Planar graphs and Euler's formula in WACE Mathematics Applications Unit 3

What this dot point is asking

You must recognise planar graphs, count vertices, edges and faces correctly, and apply Euler's formula.

Planar graphs

A graph is planar if it can be drawn in the plane so that no two edges cross. A graph drawn with crossings may still be planar if it can be redrawn without them.

When a planar graph is drawn without crossings, the edges divide the plane into regions called faces.

Counting faces

A face is a region bounded by edges. The crucial point is that the single unbounded region surrounding the whole graph counts as one face.

For example, a triangle with one vertex inside joined to all three corners has $3$ small inner faces plus the outer face, so $f = 4$.

Euler's formula

For any connected planar graph, the three counts are tied together.

Rearrange to find any one count: $f = 2 - v + e$, or $e = v + f - 2$, or $v = 2 - f + e$.

When Euler's formula applies

Euler's formula needs the graph to be connected and planar (drawn without crossings). For a graph that is not connected, the formula adjusts by the number of separate components, but in SCSA Mathematics Applications you apply it to connected planar graphs. If a face count does not satisfy $v - e + f = 2$, recheck whether you counted the outer face or whether the graph is actually connected.

Degree of a face and the edge-face relationship

Each face is bounded by a number of edges, called the degree of the face. Just as the handshake rule sums vertex degrees to twice the edges, the sum of all face degrees also equals twice the number of edges, because each edge borders exactly two faces. So $\sum (\text{face degrees}) = 2e$. This gives a second way to check a planar drawing: count the edges around every region (including the outer one) and confirm the total is $2e$.

For example, in a connected planar graph with $4$ faces totalling face degrees $3 + 3 + 4 + 6 = 16$, the number of edges is $\dfrac{16}{2} = 8$, which you can then confirm against $v - e + f = 2$.

Why planarity matters in applications

Planar graphs model situations that must be drawn or built without crossings: printed circuit boards where wires cannot overlap, road interchanges without overpasses, and map regions sharing borders. Euler's formula then links the counts of junctions, connections and enclosed regions, letting you deduce one count from the other two without redrawing. This is why SCSA pairs planarity with the formula: identify that a crossing-free drawing exists, then use $v - e + f = 2$ to reason about the structure.