Finding time complexity of a depth first traversal algorithm with a depth limit to get all node traversals starting from the root node

What would be the time complexity of a depth-first traversal algorithm on a graph, that is simply trying to retrieve all nodes being visited from starting node up until a depth limit is reached (i.e. same node can be visited multiple times if cycles are present). The idea here is to retrieve an ordered list of nodes that are reached (regardless of the fact that a given node was visited previously). How would I approach this problem? Any helpful input or a link to a resource that I could refer to would be appreciated!