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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.16193 |
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| _version_ | 1866917402209419264 |
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| author | Cabello, Sergio |
| author_facet | Cabello, Sergio |
| contents | We consider the following two algorithmic problems: given a graph $G$ and a subgraph $H\subseteq G$, decide whether $H$ is an isometric or a geodesically convex subgraph of $G$. It is relatively easy to see that the problems can be solved by computing the distances between all pairs of vertices. We provide a conditional lower bound showing that, for sparse graphs with $n$ vertices and $Θ(n)$ edges, we cannot expect to solve the problem in $O(n^{2-\varepsilon})$ time for any constant $\varepsilon>0$. We also show that the problem can be solved in subquadratic time for planar graphs and in near-linear time for graphs of bounded treewidth. Finally, we provide a near-linear time algorithm for the setting where $G$ is a plane graph and $H$ is defined by a few cycles in $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16193 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Testing whether a subgraph is convex or isometric Cabello, Sergio Data Structures and Algorithms We consider the following two algorithmic problems: given a graph $G$ and a subgraph $H\subseteq G$, decide whether $H$ is an isometric or a geodesically convex subgraph of $G$. It is relatively easy to see that the problems can be solved by computing the distances between all pairs of vertices. We provide a conditional lower bound showing that, for sparse graphs with $n$ vertices and $Θ(n)$ edges, we cannot expect to solve the problem in $O(n^{2-\varepsilon})$ time for any constant $\varepsilon>0$. We also show that the problem can be solved in subquadratic time for planar graphs and in near-linear time for graphs of bounded treewidth. Finally, we provide a near-linear time algorithm for the setting where $G$ is a plane graph and $H$ is defined by a few cycles in $G$. |
| title | Testing whether a subgraph is convex or isometric |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2502.16193 |