zbMATH — the first resource for mathematics

Tree sets. (English) Zbl 1412.06002
Author’s abstract: We study an abstract notion of tree structure which lies at the common core of various tree-like discrete structures commonly used in combinatorics: trees in graphs, order trees, nested subsets of a set, tree-decompositions of graphs and matroids etc. Unlike graph-theoretical or order trees, these tree sets can provide a suitable formalization of tree structure also for infinite graphs, matroids, and set partitions. Order trees reappear as oriented tree sets. We show how each of the above structures defines a tree set, and which additional information, if any, is needed to reconstruct it from this tree set.

06A07 Combinatorics of partially ordered sets
05B35 Combinatorial aspects of matroids and geometric lattices
05C05 Trees
Full Text: DOI arXiv
[1] Bowler, N., Diestel, R., Mazoit, F.: Tangle-tree duality in infinite graphs. In preparation · Zbl 0696.05027
[2] Diestel, R.: Abstract separation systems, arXiv:1406.3797. Order (2017). doi:10.1007/s11083-017-9424-5 · Zbl 0795.05110
[3] Diestel, R.: Graph Theory (5th edition, 2016). Springer-Verlag, 2017. Electronic edition available at http://diestel-graph-theory.com/ · Zbl 1375.05002
[4] Diestel, R., Erde, J., Eberenz, P.: Duality theorem for blocks and tangles in graphs. arXiv:1605.09139, to appear in SIAM J. Discrete Mathematics (2016) · Zbl 1366.05103
[5] Diestel, R., Hundertmark, F., Lemanczyk, S.: Profiles of separations: in graphs, matroids, and beyond. arXiv:1110.6207, to appear in Combinatorica (2017) · Zbl 1438.05040
[6] Diestel, R., Kneip, J.: Profinite tree sets. In preparation · Zbl 0812.20012
[7] Diestel, R., Oum, S.: Tangle-tree duality in abstract separation systems, arXiv:1701.02509. (2017)
[8] Diestel, R., Oum, S.: Tangle-tree duality in graphs, matroids and beyond. arXiv:1701.02651 (2017)
[9] Diestel, R., Whittle, G.: Tangles and the Mona Lisa. arXiv:1603.06652
[10] Dunwoody, MJ, Inaccessible groups and protrees, J. Pure Appl. Algebra, 88, 63-78, (1993) · Zbl 0812.20012
[11] Dunwoody, MJ, Groups acting on protrees, J. Lond. Math. Soc., 56, 125-136, (1997) · Zbl 0918.20011
[12] Gollin, P., Kneip, J.: Representations of infinite tree sets. In preparation · Zbl 0795.05110
[13] Hundertmark, F.: Profiles. An algebraic approach to combinatorial connectivity. arXiv:1110.6207 (2011) · Zbl 0696.05027
[14] Seymour, P; Thomas, R, Graph searching and a MIN-MAX theorem for tree-width, J. Comb. Theory (Ser. B), 58, 22-33, (1993) · Zbl 0795.05110
[15] Woess, W, Graphs and groups with tree-like properties, J. Comb. Theory Ser. B, 47, 361-371, (1989) · Zbl 0696.05027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.