  A natural generalization of the recognition problem for a geometric graph
class is the problem of extending a representation of a subgraph to a
representation of the whole graph. A related problem is to find representations
for multiple input graphs that coincide on subgraphs shared by the input
graphs. A common restriction is the sunflower case where the shared graph is
the same for each pair of input graphs. These problems translate to the setting
of comparability graphs where the representations correspond to transitive
orientations of their edges. We use modular decompositions to improve the
runtime for the orientation extension problem and the sunflower orientation
problem to linear time. We apply these results to improve the runtime for the
partial representation problem and the sunflower case of the simultaneous
representation problem for permutation graphs to linear time. We also give the
first efficient algorithms for these problems on circular permutation graphs.
