Geodetic Set on Graphs of Constant Pathwidth and Feedback Vertex Set Number
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917998632108032 |
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| author | Tale, Prafullkumar |
| author_facet | Tale, Prafullkumar |
| contents | In the \textsc{Geodetic Set} problem, the input consists of a graph $G$ and a positive integer $k$. The goal is to determine whether there exists a subset $S$ of vertices of size $k$ such that every vertex in the graph is included in a shortest path between two vertices in $S$. Kellerhals and Koana [IPEC 2020; J. Graph Algorithms Appl 2022] proved that the problem is $\W[1]$-hard when parameterized by the pathwidth and the feedback vertex set number of the input graph. They posed the question of whether the problem admits an $\XP$ algorithm when parameterized by the combination of these two parameters. We answer this in negative by proving that the problem remains \NP-hard on graphs of constant pathwidth and feedback vertex set number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_17862 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geodetic Set on Graphs of Constant Pathwidth and Feedback Vertex Set Number Tale, Prafullkumar Data Structures and Algorithms Computational Complexity In the \textsc{Geodetic Set} problem, the input consists of a graph $G$ and a positive integer $k$. The goal is to determine whether there exists a subset $S$ of vertices of size $k$ such that every vertex in the graph is included in a shortest path between two vertices in $S$. Kellerhals and Koana [IPEC 2020; J. Graph Algorithms Appl 2022] proved that the problem is $\W[1]$-hard when parameterized by the pathwidth and the feedback vertex set number of the input graph. They posed the question of whether the problem admits an $\XP$ algorithm when parameterized by the combination of these two parameters. We answer this in negative by proving that the problem remains \NP-hard on graphs of constant pathwidth and feedback vertex set number. |
| title | Geodetic Set on Graphs of Constant Pathwidth and Feedback Vertex Set Number |
| topic | Data Structures and Algorithms Computational Complexity |
| url | https://arxiv.org/abs/2504.17862 |