Intersection graphs of segments (the class SEG) and of other simple ge
ometric objects in the plane are considered. The results fall into two
main areas: how difficult is the membership problem for a given class
and how large are the pictures needed to draw the representations. Am
ong others, we prove that the recognition of SEG-graphs is of the same
complexity as the decision of solvability of a system of strict polyn
omial inequalities in the reals, i.e., as the decision of a special ex
istentially quantified sentence in the theory of real closed fields, a
nd thus it belongs to PSPACE. If the segments of the representation ar
e restricted to lie in a fixed number (k) of directions, we show that
the corresponding recognition problem is NP-complete for every k great
er than or equal to 2. As for the size of representations, we show tha
t the description of any segment representation, specifying the coordi
nates of the endpoints, may require exponential number of digits for c
ertain n-vertex graphs. One of our main tools is a lemma, saying that
given a representation R of a graph by segments, one may construct a l
arger graph whose each segment representation contains a homeomorphic
copy of R. (C) 1994 Academic Press, Inc.