The Complexity of Diameter on H-free graphs
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| Format: | Preprint |
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2024
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| _version_ | 1866910969960071168 |
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| author | Oostveen, Jelle J. Paulusma, Daniël van Leeuwen, Erik Jan |
| author_facet | Oostveen, Jelle J. Paulusma, Daniël van Leeuwen, Erik Jan |
| contents | The intensively studied Diameter problem is to find the diameter of a given connected graph. We investigate, for the first time in a structured manner, the complexity of Diameter for H-free graphs, that is, graphs that do not contain a fixed graph H as an induced subgraph. We first show that if H is not a linear forest with small components, then Diameter cannot be solved in subquadratic time for H-free graphs under SETH. For some small linear forests, we do show linear-time algorithms for solving Diameter. For other linear forests H, we make progress towards linear-time algorithms by considering specific diameter values. If H is a linear forest, the maximum value of the diameter of any graph in a connected H-free graph class is some constant dmax dependent only on H. We give linear-time algorithms for deciding if a connected H-free graph has diameter dmax, for several linear forests H. In contrast, for one such linear forest H, Diameter cannot be solved in subquadratic time for H-free graphs under SETH. Moreover, we even show that, for several other linear forests H, one cannot decide in subquadratic time if a connected H-free graph has diameter dmax under SETH. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_16678 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Complexity of Diameter on H-free graphs Oostveen, Jelle J. Paulusma, Daniël van Leeuwen, Erik Jan Data Structures and Algorithms Discrete Mathematics The intensively studied Diameter problem is to find the diameter of a given connected graph. We investigate, for the first time in a structured manner, the complexity of Diameter for H-free graphs, that is, graphs that do not contain a fixed graph H as an induced subgraph. We first show that if H is not a linear forest with small components, then Diameter cannot be solved in subquadratic time for H-free graphs under SETH. For some small linear forests, we do show linear-time algorithms for solving Diameter. For other linear forests H, we make progress towards linear-time algorithms by considering specific diameter values. If H is a linear forest, the maximum value of the diameter of any graph in a connected H-free graph class is some constant dmax dependent only on H. We give linear-time algorithms for deciding if a connected H-free graph has diameter dmax, for several linear forests H. In contrast, for one such linear forest H, Diameter cannot be solved in subquadratic time for H-free graphs under SETH. Moreover, we even show that, for several other linear forests H, one cannot decide in subquadratic time if a connected H-free graph has diameter dmax under SETH. |
| title | The Complexity of Diameter on H-free graphs |
| topic | Data Structures and Algorithms Discrete Mathematics |
| url | https://arxiv.org/abs/2402.16678 |