A New Temporal Interpretation of Cluster Editing
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866911454569955328 |
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| author | Bocci, Cristiano Capresi, Chiara Meeks, Kitty Sylvester, John |
| author_facet | Bocci, Cristiano Capresi, Chiara Meeks, Kitty Sylvester, John |
| contents | The NP-complete graph problem Cluster Editing seeks to transform a static graph into a disjoint union of cliques by making the fewest possible edits to the edges. We introduce a natural interpretation of this problem in temporal graphs, whose edge sets change over time. This problem is NP-complete even when restricted to temporal graphs whose underlying graph is a path, but we obtain two polynomial-time algorithms for restricted cases. In the static setting, it is well-known that a graph is a disjoint union of cliques if and only if it contains no induced copy of $P_3$; we demonstrate that no general characterisation involving sets of at most four vertices can exist in the temporal setting, but obtain a complete characterisation involving forbidden configurations on at most five vertices. This characterisation gives rise to an FPT algorithm parameterised simultaneously by the permitted number of modifications and the lifetime of the temporal graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_01103 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A New Temporal Interpretation of Cluster Editing Bocci, Cristiano Capresi, Chiara Meeks, Kitty Sylvester, John Discrete Mathematics Computational Complexity Data Structures and Algorithms Combinatorics 05C75, 05C85, 68Q25 G.2.2; F.2.2 The NP-complete graph problem Cluster Editing seeks to transform a static graph into a disjoint union of cliques by making the fewest possible edits to the edges. We introduce a natural interpretation of this problem in temporal graphs, whose edge sets change over time. This problem is NP-complete even when restricted to temporal graphs whose underlying graph is a path, but we obtain two polynomial-time algorithms for restricted cases. In the static setting, it is well-known that a graph is a disjoint union of cliques if and only if it contains no induced copy of $P_3$; we demonstrate that no general characterisation involving sets of at most four vertices can exist in the temporal setting, but obtain a complete characterisation involving forbidden configurations on at most five vertices. This characterisation gives rise to an FPT algorithm parameterised simultaneously by the permitted number of modifications and the lifetime of the temporal graph. |
| title | A New Temporal Interpretation of Cluster Editing |
| topic | Discrete Mathematics Computational Complexity Data Structures and Algorithms Combinatorics 05C75, 05C85, 68Q25 G.2.2; F.2.2 |
| url | https://arxiv.org/abs/2202.01103 |