Graph theory: shorter path with a starting vertex that touches all nodes at least once with allowed repetitions

I tried to look for this problem a bit now, but it seems that I can not find much discussion about this. At first it sounded like TSP, but I do not think so (I think it's much harder). Maybe I'm thinking too much about this though? I'm not looking for the fastest solution, but I'm afraid that a brute-force algorithm can take a long time to do this on an unmanaged graph with, say, 30 nodes. Personally, I'm trying to discover in a videogame the fastest way to traverse each generation point of a monster in a certain area (with a starting point, of course) to be able to find it faster. I already constructed a graph and calculated the distances assuming a complete graph. I suppose that, for my part, one way to reduce the execution time of such an algorithm is to realistically eliminate what the edges would not be included and probably would not be included in an optimal route. I guess adding many restrictions could help too. Could anyone know if there is a specific name for this problem, or if there is a simpler approach apart from the standard brute force? Thank you!