Showing that a graph is planar

A graph satisfying the given degree conditions may be planarly embedded as follows:

1. Unsubdivide to remove degree-$2$ vertices
2. Remove degree-$1$ vertices
3. Find a planar embedding of the remaining graph, which now contains only those vertices which originally had degrees $3,4,5,6,7$. This is always possible: because of the originally degree-$3$ vertex this "core" (minus loops and with multiple edges identified) will now always be a subgraph of $K_5-e$, which is planar
4. Reverse step 2 (add back degree-$1$ vertices; this preserves planarity)
5. Reverse step 1 (subdivide edges to restore degree-$2$ vertices; this preserves planarity)