Abstract
The new graph parameter twin-width, recently introduced by Bonnet et al., allows for an FPT algorithm for testing all FO properties of graphs. This makes classes of efficiently bounded twin-width attractive from the algorithmic point of view. In particular, such classes (of small twin-width) include proper interval graphs, and (as digraphs) posets of width k. Inspired by an existing generalization of interval graphs into so-called k-thin graphs, we define a new class of proper k-mixed-thin graphs which largely generalizes proper interval graphs. We prove that proper k-mixed-thin graphs have twin-width linear in k, and that a certain subclass of k-mixed-thin graphs is transduction-equivalent to posets of width \(k'\) such that there is a quadratic relation between k and \(k'\).
Supported by the Czech Science Foundation, project no. 20-04567S.
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Notes
- 1.
Two vertices x and y are called twins in a graph G if they have the same neighbours in \(V(G) \setminus \{x,y\}\).
- 2.
Note that one can also define the “natural” twin-width of graphs which, informally, ignores the red entries on the main diagonal (as there are no loops in a simple graph). The natural twin-width is never larger, but possibly by one lower, than the symmetric matrix twin-width. For instance, for the sequence in Fig. 1, the natural twin-width would be only 2.
References
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Balabán, J., Hliněný, P., Jedelský, J. (2022). Twin-Width and Transductions of Proper k-Mixed-Thin Graphs. In: Bekos, M.A., Kaufmann, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2022. Lecture Notes in Computer Science, vol 13453. Springer, Cham. https://doi.org/10.1007/978-3-031-15914-5_4
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