When can a network be drawn with no crossings, and how do its faces, edges and vertices relate?
Recognise planar graphs and apply Euler's formula relating vertices, edges and faces.
Planar graphs, faces, and Euler's formula v - e + f = 2, with worked checks and applications in TCE Mathematics Applications.
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What this dot point is asking
A graph drawing often has edges that cross, but the crossings may be an accident of how it was drawn. The real question is whether the same connections can be redrawn flat with no crossings at all.
Counting faces
Faces are the enclosed regions of a planar drawing, plus the single infinite region outside the graph. Beginners forget the outer face, so always add one for the outside when you count.
Why planarity matters
Planar graphs model situations where crossings are physically awkward or forbidden, such as circuit boards where wires must not touch, road networks without overpasses, or seating plans. Knowing a graph is planar tells you a clean, crossing-free layout exists.
Connected requirement
Euler's formula in the form applies to connected planar graphs. If the graph splits into separate pieces, the formula must be adjusted, so first confirm the graph is in one connected piece before applying it.
A complete answer states whether the graph is planar (and redraws it without crossings if needed), counts vertices and edges, includes the outer face when counting faces, and uses to confirm or to find a missing count.
Exam-style practice questions
Practice questions written in the style of TASC exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
2018 TASC General Mathematics5 marksA planar connected network (graph) has 5 faces and 8 edges. a) Use Euler's formula to determine the number of vertices in this network. b) Draw the planar connected network in the space below and label the vertices. c) Describe a Hamiltonian circuit that can be travelled on the network in part b).Show worked answer →
a) (2 marks) Euler's formula for a connected planar graph is v - e + f = 2. Substitute e = 8 and f = 5: v - 8 + 5 = 2, so v - 3 = 2, giving v = 5 vertices.
b) (2 marks) Draw any connected planar graph with 5 vertices and 8 edges so that no edges cross. Counting the regions it splits the plane into (including the outer region) must give 5 faces. Label the 5 vertices.
c) (1 mark) A Hamiltonian circuit visits every vertex exactly once and returns to the start. Trace such a closed route through all 5 vertices on your graph and write it as a sequence of vertex labels.
2017 TASC General Mathematics4 marksBelow is a network with 12 vertices and 20 edges. a) Use Euler's formula to calculate the number of faces of the network. c) Draw an example of a network, with 5 vertices and 7 edges, for which Euler's formula does not apply. State why Euler's formula does not apply in this case.Show worked answer →
a) (1 mark) Euler's formula v - e + f = 2 rearranges to f = 2 - v + e. With v = 12 and e = 20: f = 2 - 12 + 20 = 10 faces.
c) (3 marks) Draw a network with 5 vertices and 7 edges that is either not connected or not planar (cannot be drawn without edges crossing). Euler's formula v - e + f = 2 only holds for connected planar graphs, so if the graph is disconnected (or non-planar) the relationship breaks down. State which condition your example violates.