  Unit disk graphs are graphs that have a unit disk intersection representation
(UDR). In the recognition problem the objective is to decide whether a given
graph is a unit disk graph, which is known to be NP-hard, even for planar
graphs. In this work, we show that the recognition of unit disk graphs remains
NP-hard for outerplanar graphs and for embedded trees. We also show that given
a caterpillar graph, we can decide in linear time whether it is a unit disk
graph.
