Unary automatic graphs: An algorithmic perspective

This paper studies infinite graphs produced from a natural unfolding operation applied to finite graphs. Graphs produced via such operations are of finite degree and can be described by finite automata over the unary alphabet. We investigate algorithmic properties of such unfolded graphs given their finite presentations. In particular, we ask whether a given node belongs to an infinite component, whether two given nodes in the graph are reachable from one another, and whether the graph is connected. We give polynomial time algorithms for each of these questions. Hence, we improve on previous work, in which non-elementary or non-uniform algorithms were found.