  A unit disk graph is the intersection graph of disks of equal radii in the
plane. The class of unit disk graphs is hereditary, and therefore admits a
characterization in terms of minimal forbidden induced subgraphs. In spite of
quite active study of unit disk graphs very little is known about minimal
forbidden induced subgraphs for this class. We found only finitely many minimal
non unit disk graphs in the literature. In this paper we study in a systematic
way forbidden induced subgraphs for the class of unit disk graphs. We develop
several structural and geometrical tools, and use them to reveal infinitely
many new minimal non unit disk graphs. Further we use these results to
investigate structure of co-bipartite unit disk graphs. In particular, we give
structural characterization of those co-bipartite unit disk graphs whose edges
between parts form a C4-free bipartite graph, and show that bipartite
complements of these graphs are also unit disk graphs. Our results lead us to
propose a conjecture that the class of co-bipartite unit disk graphs is closed
under bipartite complementation.
