A pseudo-h-hamiltonian cycle in a graph is a closed walk that visits every vertex exactly h times. We present a variety of combinatorial and algorithmic results on pseudo- h-hamiltonian cycles: First, we show that deciding whether a graph is pseudo-h-hamiltonian is NP-complete for any given h ≥ 1. Surprisingly, deciding whether there exists an h ≥ 1 such that the graph is pseudo-h-hamiltonian, can be done in polynomial time. We also present sufficient conditions for pseudo- h-hamiltonicity that are based on stable sets and on toughness. Moreover, we investigate the computational complexity of finding pseudo-h-hamiltonian cycles on special graph classes like bipartite graphs, split graphs, planar graphs, cocomparability graphs; in doing this, we establish a precise separating line between easy and difficult cases of this problem.
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