Detection Of Planarity Of A Graph

Problem. Check the planarity of the following graph by the method of elementary deduction.

Solution. Step 1 : Does not apply, because the graph is connected.

Step 2 : Separating blocks of G

Step 3 : Removing self-loops and parallel edges

Step 4 : Merging the series edges.

The final graph contains three components. Largest component contains 7 vertices. Remaining two is triangle and an edge hence they are planar. The largest component contains no subgraph isomorphic to K5 or K_{3, 3} and hence it is planar.
Thus the given graph is planar.