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Selected Papers from the 1999 Symposium on Graph Drawing
DOI: 10.7155/jgaa.00044
Planar Graphs with Topological Constraints
Vol. 6, no. 1, pp. 2766, 2002. Regular paper.
Abstract We address in this paper the problem of constructing embeddings of
planar graphs satisfying declarative, userdefined topological
constraints. The constraints consist each of a cycle of
the given graph and a set of its edges to be embedded inside this cycle
and a set of its edges to be embedded outside this cycle.
Their practical importance in graph visualization applications is due
to the capability of supporting the semantics of graphs.
Additionally, embedding algorithms for planar graphs with topological
constraints can be combined with planar graph drawing algorithms that
transform a given embedding into a topology preserving drawing
according to particular drawing conventions and aesthetic
criteria.
We obtain the following main results on the planarity problem with
topological constraints. Since it turns out to be NPcomplete, we
develop a polynomial time algorithm for reducing the problem for arbitrary
planar graphs to a planarity problem with constraints for biconnected
graphs. This allows focussing on biconnected graphs when searching for
heuristics or polynomial time subproblems.
We then define a particular subproblem by restricting the maximum
number of vertices that any two distinct cycles
involved in the constraints can have in common.
Whereas the problem remains NPcomplete if this number exceeds 1, it
can otherwise be solved in polynomial time.
The embedding algorithm we develop for this purpose is based on the
reduction method.
