find a path to visit every node in graph not necessarily once

I meet a problem but when I google, there are all Hamiltonian Path Problem: How to find a path to visit every node in directed graph(not necessarily once)?

This problem is different from Hamiltonian Path.

For example:

1 --> 2 -->  3 ↔ 4 
             ↕ 
             5

I have a path $1\rightarrow 2\rightarrow3\rightarrow4\rightarrow3\rightarrow5$ to visit all nodes in this graph, however this graph has no hamiltonian path.

My questions:

-
What's the terminology of this problem?

-
Is there algorithm or theorem to determine whether a general graph has such path? If a graph has such graph, how to find the detailed
path?

Thanks

You probably haven't found anything in your searches because this problem is fairly easy to solve, without needing any fancy methods.

If the graph is strongly connected, then it is easy. Let $s$ be the starting node. Pick any vertex $v$ you haven't visited yet, and append a path $s \leadsto v$ followed by some path $v \leadsto s$ (both must exist, since the graph is strongly connected). Repeat, appending to the end of the path, until every vertex has been visited.

If the graph isn't strongly connected, decompose it into a dag of scc's. Then you can reduce the problem to finding a path through this dag (within a scc, use the algorithm of the first paragraph). You can find a path to visit every node in a dag iff the dag is itself a path, i.e., it has the structure $v_1 \to v_2 \to \dots \to v_k$, and then it is trivial to write down this path.

This is not a Hamiltonian path, since it may visit the same vertex multiple times.

StackExchange Computer Science Q#139327, answer score: 3