Abstract
We prove a general duality theorem for width parameters in combinatorial structures such as graphs and matroids. It implies the classical such theorems for path-width, tree-width, branch-width and rank-width, and gives rise to new width parameters with associated duality theorems. The dense substructures witnessing large width are presented in a unified way akin to tangles, as orientations of separation systems satisfying certain consistency axioms.
This is an extended abstract of arXiv:1406.3797, which contains all the proofs omitted here. See also arXiv:1406.3798 for further work in this direction.
Sang-il Oum: Supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2011-0011653).
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Notes
- 1.
In fact, we need even less. It would be enough to consider instead of ‘separations’ any poset with an involution that commutes with its ordering, just as the ordering of separations introduced below satisfies \((A,B)\le (C,D) \Leftrightarrow (B,A)\ge (D,C)\). It is only for the sake of readability that we are writing this paper in terms of separations, as readers are likely to have graphs or matroids in mind.
- 2.
Our notational convention will be that we think of \((A,B)\) as pointing towards \(B\).
- 3.
It is a good idea to work with this formal definition of consistency, since the more intuitive notion of ‘pointing away from each other’ can be counterintuitive. For example, we shall need that no consistent set of separations of \(V\) contains a separation of the form \((V,A)\); this follows readily from the formal definition, as \((A,V)\le (V,A)\), but is less obvious from the informal.
- 4.
This will help us show that \((T,\alpha ')\) is over \(\mathcal F\) if \((T,\alpha )\) is.
- 5.
This will help us show that \((T,\alpha ')\) is rooted in \(S^-\) if \((T,\alpha )\) is.
- 6.
For example, we do not need Menger’s theorem, as all the other proofs do.
- 7.
In our matroid terminology we follow Oxley [9].
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Diestel, R., Oum, Si. (2014). Unifying Duality Theorems for Width Parameters in Graphs and Matroids (Extended Abstract). In: Kratsch, D., Todinca, I. (eds) Graph-Theoretic Concepts in Computer Science. WG 2014. Lecture Notes in Computer Science, vol 8747. Springer, Cham. https://doi.org/10.1007/978-3-319-12340-0_1
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