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Documents authored by Verciani, Francesco

Document
DOI: 10.4230/LIPIcs.GD.2025.12
Flipping Odd Matchings in Geometric and Combinatorial Settings

Authors: Oswin Aichholzer, Sofia Brenner, Joseph Dorfer, Hung P. Hoang, Daniel Perz, Christian Rieck, and Francesco Verciani

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
We study the problem of reconfiguring odd matchings, that is, matchings that cover all but a single vertex. Our reconfiguration operation is a so-called flip where the unmatched vertex of the first matching gets matched, while consequently another vertex becomes unmatched. We consider two distinct settings: the geometric setting, in which the vertices are points embedded in the plane and all occurring odd matchings are crossing-free, and a combinatorial setting, in which we consider odd matchings in general graphs. For the latter setting, we provide a complete polynomial time checkable characterization of graphs in which any two odd matchings can be reconfigured into each another. This complements the previously known result that the flip graph is always connected in the geometric setting [Oswin Aichholzer et al., 2025]. In the combinatorial setting, we prove that the diameter of the flip graph, if connected, is linear in the number of vertices. Furthermore, we establish that deciding whether there exists a flip sequence of length k transforming one given matching into another is NP-complete in both the combinatorial and the geometric settings. To prove the latter, we introduce a framework that allows us to transform partial order types into general position with only polynomial overhead. Finally, we demonstrate that when parameterized by the flip distance k, the problem is fixed-parameter tractable (FPT) in the geometric setting when restricted to convex point sets.
Cite as

Oswin Aichholzer, Sofia Brenner, Joseph Dorfer, Hung P. Hoang, Daniel Perz, Christian Rieck, and Francesco Verciani. Flipping Odd Matchings in Geometric and Combinatorial Settings. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 12:1-12:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{aichholzer_et_al:LIPIcs.GD.2025.12,
  author =	{Aichholzer, Oswin and Brenner, Sofia and Dorfer, Joseph and Hoang, Hung P. and Perz, Daniel and Rieck, Christian and Verciani, Francesco},
  title =	{{Flipping Odd Matchings in Geometric and Combinatorial Settings}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{12:1--12:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.12},
  URN =		{urn:nbn:de:0030-drops-249983},
  doi =		{10.4230/LIPIcs.GD.2025.12},
  annote =	{Keywords: Odd matchings, reconfiguration, flip graph, geometric, combinatorial, connectivity, NP-hardness, FPT}
}
Document
DOI: 10.4230/LIPIcs.GD.2024.30
Flips in Colorful Triangulations

Authors: Rohan Acharya, Torsten Mütze, and Francesco Verciani

Published in: LIPIcs, Volume 320, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

Abstract
The associahedron is the graph G_N that has as nodes all triangulations of a convex N-gon, and an edge between any two triangulations that differ in a flip operation. A flip removes an edge shared by two triangles and replaces it by the other diagonal of the resulting 4-gon. In this paper, we consider a large collection of induced subgraphs of G_N obtained by Ramsey-type colorability properties. Specifically, coloring the points of the N-gon red and blue alternatingly, we consider only colorful triangulations, namely triangulations in which every triangle has points in both colors, i.e., monochromatic triangles are forbidden. The resulting induced subgraph of G_N on colorful triangulations is denoted by F_N. We prove that F_N has a Hamilton cycle for all N ≥ 8, resolving a problem raised by Sagan, i.e., all colorful triangulations on N points can be listed so that any two cyclically consecutive triangulations differ in a flip. In fact, we prove that for an arbitrary fixed coloring pattern of the N points with at least 10 changes of color, the resulting subgraph of G_N on colorful triangulations (for that coloring pattern) admits a Hamilton cycle. We also provide an efficient algorithm for computing a Hamilton path in F_N that runs in time O(1) on average per generated node. This algorithm is based on a new and algorithmic construction of a tree rotation Gray code for listing all n-vertex k-ary trees that runs in time O(k) on average per generated tree.
Cite as

Rohan Acharya, Torsten Mütze, and Francesco Verciani. Flips in Colorful Triangulations. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 30:1-30:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{acharya_et_al:LIPIcs.GD.2024.30,
  author =	{Acharya, Rohan and M\"{u}tze, Torsten and Verciani, Francesco},
  title =	{{Flips in Colorful Triangulations}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{30:1--30:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-343-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{320},
  editor =	{Felsner, Stefan and Klein, Karsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.30},
  URN =		{urn:nbn:de:0030-drops-213143},
  doi =		{10.4230/LIPIcs.GD.2024.30},
  annote =	{Keywords: Flip graph, associahedron, triangulation, binary tree, vertex coloring, Hamilton cycle, Gray code}
}
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