Lesson DC1-DC3

DC1-DC3 Flows, cuts and the maximum flow-minimum cut theorem Quiz: AQA Further Maths, Unit 5

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson DC1-DC3, Flows, cuts and the maximum flow-minimum cut theorem: 20 multiple choice questions for the AQA Further Maths (7367), Unit 5: Optional application 3: discrete mathematics, written with Revision Ninja.

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The 20 questions

  1. In a flow network, what does conservation of flow require at every node other than the source and the sink?

    • The flow on every arc entering the node equals its capacity
    • The flow into the node equals the flow out of the node
    • The flow into the node equals the total capacity of its outgoing arcs
    • The flow out of the node is always twice the flow into the node

  2. What is the capacity of a directed arc in a flow network?

    • The maximum flow that the arc can carry in its stated direction
    • The weight of the arc used in a shortest-path calculation
    • The total flow entering the network from the source
    • The minimum flow that must be sent along the arc

  3. What is a cut in a directed network from source S to sink T?

    • A partition whose capacity is the sum of all arcs in both sets
    • A split of the nodes into a set with S and a set with T; its capacity is the total capacity of arcs running from the S set to the T set
    • A set of nodes that contains every arc of the network
    • A path from S to T whose arcs have the greatest total capacity

  4. What does the maximum flow-minimum cut theorem state?

    • The minimum flow equals the minimum capacity of any cut separating source and sink
    • The maximum flow from source to sink equals the minimum capacity of any cut separating them
    • The maximum flow equals the maximum capacity of any cut separating source and sink
    • The maximum flow equals the sum of all arc capacities in the network

  5. A directed network has source S and sink T, with arcs SA (capacity 4), SB (capacity 3), AB (capacity 2), AT (capacity 3) and BT (capacity 4). What is the maximum flow from S to T?

    • 5
    • 8
    • 6
    • 7

  6. A directed network has source S and sink T, with arcs SA (capacity 4), SB (capacity 3), AB (capacity 2), AT (capacity 3) and BT (capacity 4). What is the capacity of the cut separating {S, A, B} from {T}?

    • 7
    • 8
    • 10
    • 5

  7. A directed network has source S and sink T, with arcs SA (capacity 4), SB (capacity 3), AB (capacity 2), AT (capacity 3) and BT (capacity 4). What is the capacity of the cut separating {S, A} from {B, T}?

    • 6
    • 8
    • 11
    • 7

  8. A directed network has source S and sink T, with arcs SA (capacity 4), SB (capacity 3), AB (capacity 2), AT (capacity 3) and BT (capacity 4). Which statement about the cuts of this network is correct?

    • The cut with the largest capacity gives the maximum flow of 8
    • Some cut has capacity 5, so the maximum flow is at most 5
    • Every cut has capacity exactly 7, so the maximum flow is unique
    • Every cut has capacity at least 7, and a cut of capacity 7 exists, so the maximum flow is 7

  9. A directed flow sends 3 along SA and 1 along SB. Node A receives 3 and sends 2 along AB and 1 along AT, with the arc capacities of SA = 4, AB = 2 and AT = 3. Which statement is correct?

    • The flow is invalid, since the flow out of S must equal the capacity of SA
    • The flow is valid only if the flow into T is also exactly 4
    • The flow is invalid, since A sends less than it receives
    • The flow is valid, since the flow into A equals the flow out of A and every arc is within its capacity

  10. What is the value of a flow in a network from source S to sink T?

    • The flow carried by the single largest arc in the network
    • The total capacity of all arcs leaving the sink
    • The net flow out of the source, which equals the net flow into the sink
    • The sum of the flows on all arcs in the network

  11. A flow in a network is maximal. Which statement must be true for a minimum cut?

    • Every arc in the network carries flow equal to its capacity
    • Every arc in the network carries zero flow
    • The minimum cut contains only backward arcs with positive flow
    • Every forward arc crossing the minimum cut from the source side to the sink side is saturated, carrying flow equal to its capacity

  12. What is a backward arc of a cut?

    • An arc going from the source side of the cut to the sink side
    • An arc whose capacity is zero
    • An arc going from the sink side of the cut back to the source side, which does not add to the cut capacity
    • An arc that starts and ends at the same node

  13. A network has a cut of capacity 10 and a feasible flow of value 9. What can be concluded?

    • The maximum flow lies between 9 and 10 inclusive
    • The maximum flow is at least 11
    • The maximum flow must be 9, since it equals the flow already found
    • The maximum flow is exactly 10

  14. A cut in a network has capacity 12. Which statement is correct?

    • It guarantees that the flow value is exactly 12
    • It limits the flow that can pass from the source side to the sink side, so the flow value cannot exceed 12
    • It is a lower bound on the maximum flow in the network
    • It is the total flow leaving the sink

  15. A directed network has source S and sink T, with arcs SA (capacity 4), SB (capacity 3), AB (capacity 2), AT (capacity 3) and BT (capacity 4). If the capacity of arc BT is reduced from 4 to 2, what is the new maximum flow from S to T?

    • 5
    • 3
    • 7
    • 6

  16. A directed network has source S and sink T, with arcs SA (capacity 4), SB (capacity 3), AB (capacity 2), AT (capacity 3) and BT (capacity 4). After BT is reduced to capacity 2, which set of nodes gives a minimum cut, and what is its capacity?

    • {S, B} with capacity 6
    • {S, A, B} with capacity 5
    • {S} with capacity 7
    • {S, A} with capacity 8

  17. A network with source S and sink T has maximum flow 12. Which statement is correct?

    • Some cut separating S from T has capacity less than 12
    • Every cut separating S from T has capacity exactly 12
    • Every cut separating S from T has capacity at least 12, and some cut has capacity exactly 12
    • Every arc carries a flow equal to 12 divided by the number of arcs

  18. A directed network has source S and sink T. The arcs are SA (capacity 10), SB (capacity 5), AB (capacity 15), AT (capacity 5) and BT (capacity 10). What is the maximum flow from S to T?

    • 10
    • 20
    • 15
    • 25

  19. Why does an augmenting-flow method stop when no augmenting path from source to sink remains?

    • Because the source has no unused arcs left, so no flow can leave it
    • Because the flow value has reached the sum of all arc capacities in the network
    • Because every node then has equal flow in and out, which means the network is balanced
    • Because the saturated forward arcs of a cut form a barrier that no further flow can cross from source to sink

  20. What term describes a forward arc whose flow equals its capacity?

    • Reversed
    • Balanced
    • Saturated
    • Empty

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