We address the problem of constrained exploration of an unknown graph G = (V, E) from a given start node s with either a tethered robot or a robot with a fuel tank of limited capacity, the former being a tighter constraint. In both variations of the problem, the robot can only move along the edges of the graph, i.e, it cannot jump between non-adjacent vertices. In the tethered robot case, if the tether (rope) has length l, then the robot must remain within distance l from the start node s. In

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... e second variation, a fuel tank of limited capacity forces the robot to return to s after traversing C edges. The efficiency of algorithms for both variations of the problem is measured by the number of edges traversed during the exploration. We present an algorithm for a tethered robot which explores the graph in Θ(|E|) edge traversals. The problem of exploration using a robot with a limited fuel tank capacity can be solved with a simple reduction from the tethered robot case and also yields a Θ(|E|) algorithm. This improves on the previous best known bound of O(|E| + |V | log 2 |V |) in [4] . Since the lower bound for the graph exploration problems is |E|, our algorithm is optimal within a constant factor, thus answering the open problem of Awerbuch, Betke, Rivest, and Singh [3] .