A mobile agent, starting from a node s of a simple undirected graph \(G=(V,E)\) , has to explore all nodes and edges of G. To do so, the agent uses a deterministic algorithm that allows it to gain information on G as it traverses its edges. During its exploration, the agent must always respect the constraint of knowing a path of length at most D to go back to node s. The upper bound D is fixed as being equal to \((1+\alpha )r\) , where r is the eccentricity of node s and \(\alpha \) is any positive real constant. This task has been introduced by Duncan et al. (ACM Trans Algorithms 2(3):380–402, 2006) and is known as distance-constrained exploration. The penalty of an exploration algorithm running in G is the number of edge traversals made by the agent in excess of |E|. In Panaite and Pelc (J Algorithms 33(2):281–295, 1999), Panaite and Pelc gave an algorithm for solving exploration without any constraint on the moves that is guaranteed to work in every graph G with a (small) penalty in \(\mathcal {O}(|V|)\) . Can we obtain a distance-constrained exploration algorithm with the same guarantee? In this paper, we provide a negative answer to this question. We also observe that an algorithm working in every graph G with a linear penalty in |V| cannot be obtained for the task of fuel-constrained exploration, another variant studied in the literature. This solves an open problem posed by Duncan et al. (ACM Trans Algorithms 2(3):380–402, 2006) and shows a fundamental separation with the task of exploration without constraint on the moves.
