Computing Vertex and Edge Connectivity of Graphs Embedded with Crossings Article Swipe
Thérèse Biedl
,
Prosenjit Bose
,
Karthik Murali
·
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2407.00586
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2407.00586
Vertex connectivity and edge connectivity are fundamental concepts in graph theory that have been widely studied from both structural and algorithmic perspectives. The focus of this paper is on computing these two parameters for graphs embedded on the plane with crossings. For planar graphs -- which can be embedded on the plane without any crossings -- it has long been known that vertex and edge connectivity can be computed in linear time. Recently, the algorithm for vertex connectivity was extended from planar graphs to 1-plane graphs (where each edge is crossed at most once) without $\times$-crossings -- these are crossings whose endpoints induce a matching. The key insight, for both these classes of graphs, is that any two vertices/edges of a minimum vertex/edge cut have small face-distance (distance measured by number of faces) in the embedding. In this paper, we attempt at a comprehensive generalization of this idea to a wider class of graphs embedded on the plane. Our method works for all those embedded graphs where every pair of crossing edges is connected by a path whose vertices and edges have a small face-distance from the crossing point. Important examples of such graphs include optimal 2-planar and optimal 3-planar graphs, $d$-map graphs, $d$-framed graphs, graphs with bounded crossing number, and $k$-plane graphs with bounded number of $\times$-crossings. For all these graph classes, we get a linear-time algorithm for computing vertex and edge connectivity.
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Metadata
- Type
- preprint
- Language
- en
- Landing Page
- https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.24
- OA Status
- green
- Cited By
- 1
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4400325795
All OpenAlex metadata
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- OpenAlex ID
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https://openalex.org/W4400325795Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2407.00586Digital Object Identifier
- Title
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Computing Vertex and Edge Connectivity of Graphs Embedded with CrossingsWork title
- Type
-
preprintOpenAlex work type
- Language
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enPrimary language
- Publication year
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2024Year of publication
- Publication date
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2024-06-30Full publication date if available
- Authors
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Thérèse Biedl, Prosenjit Bose, Karthik MuraliList of authors in order
- Landing page
-
https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.24Publisher landing page
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YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
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https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.24Direct OA link when available
- Concepts
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Parameterized complexity, Vertex (graph theory), Enhanced Data Rates for GSM Evolution, Algorithm, Combinatorics, Computer science, Mathematics, Graph, Artificial intelligenceTop concepts (fields/topics) attached by OpenAlex
- Cited by
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1Total citation count in OpenAlex
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2025: 1Per-year citation counts (last 5 years)
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10Other works algorithmically related by OpenAlex
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| abstract_inverted_index.concepts | 7 |
| abstract_inverted_index.crossing | 170, 187, 208 |
| abstract_inverted_index.embedded | 35, 48, 154, 164 |
| abstract_inverted_index.examples | 190 |
| abstract_inverted_index.extended | 79 |
| abstract_inverted_index.insight, | 107 |
| abstract_inverted_index.measured | 128 |
| abstract_inverted_index.vertices | 178 |
| abstract_inverted_index.$k$-plane | 211 |
| abstract_inverted_index.(distance | 127 |
| abstract_inverted_index.Important | 189 |
| abstract_inverted_index.Recently, | 72 |
| abstract_inverted_index.algorithm | 74, 227 |
| abstract_inverted_index.computing | 29, 229 |
| abstract_inverted_index.connected | 173 |
| abstract_inverted_index.crossings | 54, 99 |
| abstract_inverted_index.endpoints | 101 |
| abstract_inverted_index.matching. | 104 |
| abstract_inverted_index.$d$-framed | 203 |
| abstract_inverted_index.crossings. | 40 |
| abstract_inverted_index.embedding. | 135 |
| abstract_inverted_index.parameters | 32 |
| abstract_inverted_index.structural | 18 |
| abstract_inverted_index.algorithmic | 20 |
| abstract_inverted_index.fundamental | 6 |
| abstract_inverted_index.linear-time | 226 |
| abstract_inverted_index.vertex/edge | 122 |
| abstract_inverted_index.connectivity | 1, 4, 65, 77 |
| abstract_inverted_index.comprehensive | 143 |
| abstract_inverted_index.connectivity. | 233 |
| abstract_inverted_index.face-distance | 126, 184 |
| abstract_inverted_index.perspectives. | 21 |
| abstract_inverted_index.generalization | 144 |
| abstract_inverted_index.vertices/edges | 118 |
| abstract_inverted_index.$\times$-crossings | 95 |
| abstract_inverted_index.$\times$-crossings. | 217 |
| cited_by_percentile_year | |
| countries_distinct_count | 0 |
| institutions_distinct_count | 3 |
| citation_normalized_percentile |