4 research outputs found
Embedding Graphs into Embedded Graphs
A (possibly degenerate) drawing of a graph G in the plane is approximable by an embedding if it can be turned into an embedding by an arbitrarily small perturbation.
We show that testing, whether a drawing of a planar graph G in the plane is approximable by an embedding, can be carried out in polynomial time, if a desired embedding of G belongs to a fixed isotopy class, i.e., the rotation system (or equivalently the faces) of the embedding of G and the choice of outer face are fixed.
In other words, we show that c-planarity with embedded pipes is tractable for graphs with fixed embeddings.
To the best of our knowledge an analogous result was previously known essentially only when G is a cycle
Subexponential-Time and FPT Algorithms for Embedded Flat Clustered Planarity
The C-Planarity problem asks for a drawing of a ,
i.e., a graph whose vertices belong to properly nested clusters, in which each
cluster is represented by a simple closed region with no edge-edge crossings,
no region-region crossings, and no unnecessary edge-region crossings. We study
C-Planarity for , graphs with a fixed
combinatorial embedding whose clusters partition the vertex set. Our main
result is a subexponential-time algorithm to test C-Planarity for these graphs
when their face size is bounded. Furthermore, we consider a variation of the
notion of in which, for each face,
including the outer face, there is a bag that contains every vertex of the
face. We show that C-Planarity is fixed-parameter tractable with the
embedded-width of the underlying graph and the number of disconnected clusters
as parameters.Comment: 14 pages, 6 figure