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Structural properties of recursively partitionable graphs with connectivity 2. (English) Zbl 1400.05197

Summary: A connected graph \(G\) is said to be arbitrarily partitionable (AP for short) if for every partition \((n_1,\dots, n_p)\) of \(|V(G)|\) there exists a partition \((V_1,\dots, V_p)\) of \(V(G)\) such that each \(V_i\) induces a connected subgraph of \(G\) on \(n_i\) vertices. Some stronger versions of this property were introduced, namely the ones of being online arbitrarily partitionable and recursively arbitrarily partitionable (OL-AP and R-AP for short, respectively), in which the subgraphs induced by a partition of \(G\) must not only be connected but also fulfil additional conditions. In this paper, we point out some structural properties of OL-AP and R-AP graphs with connectivity 2. In particular, we show that deleting a cut pair of these graphs results in a graph with a bounded number of components, some of whom have a small number of vertices. We obtain these results by studying a simple class of 2-connected graphs called balloons.

MSC:

05C75 Structural characterization of families of graphs
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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[1] D. Barth, O. Baudon and J. Puech, Decomposable trees: a polynomial algorithm for tripodes, Discrete Appl. Math. 119 (2002) 205-216. doi:10.1016/S0166-218X(00)00322-X · Zbl 1002.68107
[2] D. Barth and H. Fournier, A degree bound on decomposable trees, Discrete Math. 306 (2006) 469-477. doi:10.1016/j.disc.2006.01.006 · Zbl 1092.05054
[3] O. Baudon, F. Foucaud, J. Przyby lo and M. Woźniak, On the structure of arbitrarily partitionable graphs with given connectivity, Discrete Appl. Math. 162 (2014) 381-385. doi:10.1016/j.dam.2013.09.007 · Zbl 1300.05245
[4] O. Baudon, F. Gilbert and M. Woźniak, Recursively arbitrarily vertex-decomposable suns, Opuscula Math. 31 (2011) 533-547. doi:10.7494/OpMath.2011.31.4.533 · Zbl 1234.05182
[5] O. Baudon, F. Gilbert and M. Woźniak, Recursively arbitrarily vertex-decomposable graphs, Opuscula Math. 32 (2012) 689-706. doi:10.7494/OpMath.2012.32.4.689 · Zbl 1259.05135
[6] J. Bensmail, On the longest path in a recursively partitionable graph, Opuscula Math. 33 (2013) 631-640. doi:10.7494/OpMath.2013.33.4.631 · Zbl 1276.05064
[7] S. Cichacz, A. Görlich, A. Marczyk, J. Przyby lo and M. Woźniak, Arbitrarily vertex decomposable caterpillars with four or five leaves, Discuss. Math. Graph Theory 26 (2006) 291-305. doi:10.7151/dmgt.1321
[8] R. Diestel, Graph Theory (Springer, Berlin Heidelberg, 2005).
[9] E. Győri, On division of graphs to connected subgraphs, in: Combinatorics, Proc. Fifth Hungariam Colloq., Keszthely, 1976, Vol. I, Colloq. Math. Soc. János Bolyai 18 (1978) 485-494.
[10] M. Horňák, Zs. Tuza and M. Woźniak, On-line arbitrarily vertex decomposable trees, Discrete Appl. Math. 155 (2007) 1420-1429. doi:10.1016/j.dam.2007.02.011
[11] M. Horňák and M. Woźniak, Arbitrarily vertex decomposable trees are of maximum degree at most six, Opuscula Math. 23 (2003) 49-62.
[12] L. Lovász, A homology theory for spanning trees of a graph, Acta Math. Hungar. 30 (1977) 241-251.
[13] A. Marczyk, An Ore-type condition for arbitrarily vertex decomposable graphs, Discrete Math. 309 (2009) 3588-3594. doi:10.1016/j.disc.2007.12.066 · Zbl 1179.05089
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