Geodesic
Large Contractible Subgraphs of a 3-Connected Graph
Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 1, pp. 83-101.

Voir la notice de l'article provenant de la source Library of Science

Let m ≥ 5 be a positive integer and let G be a 3-connected graph on at least 2m + 1 vertices. We prove that G has a contractible set W such that m ≤ |W| ≤ 2m − 4. (Recall that a set W ⊂ V (G) of a 3-connected graph G is contractible if the graph G(W) is connected and the graph G − W is 2-connected.) A particular case for m = 4 is that any 3-connected graph on at least 11 vertices has a contractible set of 5 or 6 vertices.
Keywords: connectivity, 3-connected graph, contractible subgraph 
@article{DMGT_2021_41_1_a5,
     author = {Karpov, Dmitri V.},
     title = {Large {Contractible} {Subgraphs} of a {3-Connected} {Graph}},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {83--101},
     publisher = {mathdoc},
     volume = {41},
     number = {1},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2021_41_1_a5/}
}
TY  - JOUR
AU  - Karpov, Dmitri V.
TI  - Large Contractible Subgraphs of a 3-Connected Graph
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2021
SP  - 83
EP  - 101
VL  - 41
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2021_41_1_a5/
LA  - en
ID  - DMGT_2021_41_1_a5
ER  - 
%0 Journal Article
%A Karpov, Dmitri V.
%T Large Contractible Subgraphs of a 3-Connected Graph
%J Discussiones Mathematicae. Graph Theory
%D 2021
%P 83-101
%V 41
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2021_41_1_a5/
%G en
%F DMGT_2021_41_1_a5
Karpov, Dmitri V. Large Contractible Subgraphs of a 3-Connected Graph. Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 1, pp. 83-101. http://geodesic.mathdoc.fr/item/DMGT_2021_41_1_a5/

[1] W.T. Tutte, Connectivity in Graphs (Univ. Toronto Press, 1966).

[2] W. Hohberg, The decomposition of graphs into k-connected components, Discrete Math. 109 (1992) 133–145. doi: 10.1016/0012-365X(92)90284-M

[3] W. Mader, On vertices of degree n in minimally n-connected graphs and digraphs, Combinatorics, Paul Erdős is Eighty 2 (1996) 423–449.

[4] W. McCuaig and K. Ota, Contractible triples in 3- connected graphs, J. Comb. Theory Ser. B 60 (1994) 308–314. doi: 10.1006/jctb.1994.1022

[5] M. Kriesell, Contractible subgraphs in 3- connected graphs, J. Comb. Theory Ser. B 80 (2000) 32–48. doi: 10.1006/jctb.2000.1960

[6] M. Kriesell, On small contractible subgraphs in 3- connected graphs of small average degree, Graphs Combin. 23 (2007) 545–557. doi: 10.1007/s00373-007-0749-5

[7] F. Harary, Graph Theory (Addison-Wesley, 1969). doi: 10.21236/AD0705364

[8] D.V. Karpov and A.V. Pastor, On the structure of a k-connected graph, J. Math. Sci. 113 (2003) 584–597. doi: 10.1023/A:1021146226285

[9] D.V. Karpov and A.V. Pastor, The structure of a decomposition of a triconnected graph, J. Math. Sci. 184 (2012) 601–628. doi: 10.1007/s10958-012-0885-1

[10] D.V. Karpov, Blocks in k-connected graphs, J. Math. Sci. 126 (2005) 1167–1181. doi: 10.1007/s10958-005-0084-4

[11] D.V. Karpov, Cutsets in a k-connected graph, J. Math. Sci. 145 (2007) 4953–4966. doi: 10.1007/s10958-007-0330-z

[12] D.V. Karpov, The decomposition tree of a biconnected graph, J. Math. Sci. 204 (2015) 232–243. doi: 10.1007/s10958-014-2198-z

[13] D.V. Karpov, Minimal biconnected graphs, J. Math. Sci. 204 (2015) 244–257. doi: 10.1007/s10958-014-2199-y