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Graphs and networks: Edexcel A‑Level Further Maths questions

Every Edexcel A-Level Further Maths exam question on D.2 Graphs and networks, 2018–2025, with its mark scheme and examiner’s report.

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824 questions

  1. Q3

    June 2025 · Decision 1

    10 marksD.2 Graphs and networks
    The whole paper ↗
    June 2025 Decision 1, question 3: A G F E D C B 17 40 9 11 12 14 11 31 25 8 24 47 35 35 Figure 1 Direct roads between seven villages, A, B, C, D, E, F and G, are represented in Figure 1. The wei

  2. Q4

    June 2025 · Decision 1

    13 marksD.2 Graphs and networks
    The whole paper ↗
    June 2025 Decision 1, question 4: Figure 2 [The total weight of the network is 268] Figure 2 represents a network of corridors between 10 offices, A, B, C, D, E, F, G, H, J and K, in a building.

  3. Q1

    June 2025 · AS Decision 1

    7 marksD.1 Algorithms and sortingD.2 Graphs and networks
    The whole paper ↗
    June 2025 AS Decision 1, question 1: 1 16 10 25 30 13 12 28 22 23 20 The list of ten numbers above is to be sorted into ascending order. (a) Carry out a bubble sort, starting at the left-hand end o

  4. Q3

    June 2025 · AS Decision 1

    14 marksD.2 Graphs and networks
    The whole paper ↗
    June 2025 AS Decision 1, question 3: 3 10 3 8 12 6 9 8 4 6 9 2 4 10 15 14 30 3 E G H D B A C F J Figure 2 [The total weight of the network is 153] Figure 2 models a network of roads in a town, wher

  5. Q2

    June 2024 · Decision 1

    13 marksD.2 Graphs and networks
    The whole paper ↗
    June 2024 Decision 1, question 2: The table below represents a network of shortest distances, in miles, to travel between nine castles, A, B, C, D, E, F, G, H and J. A B C D E F G H J A – 50 59

  6. Q3

    June 2024 · Decision 1

    13 marksD.2 Graphs and networks
    The whole paper ↗
    June 2024 Decision 1, question 3: A B 25 47 24 5 17 18 20 34 26 4 12 37 41 25 27 15 21 9 6 E C D F G H J K Figure 1 [The total weight of the network is 413] Figure 1 represents a network of cycl

  7. Q4

    June 2024 · Decision 1

    8 marksD.2 Graphs and networks
    The whole paper ↗
    June 2024 Decision 1, question 4: (a) Explain why it is not possible to draw a graph with exactly six nodes with degrees 1, 2, 3, 4, 5 and 6 (1) A tree, T, has exactly six nodes. The degrees of

  8. Q3

    June 2024 · AS Decision 1

    11 marksD.2 Graphs and networks
    No published mark scheme
    The whole paper ↗
    June 2024 AS Decision 1, question 3: A B 2 9 14 7 6 1 15 23 6 5 3 10 13 2 6 12 5 x y D E J C F K M L H G Figure 3 [The total weight of the network is 139 + x + y] (a) Explain what is meant by the t

  9. Q1

    June 2023 · Decision 1

    7 marksD.2 Graphs and networks
    The whole paper ↗
    June 2023 Decision 1, question 1: A B D C E Figure 1 Figure 1 shows the graph G. (a) State whether G is Eulerian, semi-Eulerian, or neither, giving a reason for your answer. (1) (b) Write down a

  10. Q3

    June 2023 · Decision 1

    8 marksD.2 Graphs and networks
    The whole paper ↗
    June 2023 Decision 1, question 3: A D G B E H C F J 32 37 22 34 32 35 41 23 7 7 8 43 45 50 28 21 11 10 30 42 Figure 4 Figure 4 represents a network with nodes, A, B, C, D, E, F, G, H and J. The

  11. Q5

    June 2023 · Decision 1

    13 marksD.2 Graphs and networks
    The whole paper ↗
    June 2023 Decision 1, question 5: A D G B E 34 18 23 21 33 20 50 12 8 65 42 10 17 22 48 H C F J Figure 5 [The total weight of the network is 423] Direct roads between nine towns, A, B, C, D, E,

  12. Q3

    June 2023 · AS Decision 1

    11 marksD.2 Graphs and networks
    The whole paper ↗
    June 2023 AS Decision 1, question 3: A D G J B E H C F 50 61 23 34 41 24 49 15 43 11 32 45 28 40 58 6 6 30 Figure 2 Figure 2 represents a network of train tracks. The number on each edge represents

  13. Q5

    June 2023 · AS Decision 1

    8 marksD.2 Graphs and networks
    The whole paper ↗
    June 2023 AS Decision 1, question 5: A D B E C F 3x + 2 2x + 1 5x – 6 5x – 8 2x – 2 x + 4 x + 5 x + 7 Figure 4 [The weight of the network is 20x + 3] Figure 4 shows a graph G that contains 8 arcs a

  14. Q2

    June 2022 · Decision 1

    13 marksD.2 Graphs and networks
    The whole paper ↗
    June 2022 Decision 1, question 2: A B C D E F 10 10 10 31 17 6 G H J K 23 25 8 14 14 18 9 3 4 16 23 33 25 Figure 1 [The total weight of the network is 299] Figure 1 represents a network of cycle

  15. Q3

    June 2022 · Decision 1

    10 marksD.2 Graphs and networks
    The whole paper ↗
    June 2022 Decision 1, question 3: The initial distance matrix (Table 1) shows the lengths, in metres, of the corridors connecting six classrooms, A, B, C, D, E and F, in a school. For safety rea

  16. Q1

    June 2022 · AS Decision 1

    9 marksD.1 Algorithms and sortingD.2 Graphs and networks
    The whole paper ↗
    June 2022 AS Decision 1, question 1: 55 44 34 59 28 37 41 52 33 42 47 The list of eleven numbers shown above is to be sorted into ascending order. (a) Carry out a quick sort to produce the sorted l

  17. Q3

    June 2022 · AS Decision 1

    14 marksD.2 Graphs and networks
    The whole paper ↗
    June 2022 AS Decision 1, question 3: A C B D E F 4 G H 10 12 7 15 13 3 8 11 6 4 5 10 12 Figure 2 [The total weight of the network is 120] (a) Explain what is meant by the term “path”. (2) (b) State

  18. Q1

    November 2021 · Decision 1

    4 marksD.2 Graphs and networks
    The whole paper ↗
    November 2021 Decision 1, question 1: A B C D E U V W X Y Figure 1 A Hamiltonian cycle for the graph in Figure 1 begins C, V, E, X, A, W, …. (a) Complete the Hamiltonian cycle. (1) (b) Hence use the

  19. Q3

    November 2021 · Decision 1

    8 marksD.2 Graphs and networks
    The whole paper ↗
    November 2021 Decision 1, question 3: A B C D E F G H A – 24 42 48 34 37 32 22 B 24 – 40 35 30 41 39 44 C 42 40 – 21 26 45 38 36 D 48 35 21 – 32 37 29 27 E 34 30 26 32 – 34 40 28 F 37 41 45 37 34 –

  20. Q4

    November 2021 · Decision 1

    8 marksD.2 Graphs and networks
    The whole paper ↗
    November 2021 Decision 1, question 4: 150 191 230 63 57 95 72 289 84 D E A B C 160 F 125 132 Figure 3 [The total weight of the network is 1648] Direct roads between six cities, A, B, C, D, E and F,
All 29 questions on Graphs and networks, as a list
  1. June 2025, Decision 1, Q3 (10 marks): A G F E D C B 17 40 9 11 12 14 11 31 25 8 24 47 35 35 Figure 1 Direct roads between seven villages, A, B, C, D, E, F and G, are represented in Figure 1. The wei
  2. June 2025, Decision 1, Q4 (13 marks): Figure 2 [The total weight of the network is 268] Figure 2 represents a network of corridors between 10 offices, A, B, C, D, E, F, G, H, J and K, in a building.
  3. June 2025, AS Decision 1, Q1 (7 marks): 1 16 10 25 30 13 12 28 22 23 20 The list of ten numbers above is to be sorted into ascending order. (a) Carry out a bubble sort, starting at the left-hand end o
  4. June 2025, AS Decision 1, Q3 (14 marks): 3 10 3 8 12 6 9 8 4 6 9 2 4 10 15 14 30 3 E G H D B A C F J Figure 2 [The total weight of the network is 153] Figure 2 models a network of roads in a town, wher
  5. June 2024, Decision 1, Q2 (13 marks): The table below represents a network of shortest distances, in miles, to travel between nine castles, A, B, C, D, E, F, G, H and J. A B C D E F G H J A – 50 59
  6. June 2024, Decision 1, Q3 (13 marks): A B 25 47 24 5 17 18 20 34 26 4 12 37 41 25 27 15 21 9 6 E C D F G H J K Figure 1 [The total weight of the network is 413] Figure 1 represents a network of cycl
  7. June 2024, Decision 1, Q4 (8 marks): (a) Explain why it is not possible to draw a graph with exactly six nodes with degrees 1, 2, 3, 4, 5 and 6 (1) A tree, T, has exactly six nodes. The degrees of
  8. June 2024, AS Decision 1, Q3 (11 marks): A B 2 9 14 7 6 1 15 23 6 5 3 10 13 2 6 12 5 x y D E J C F K M L H G Figure 3 [The total weight of the network is 139 + x + y] (a) Explain what is meant by the t
  9. June 2023, Decision 1, Q1 (7 marks): A B D C E Figure 1 Figure 1 shows the graph G. (a) State whether G is Eulerian, semi-Eulerian, or neither, giving a reason for your answer. (1) (b) Write down a
  10. June 2023, Decision 1, Q3 (8 marks): A D G B E H C F J 32 37 22 34 32 35 41 23 7 7 8 43 45 50 28 21 11 10 30 42 Figure 4 Figure 4 represents a network with nodes, A, B, C, D, E, F, G, H and J. The
  11. June 2023, Decision 1, Q5 (13 marks): A D G B E 34 18 23 21 33 20 50 12 8 65 42 10 17 22 48 H C F J Figure 5 [The total weight of the network is 423] Direct roads between nine towns, A, B, C, D, E,
  12. June 2023, AS Decision 1, Q3 (11 marks): A D G J B E H C F 50 61 23 34 41 24 49 15 43 11 32 45 28 40 58 6 6 30 Figure 2 Figure 2 represents a network of train tracks. The number on each edge represents
  13. June 2023, AS Decision 1, Q5 (8 marks): A D B E C F 3x + 2 2x + 1 5x – 6 5x – 8 2x – 2 x + 4 x + 5 x + 7 Figure 4 [The weight of the network is 20x + 3] Figure 4 shows a graph G that contains 8 arcs a
  14. June 2022, Decision 1, Q2 (13 marks): A B C D E F 10 10 10 31 17 6 G H J K 23 25 8 14 14 18 9 3 4 16 23 33 25 Figure 1 [The total weight of the network is 299] Figure 1 represents a network of cycle
  15. June 2022, Decision 1, Q3 (10 marks): The initial distance matrix (Table 1) shows the lengths, in metres, of the corridors connecting six classrooms, A, B, C, D, E and F, in a school. For safety rea
  16. June 2022, AS Decision 1, Q1 (9 marks): 55 44 34 59 28 37 41 52 33 42 47 The list of eleven numbers shown above is to be sorted into ascending order. (a) Carry out a quick sort to produce the sorted l
  17. June 2022, AS Decision 1, Q3 (14 marks): A C B D E F 4 G H 10 12 7 15 13 3 8 11 6 4 5 10 12 Figure 2 [The total weight of the network is 120] (a) Explain what is meant by the term “path”. (2) (b) State
  18. November 2021, Decision 1, Q1 (4 marks): A B C D E U V W X Y Figure 1 A Hamiltonian cycle for the graph in Figure 1 begins C, V, E, X, A, W, …. (a) Complete the Hamiltonian cycle. (1) (b) Hence use the
  19. November 2021, Decision 1, Q3 (8 marks): A B C D E F G H A – 24 42 48 34 37 32 22 B 24 – 40 35 30 41 39 44 C 42 40 – 21 26 45 38 36 D 48 35 21 – 32 37 29 27 E 34 30 26 32 – 34 40 28 F 37 41 45 37 34 –
  20. November 2021, Decision 1, Q4 (8 marks): 150 191 230 63 57 95 72 289 84 D E A B C 160 F 125 132 Figure 3 [The total weight of the network is 1648] Direct roads between six cities, A, B, C, D, E and F,
  21. November 2021, Decision 1, Q6 (10 marks): A B C D E F G H 32 33 75 16 50 17 15 95 30 41 25 10 50 70 105 113 Figure 4 In Figure 4 the weights on the arcs represent distances. (a) (i) Use Dijkstra’s algor
  22. November 2020, Decision 1, Q1 (6 marks): The table below shows the lengths, in km, of the roads in a network connecting seven towns, A, B, C, D, E, F and G. A B C D E F G A – 24 – 22 35 – – B 24 – 25 2
  23. November 2020, Decision 1, Q3 (9 marks): A B C D E 1 7 8 3 6 4 10 1 Figure 2 Direct roads between five villages, A, B, C, D and E, are shown in Figure 2. The weight on each arc is the time, in minutes,
  24. November 2020, Decision 1, Q6 (11 marks): C G E D B A F 19 24 65 42 21 y 15 24 34 45 x 31 Figure 4 [The total weight of the network is 320 + x + y] (a) State, with justification, whether the graph in Fi
  25. November 2020, AS Decision 1, Q3 (11 marks): A B C D E F G H 30 25 8 42 15 23 10 12 2x + 10 3x – 2 17 29 20 7 Figure 2 [The weight of the network is 5x + 246] (a) Explain why it is not possible to draw a g
  26. June 2019, Decision 1, Q2 (14 marks): H C G E J D B A F 38 75 42 15 10 42 20 12 10 17 14 23 6 39 7 Figure 1 [The total weight of the network is 370] Figure 1 represents a network of corridors in a b
  27. June 2019, Decision 1, Q3 (14 marks): A B C D E 15 5 7 3 3 9 18 4 Figure 2 The network in Figure 2 shows the direct roads linking five villages, A, B, C, D and E. The number on each arc represents t
  28. June 2019, AS Decision 1, Q1 (6 marks): (a) Draw the graph K5 (1) (b) (i) In the context of graph theory explain what is meant by ‘semi-Eulerian’. (ii) Draw two semi-Eulerian subgraphs of K5, each hav
  29. June 2019, AS Decision 1, Q4 (10 marks): B A C D E F G H 10 17 12 16 13 25 3x + y x + y 9 5 9 7 7 3 2 Figure 1 [The total weight of the network is 135 + 4x + 2y] The weights on the arcs in Figure 1 rep

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