Let $e$ be the number of edges and $v \geq 3$ the number of vertices in a graph $G$. We know that if $G$ is planar, then $e \leq 3v-6$. My question is the opposite. Is there some sort of inequality that guarantees planarity?
For example, I know all trees are planar $e = v-1$. So, we could say: If $G$ is a connected graph with $e \leq v-1$, then $G$ is planar.
But, are there better known bounds that guarantee planarity? Also, I'm curious about the same question for outerplanar graphs. Again, in that case, we have an inequality that can determine when a graph is not outerplanar, but I wonder if there is one that could guarantee that a graph is outerplanar.
Assume connectedness if necessary.