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Documents authored by Muršič, Peter

Document
DOI: 10.4230/LIPIcs.SWAT.2024.7
The Simultaneous Interval Number: A New Width Parameter that Measures the Similarity to Interval Graphs

Authors: Jesse Beisegel, Nina Chiarelli, Ekkehard Köhler, Martin Milanič, Peter Muršič, and Robert Scheffler

Published in: LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

Abstract
We propose a novel way of generalizing the class of interval graphs, via a graph width parameter called simultaneous interval number. This parameter is related to the simultaneous representation problem for interval graphs and defined as the smallest number d of labels such that the graph admits a d-simultaneous interval representation, that is, an assignment of intervals and label sets to the vertices such that two vertices are adjacent if and only if the corresponding intervals, as well as their label sets, intersect. We show that this parameter is NP-hard to compute and give several bounds for the parameter, showing in particular that it is sandwiched between pathwidth and linear mim-width. For classes of graphs with bounded parameter values, assuming that the graph is equipped with a simultaneous interval representation with a constant number of labels, we give FPT algorithms for the clique, independent set, and dominating set problems, and hardness results for the independent dominating set and coloring problems. The FPT results for independent set and dominating set are for the simultaneous interval number plus solution size. In contrast, both problems are known to be 𝖶[1]-hard for linear mim-width plus solution size.
Cite as

Jesse Beisegel, Nina Chiarelli, Ekkehard Köhler, Martin Milanič, Peter Muršič, and Robert Scheffler. The Simultaneous Interval Number: A New Width Parameter that Measures the Similarity to Interval Graphs. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{beisegel_et_al:LIPIcs.SWAT.2024.7,
  author =	{Beisegel, Jesse and Chiarelli, Nina and K\"{o}hler, Ekkehard and Milani\v{c}, Martin and Mur\v{s}i\v{c}, Peter and Scheffler, Robert},
  title =	{{The Simultaneous Interval Number: A New Width Parameter that Measures the Similarity to Interval Graphs}},
  booktitle =	{19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
  pages =	{7:1--7:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-318-8},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{294},
  editor =	{Bodlaender, Hans L.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.7},
  URN =		{urn:nbn:de:0030-drops-200470},
  doi =		{10.4230/LIPIcs.SWAT.2024.7},
  annote =	{Keywords: Interval graph, simultaneous representation, width parameter, algorithm, parameterized complexity}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2017.28
Induced Embeddings into Hamming Graphs

Authors: Martin Milanic, Peter Mursic, and Marcelo Mydlarz

Published in: LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Abstract
Let d be a positive integer. Can a given graph G be realized in R^d so that vertices are mapped to distinct points, two vertices being adjacent if and only if the corresponding points lie on a common line that is parallel to some axis? Graphs admitting such realizations have been studied in the literature for decades under different names. Peterson asked in [Discrete Appl. Math., 2003] about the complexity of the recognition problem. While the two-dimensional case corresponds to the class of line graphs of bipartite graphs and is well-understood, the complexity question has remained open for all higher dimensions. In this paper, we answer this question. We establish the NP-completeness of the recognition problem for any fixed dimension, even in the class of bipartite graphs. To do this, we strengthen a characterization of induced subgraphs of 3-dimensional Hamming graphs due to Klavžar and Peterin. We complement the hardness result by showing that for some important classes of perfect graphs –including chordal graphs and distance-hereditary graphs– the minimum dimension of the Euclidean space in which the graph can be realized, or the impossibility of doing so, can be determined in linear time.
Cite as

Martin Milanic, Peter Mursic, and Marcelo Mydlarz. Induced Embeddings into Hamming Graphs. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{milanic_et_al:LIPIcs.MFCS.2017.28,
  author =	{Milanic, Martin and Mursic, Peter and Mydlarz, Marcelo},
  title =	{{Induced Embeddings into Hamming Graphs}},
  booktitle =	{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
  pages =	{28:1--28:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-046-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{83},
  editor =	{Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.28},
  URN =		{urn:nbn:de:0030-drops-81289},
  doi =		{10.4230/LIPIcs.MFCS.2017.28},
  annote =	{Keywords: gridline graph, Hamming graph, induced embedding, NP-completeness, chordal graph}
}
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