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DOI: 10.4230/LIPIcs.MFCS.2025.16

Isometric-Universal Graphs for Trees

Authors: Edgar Baucher, François Dross, and Cyril Gavoille

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract

We consider the problem of finding the smallest graph that contains two input trees each with at most n vertices preserving their distances. In other words, we look for an isometric-universal graph with the minimum number of vertices for two given trees. We prove that this problem can be solved in time O(n^{5/2}log{n}). We extend this result to forests instead of trees, and propose an algorithm with running time O(n^{7/2}log{n}). As a key ingredient, we show that a smallest isometric-universal graph of two trees essentially is a tree. Furthermore, we prove that these results cannot be extended. Firstly, we show that deciding whether there exists an isometric-universal graph with t vertices for three forests is NP-complete. Secondly, we show that any smallest isometric-universal graph cannot be a tree for some families of three trees. This latter result has implications for greedy strategies solving the smallest isometric-universal graph problem.

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Edgar Baucher, François Dross, and Cyril Gavoille. Isometric-Universal Graphs for Trees. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{baucher_et_al:LIPIcs.MFCS.2025.16,
  author =	{Baucher, Edgar and Dross, Fran\c{c}ois and Gavoille, Cyril},
  title =	{{Isometric-Universal Graphs for Trees}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{16:1--16:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.16},
  URN =		{urn:nbn:de:0030-drops-241237},
  doi =		{10.4230/LIPIcs.MFCS.2025.16},
  annote =	{Keywords: tree, forest, isometric subgraph, universal graph, distance-preserving}
}

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DOI: 10.4230/LIPIcs.SWAT.2016.16

Colouring Diamond-free Graphs

Authors: Konrad K. Dabrowski, François Dross, and Daniël Paulusma

Published in: LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

Abstract

The Colouring problem is that of deciding, given a graph G and an integer k, whether G admits a (proper) k-colouring. For all graphs H up to five vertices, we classify the computational complexity of Colouring for (diamond,H)-free graphs. Our proof is based on combining known results together with proving that the clique-width is bounded for (diamond,P_1+2P_2)-free graphs. Our technique for handling this case is to reduce the graph under consideration to a k-partite graph that has a very specific decomposition. As a by-product of this general technique we are also able to prove boundedness of clique-width for four other new classes of (H_1,H_2)-free graphs. As such, our work also continues a recent systematic study into the (un)boundedness of clique-width of (H_1,H_2)-free graphs, and our five new classes of bounded clique-width reduce the number of open cases from 13 to 8.

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Konrad K. Dabrowski, François Dross, and Daniël Paulusma. Colouring Diamond-free Graphs. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{dabrowski_et_al:LIPIcs.SWAT.2016.16,
  author =	{Dabrowski, Konrad K. and Dross, Fran\c{c}ois and Paulusma, Dani\"{e}l},
  title =	{{Colouring Diamond-free Graphs}},
  booktitle =	{15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)},
  pages =	{16:1--16:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-011-8},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{53},
  editor =	{Pagh, Rasmus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.16},
  URN =		{urn:nbn:de:0030-drops-60380},
  doi =		{10.4230/LIPIcs.SWAT.2016.16},
  annote =	{Keywords: colouring, clique-width, diamond-free, graph class, hereditary graph class}
}

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Isometric-Universal Graphs for Trees

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Colouring Diamond-free Graphs

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