a network with 1 algorithm does not need to start from the point 1 correctly If there are more than four odd 1 a single source and a single sink is a where the route is to start You can start the vertices, writing out all the possible pairings non-negative number assigned to nearest neighbour algorithm from any point. This becomes very tedious. In such cases it is that arc. The flow must satisfy gives a tour, ending up at the starting point, usually possible to spot some obvious the feasibility condition and which can be rewritten so that any point can be pairings, and if there are four vertices left the conservation condition. the starting point of the route. over the possible pairings can be checked Capacity The capacity of an arc in a network is easily. Remember that a direct link is not always the 2 the upper limit to the flow along that 2 shortest route Sometimes going via a vertex Remember that the direct route is not arc. which is already in the tour produces a quicker 2 always the shortest When pairing up odd Make sure you understand what is meant by a cut route than a direct link. If the graph is not vertices, check that you have looked at the 3 and its capacity Remember in particular that a cut already complete, a matrix of shortest distances shortest route for each pair. This may not must divide a network into two parts, one of which between all pairs of vertices should be drawn up. always be the direct route! contains the source and the other the sink. A cut The nearest neighbour algorithm should be When looking for a suitable route, note may include arcs which are directed from the part performed on a complete graph or a matrix of 3 which edges need to be traversed twice containing the sink to the part containing the source shortest distances – this is the ‘classical problem’ There are usually many possible optimal – remember that such arcs must be included if you as opposed to the ‘practical problem’. routes for the route inspection problem. are listing the arcs in the cut, but that you do not If you are asked to give a route make sure that Make a list of the edges which need to be include the capacities of such arcs when finding the 3 you include all vertices visited, including ones traversed twice (if the vertices pairings capacity of the cut. already visited The nearest neighbour algorithm involve going via another vertex, write down Remember the maximum flow – minimum cut gives you the order of visiting the vertices. all edges involved). This should help you to 4 theorem If you have checked all possible cuts and However, in the practical problem (rather than find a suitable route. identified the minimum cut, this tells you the the classical problem), in some cases this involves maximum flow for the network. going via a vertex already visited. If you are asked for the route rather than just the order, Remember that forward arcs in the minimum cut make sure that you include all vertices visited. 5 must be saturated 4 Remember that the greatest lower bound and Make sure that you can deal with multiple sources the least upper bound are used. The greatest 6 and sinks Remember to add a supersource and (largest) lower bound and least (smallest) upper supersink in networks with multiple sources and bound are nearest to the optimal solution. sinks. Further Maths AS Discrete Year 12

Remember that the optimum solution to Critical Path Analysis Linear Programming 6 an IP problem is not always the integer Make sure that you consider all preceding Make sure that you are fluent with plotting solution closest to the LP solution You will 1 activities You must consider all of the 1 straight-line graphs This is a skill you will need to check the integer solutions close to preceding activities allocating the earliest need throughout your A level Maths and the LP solution to see which is best. At this time to an activity, and you must consider Further Maths work, so you need to make level of work this is the only method all of the following activities before sure you have really mastered it. available to you. Checking the two or three allocating the latest time for the activity. If points closest to the IP solution will be Define your variables carefully In some you follow the algorithm for calculating enough to find the optimum solution in the 2 questions the variables will be defined for early times and late times correctly, this problems that you will meet. you, but in others you need to define them should not be a problem. yourself. It is important to be precise – for 7 Make sure that the integer points you are Remember that there may be more than example “Let x be the energy drink” would considering are within the feasible region 2 one critical path Any activity for which the earn no marks. You must write “Let x be the It is not always possible to tell from a graph late time minus the early time is equal to number of litres of the energy drink which if an integer point is in the feasible region, the duration of the activity is critical. are to be produced” especially if large numbers are involved. Sometimes there may be branches in the Check that the point you are considering Use graph paper Always use graph paper to critical path, so that more than one chain of satisfies all the constraints. A good 3 plot your graph, otherwise you may not be critical activities can be traced from the approach is to take the integer values of x able to read of the vertices with sufficient start event to the finish event of the either side of the exact vertex, and accuracy. project. Note that a critical activity must be substitute each of these into each of the part of such a chain. If you think an activity Make sure that you understand the constraints, to find the largest (or in the is critical, but it is not part of a chain from 4 terminology Make sure that you are familiar case of a minimisation problem, smallest) the start event to the finish event of the with all the terms in the glossary. possible integer value of y which satisfies project, you have made a mistake. all the constraints. 5 Be careful to shade out the region on the correct side of a constraint line on the graph 8 Be careful to formulate minimisation of an LP problem Check a point to see problems correctly Read questions whether or not it satisfies the constraint. If it carefully. Minimisation problems often does, shade out the region on the other side relate to minimising costs. The constraints of the constraint line to the point. If it will involve mostly or inequalities, doesn’t satisfy the constraint, shade out the whereas maximisation problems involve region on the same side as the point. See the mostly or inequalities. Notes and Examples for an example. M r Further Maths AS Discrete Year 12

Draw graphs carefully and interpret them Game Theory 4 correctly When using a graphical method Binary Operations Make sure that you know how to find for a 2 × n game, you do need to be fairly Make sure you know the meanings of 1 play-safe strategies Remember that for the accurate with the graph, as otherwise it 1 associative and commutative first player, you need to find the row may not be clear which point you need to minima and take the maximum of these consider. Remember that your graph only For an associative operation acting on a set (the maximin strategy), whereas for the need cover values of p between 0 and 1. (a b) c = a (b c) for all a, b, c in the set second player, you need to find the column Work out the expected pay-offs for p = 0 maxima and take the minimum of these and p = 1, and use these to draw the lines. For a commutative operation acting on a (the minimax strategy). Make sure that you Remember that the shaded area is the area set a b = b a for all a, b in the set indicate all the row minima and column under ALL the lines, and that you need the Make sure that you prove properties for all maxima. highest point of the shaded area. 2 cases If you are asked to prove that a given Check for stable solutions Remember that 5 Be careful if you need to work with a operation on a given set is commutative or 2 you should check for stable solutions transposed pay-off matrix To solve n × 2 associative, or that the set is closed, you before starting to look for the optimal games, you need to transpose the matrix. must prove it for all possible cases. For a mixed strategy. In examination questions, Remember that you must change the sign small, finite set, a Cayley table will allow you you will often be asked to “show that the of all the outcomes, since you are now to check for some properties. game has a stable solution” or “show that considering the game from the point of the game does not have a stable solution”. view of the second player. Remember that To do this, you need to clearly indicate all the value of the original game is the the row minima and column maxima, and negative of the value of the transposed state explicitly that the maximum of the game. row minima either is, or is not, equal to the minimum of the column maxima. Remember to look for dominance If one or 3 more row and / or column can be ignored due to dominance, the problem is simplified. For example, a 3 × 3 game would normally require linear programming methods, but if dominance allows you to ignore a row or column, you will be able to use the graphical method. Remember to check both rows and columns