  Function graphs are graphs representable by intersections of continuous
real-valued functions on the interval [0,1] and are known to be exactly the
complements of comparability graphs. As such they are recognizable in
polynomial time. Function graphs generalize permutation graphs, which arise
when all functions considered are linear.
  We focus on the problem of extending partial representations, which
generalizes the recognition problem. We observe that for permutation graphs an
easy extension of Golumbic's comparability graph recognition algorithm can be
exploited. This approach fails for function graphs. Nevertheless, we present a
polynomial-time algorithm for extending a partial representation of a graph by
functions defined on the entire interval [0,1] provided for some of the
vertices. On the other hand, we show that if a partial representation consists
of functions defined on subintervals of [0,1], then the problem of extending
this representation to functions on the entire interval [0,1] becomes
NP-complete.
