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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2026.137

Determining the Outerthickness of Graphs Is NP-Hard

Authors: Pin-Hsian Lee, Te-Cheng Liu, and Meng-Tsung Tsai

Published in: LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

Abstract

We give a short, self-contained, and easily verifiable proof that determining the outerthickness of a general graph is NP-hard. This resolves a long-standing open problem on the computational complexity of outerthickness. Moreover, our hardness result applies to a more general covering problem P_{ℱ, k}, defined as follows. Let ℱ be a proper graph class. Let k ≥ 1 be an integer parameter. Given an undirected simple graph G = (V, E), the task is to cover the edge set E(G) by at most k subsets E₁,…,E_k such that each subgraph (V(G),E_i) for i ∈ [k] belongs to ℱ. Note that if ℱ is monotone (in particular, when ℱ is the class of all outerplanar graphs), any such cover can be converted into an edge partition by deleting overlaps; hence, in this case, covering and partitioning are equivalent. Our result shows that for every proper graph class ℱ that satisfies all of the following conditions: (a) ℱ is closed under topological minors, (b) ℱ is closed under 1-sums, and (c) ℱ contains a cycle of length 3, the problem P_{ℱ, k} is NP-hard for every integer k ≥ 3. In particular: - For ℱ equal to the class of all outerplanar graphs, our result settles the long-standing open problem on the complexity of determining outerthickness. - For ℱ equal to the class of all planar graphs, our result complements Mansfield’s NP-hardness result (1983) for the thickness, which applies only to the case k = 2. It is also worth noting that each of the three conditions above is necessary. If ℱ is the class of all eulerian graphs, then condition (a) fails. If ℱ is the class of all pseudoforests, then condition (b) fails. If ℱ is the class of all forests, then condition (c) fails. For each of these three classes ℱ, the problem P_{ℱ, k} is solvable in polynomial time for every integer k ≥ 3, showing that none of the three conditions can be dropped unless P = NP.

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Pin-Hsian Lee, Te-Cheng Liu, and Meng-Tsung Tsai. Determining the Outerthickness of Graphs Is NP-Hard. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 137:1-137:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{lee_et_al:LIPIcs.ICALP.2026.137,
  author =	{Lee, Pin-Hsian and Liu, Te-Cheng and Tsai, Meng-Tsung},
  title =	{{Determining the Outerthickness of Graphs Is NP-Hard}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{137:1--137:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.137},
  URN =		{urn:nbn:de:0030-drops-265265},
  doi =		{10.4230/LIPIcs.ICALP.2026.137},
  annote =	{Keywords: outerthickness, outerplanar graphs, edge partition}
}

Document

DOI: 10.4230/LIPIcs.WADS.2025.43

On the Complexity of Finding 1-Center Spanning Trees

Authors: Pin-Hsian Lee, Meng-Tsung Tsai, and Hung-Lung Wang

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract

We consider the problem of finding a spanning tree T of a given undirected graph G such that any other spanning tree can be obtained from T by removing k edges and subsequently adding k edges, where k is minimized over all spanning trees of G. We refer to this minimum k as the treeradius of G. Treeradius is an interesting graph parameter with natural interpretations: (1) It is the smallest radius of a Hamming ball centered at an extreme point of the spanning tree polytope that covers the entire polytope. (2) Any graph with bounded treeradius also has bounded treewidth. Consequently, if a problem admits a fixed-parameter algorithm parameterized by treewidth, it also admits a fixed-parameter algorithm parameterized by treeradius. In this paper, we show that computing the exact treeradius for n-vertex graphs requires 2^Ω(n) time under the Exponential Time Hypothesis (ETH) and does not admit a PTAS, with an inapproximability bound of 1153/1152, unless P = NP. This hardness result is surprising, as treeradius has significantly higher ETH complexity compared to analogous problems on shortest path polytopes and star subgraph polytopes.

Cite as

Pin-Hsian Lee, Meng-Tsung Tsai, and Hung-Lung Wang. On the Complexity of Finding 1-Center Spanning Trees. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 43:1-43:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lee_et_al:LIPIcs.WADS.2025.43,
  author =	{Lee, Pin-Hsian and Tsai, Meng-Tsung and Wang, Hung-Lung},
  title =	{{On the Complexity of Finding 1-Center Spanning Trees}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{43:1--43:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.43},
  URN =		{urn:nbn:de:0030-drops-242743},
  doi =		{10.4230/LIPIcs.WADS.2025.43},
  annote =	{Keywords: Treeradius, Spanning tree polytope, Shortest s, t-path polytope}
}

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Documents authored by Lee, Pin-Hsian

Determining the Outerthickness of Graphs Is NP-Hard

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On the Complexity of Finding 1-Center Spanning Trees

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