Fig. 1

Node-safe walks. In a, the walk (a, b, c, d) is node-safe, because every circular walk covering node e contains (a, b, c, d) as sub-walk (we draw one such circular walk in orange). In b, the walk (a, b, c, d) is not node-safe, because the graph admits two circular walks covering all nodes (in blue and red) that do not contain it as sub-walk; it does not satisfy condition (b) of Theorem 2. In c the walk (a, b, c, d) is not safe because there is a node-covering circular walk not containing it as sub-walk (in green); it does not satisfy condition (a) of Theorem 2