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Drawn Tree Decomposition: New Approach for Graph Drawing Problems

Authors Siddharth Gupta , Guy Sa'ar, Meirav Zehavi

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Author Details

Siddharth Gupta
  • BITS Pilani, Goa Campus, India
Guy Sa'ar
  • Ben Gurion University of the Negev, Beersheba, Israel
Meirav Zehavi
  • Ben Gurion University of the Negev, Beersheba, Israel

Funding

  • Gupta, Siddharth: Supported by Engineering and Physical Sciences Research Council (EPSRC) grant EP/V007793/1.
  • Sa'ar, Guy: Supported in part by the Israeli Smart Transportation Research Center and by the Lynne and William Frankel Center for Computing Science at Ben-Gurion University.
  • Zehavi, Meirav: Supported by the European Research Council (ERC) grant titled PARAPATH.

Cite AsGet BibTex

Siddharth Gupta, Guy Sa'ar, and Meirav Zehavi. Drawn Tree Decomposition: New Approach for Graph Drawing Problems. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 23:1-23:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.IPEC.2023.23

Abstract

Over the past decade, we witness an increasing amount of interest in the design of exact exponential-time and parameterized algorithms for problems in Graph Drawing. Unfortunately, we still lack knowledge of general methods to develop such algorithms. An even more serious issue is that, here, "standard" parameters very often yield intractability. In particular, for the most common structural parameter, namely, treewidth, we frequently observe NP-hardness already when the input graphs are restricted to have constant (often, being just 1 or 2) treewidth. Our work deals with both drawbacks simultaneously. We introduce a novel form of tree decomposition that, roughly speaking, does not decompose (only) a graph, but an entire drawing. As such, its bags and separators are of geometric (rather than only combinatorial) nature. While the corresponding parameter - like treewidth - can be arbitrarily smaller than the height (and width) of the drawing, we show that - unlike treewidth - it gives rise to efficient algorithms. Specifically, we get slice-wise polynomial (XP) time algorithms parameterized by our parameter. We present a general scheme for the design of such algorithms, and apply it to several central problems in Graph Drawing, including the recognition of grid graphs, minimization of crossings and bends, and compaction. Other than for the class of problems we discussed in the paper, we believe that our decomposition and scheme are of independent interest and can be further extended or generalized to suit even a wider class of problems. Additionally, we discuss classes of drawings where our parameter is bounded by 𝒪(√n) (where n is the number of vertices of the graph), yielding subexponential-time algorithms. Lastly, we prove which relations exist between drawn treewidth and other width measures, including treewidth, pathwidth, (dual) carving-width and embedded-width.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Human-centered computing → Graph drawings
  • Theory of computation → Computational geometry
Keywords
  • Graph Drawing
  • Parameterized Complexity
  • Tree decomposition

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