Kuratowski's theoremAQA A-Level Further Maths: Mind map

Exam questions on Kuratowski's theorem

1. The complete graph $K_5$ has five vertices, each joined to every other vertex.
   A new vertex $w$ is placed in the middle of one edge of $K_5$, replacing that edge by two edges, to give a graph $H$. State the number of edges of $H$ and explain whether $H$ is planar.2 marks
2. A graph $G$ has vertices $A,B,C,D,E,F$ and nine edges: $AB$, $BC$, $CD$, $DE$, $EF$, $FA$, $AD$, $BE$ and $CF$.
   Use Kuratowski's theorem to decide whether $G$ is planar.2 marks
3. A graph $H$ has vertices 1 to 7 and ten edges: 1–2, 1–6, 1–7, 4–7, 2–3, 3–4, 3–6, 2–5, 4–5 and 5–6.
   Show that $H$ is a subdivision of $K_{3,3}$.3 marks