Isometric-Universal Graphs for Trees
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911007031427072 |
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| author | Baucher, Edgar Dross, François Gavoille, Cyril |
| author_facet | Baucher, Edgar Dross, François Gavoille, Cyril |
| contents | We consider the problem of finding the smallest graph that contains two input trees each with at most $n$ vertices preserving their distances. In other words, we look for an isometric-universal graph with the minimum number of vertices for two given trees. We prove that this problem can be solved in time $O(n^{5/2}\log{n})$. We extend this result to forests instead of trees, and propose an algorithm with running time $O(n^{7/2}\log{n})$. As a key ingredient, we show that a smallest isometric-universal graph of two trees essentially is a tree. Furthermore, we prove that these results cannot be extended. Firstly, we show that deciding whether there exists an isometric-universal graph with $t$ vertices for three forests is NP-complete. Secondly, we show that any smallest isometric-universal graph cannot be a tree for some families of three trees. This latter result has implications for greedy strategies solving the smallest isometric-universal graph problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_11704 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Isometric-Universal Graphs for Trees Baucher, Edgar Dross, François Gavoille, Cyril Data Structures and Algorithms We consider the problem of finding the smallest graph that contains two input trees each with at most $n$ vertices preserving their distances. In other words, we look for an isometric-universal graph with the minimum number of vertices for two given trees. We prove that this problem can be solved in time $O(n^{5/2}\log{n})$. We extend this result to forests instead of trees, and propose an algorithm with running time $O(n^{7/2}\log{n})$. As a key ingredient, we show that a smallest isometric-universal graph of two trees essentially is a tree. Furthermore, we prove that these results cannot be extended. Firstly, we show that deciding whether there exists an isometric-universal graph with $t$ vertices for three forests is NP-complete. Secondly, we show that any smallest isometric-universal graph cannot be a tree for some families of three trees. This latter result has implications for greedy strategies solving the smallest isometric-universal graph problem. |
| title | Isometric-Universal Graphs for Trees |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2506.11704 |