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arXiv · 1512.04839

On the Planar Split Thickness of Graphs

David Eppstein·Philipp Kindermann·Stephen Kobourov·Giuseppe Liotta·Anna Lubiw·Aude Maignan·Debajyoti Mondal·Hamideh Vosoughpour·Sue Whitesides·Stephen Wismath
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Abstract

Motivated by applications in graph drawing and information visualization, we examine the planar split thickness of a graph, that is, the smallest $k$ such that the graph is $k$-splittable into a planar graph. A $k$-split operation substitutes a vertex $v$ by at most $k$ new vertices such that each neighbor of $v$ is connected to at least one of the new vertices. We first examine the planar split thickness of complete graphs, complete bipartite graphs, multipartite graphs, bounded degree graphs, and genus-1 graphs. We then prove that it is NP-hard to recognize graphs that are $2$-splittable into a planar graph, and show that one can approximate the planar split thickness of a graph within a constant factor. If the treewidth is bounded, then we can even verify $k$-splittability in linear time, for a constant $k$.

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BibTeXRIS

David Eppstein, Philipp Kindermann, Stephen Kobourov, Giuseppe Liotta, Anna Lubiw, Aude Maignan, Debajyoti Mondal, Hamideh Vosoughpour, Sue Whitesides, Stephen Wismath. 2017-06-06. On the Planar Split Thickness of Graphs. https://doi.org/10.1007/s00453-017-0328-y

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