Abstract
The dichotomy conjecture for the parameterized embedding problem states that the problem of deciding whether a given graph G from some class \(\mathcal K\) of “pattern graphs” can be embedded into a given graph H (that is, is isomorphic to a subgraph of H) is fixed-parameter tractable if \(\mathcal K\) is a class of graphs of bounded tree width and \(W [1]\)-complete otherwise.
Towards this conjecture, we prove that the embedding problem is \(W [1]\)-complete if \(\mathcal K\) is the class of all grids or the class of all walls.
Full version available at https://arxiv.org/abs/1703.06423.
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Notes
- 1.
There is a minor issue here regarding the computability of the class \(\mathcal K\): if we want to include classes \(\mathcal K\) that are not recursively enumerable here then we need the nonuniform notion of fixed-parameter tractability [11].
- 2.
If \(\mathcal K\) is not recursively enumerable, there is still a “non-uniform” hardness result. See [13] for a discussion.
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Acknowledgement
Yijia Chen is partially supported by the Sino-German Center for Research Promotion (CDZ 996) and National Nature Science Foundation of China (Project 61373029). Bingkai Lin is partially supported by the JSPS KAKENHI Grant (JP16H07409) and JST ERATO Grant (JPMJER1201) of Japan.
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Chen, Y., Grohe, M., Lin, B. (2017). The Hardness of Embedding Grids and Walls. In: Bodlaender, H., Woeginger, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2017. Lecture Notes in Computer Science(), vol 10520. Springer, Cham. https://doi.org/10.1007/978-3-319-68705-6_14
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