1 research outputs found
Bounded, minimal, and short representations of unit interval and unit circular-arc graphs
We consider the unrestricted, minimal, and bounded representation problems
for unit interval (UIG) and unit circular-arc (UCA) graphs. In the unrestricted
version, a proper circular-arc (PCA) model is given and the goal is to
obtain an equivalent UCA model . We show a linear time algorithm with
negative certification that can also be implemented to run in logspace. In the
bounded version, is given together with some lower and upper bounds
that the beginning points of must satisfy. We develop a linear space
time algorithm for this problem. Finally, in the minimal version, the
circumference of the circle and the length of the arcs in must be
simultaneously as minimum as possible. We prove that every UCA graph admits
such a minimal model, and give a polynomial time algorithm to find it. We also
consider the minimal representation problem for UIG graphs. As a bad result, we
show that the previous linear time algorithm fails to provide a minimal model
for some input graphs. We fix this algorithm but, unfortunately, it runs in
linear space time. Finally, we apply the minimal representation
algorithms so as to find the minimum powers of paths and cycles that contain a
given UIG and UCA models, respectively.Comment: 33 pages, 3 figure