Characteristics of Traversable Graph

Theorem
A finite graph is traversable iff it is connected and no more than two vertices are odd.

Proof
Let $$G$$ be a graph.

Suppose all the vertices are even, that is, there are no odd vertices.

Then $G$ is Eulerian, and the result holds.

Similarly, by the same result, if $$G$$ is Eulerian, it is by definition traversable.

So the question of graphs all of whose vertices are even is settled.

Necessary Condition
Suppose a graph $$G$$ is semi-Eulerian.