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On the Parameterized Complexity of the Structure of Lineal Topologies (Depth-First Spanning Trees) of Finite Graphs: The Number of Leaves

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Algorithms and Complexity (CIAC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13898))

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Abstract

A lineal topology \(\mathcal {T} = (G, r, T)\) of a graph G is an r-rooted depth-first spanning (DFS) tree T of G. Equivalently, this is a spanning tree of G such that every edge uv of G is either an edge of T or is between a vertex u and an ancestor v on the unique path in T from u to r. We consider the parameterized complexity of finding a lineal topology that satisfies upper or lower bounds on the number of leaves of T, parameterized by the bound. This immediately yields four natural parameterized problems: (i) \(\le k\) leaves, (ii) \(\ge k\) leaves, (iii) \(\le n-k\) leaves, and (iv) \(\ge n-k\) leaves, where \(n=|G|\). We show that all four problems are NP-hard, considered classically. We show that (i) is para-NP-hard, (ii) is hard for W[1], (iii) is FPT, and (iv) is FPT. Our work is motivated by possible applications in graph drawing and visualization.

Supported by Research Council of Norway (NFR, no. 274526 and 314528).

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Acknowledgement

We acknowledge support from the Research Council of Norway (NFR, no. 274526 and 314528). We also thank Nello Blaser and Benjamin Bergougnoux for helping to review the initial versions of the paper.

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Authors and Affiliations

  1. Department of Informatics, University of Bergen, Bergen, Norway

    Emmanuel Sam, Michael Fellows, Frances Rosamond & Petr A. Golovach

  2. Western Sydney University, Locked Bag 1797, Penrith, NSW, 2751, Australia

    Michael Fellows & Frances Rosamond

Authors
  1. Emmanuel Sam
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  2. Michael Fellows
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  3. Frances Rosamond
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  4. Petr A. Golovach
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Correspondence to Emmanuel Sam .

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  1. University of Cyprus, Nicosia, Cyprus

    Marios Mavronicolas

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Sam, E., Fellows, M., Rosamond, F., Golovach, P.A. (2023). On the Parameterized Complexity of the Structure of Lineal Topologies (Depth-First Spanning Trees) of Finite Graphs: The Number of Leaves. In: Mavronicolas, M. (eds) Algorithms and Complexity. CIAC 2023. Lecture Notes in Computer Science, vol 13898. Springer, Cham. https://doi.org/10.1007/978-3-031-30448-4_25

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  • DOI: https://doi.org/10.1007/978-3-031-30448-4_25

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