Every large $k$-connected graph-minor induces a $k$-tangle in its ambient graph. The converse holds for $k\leqslant 3$, but fails for $k\geqslant 4$. This raises the question whether '$k$-connected' can be relaxed to obtain a characterisation of $k$-tangles through highly cohesive graph-minors. We show that this can be achieved for $k=4$ by proving that internally 4-connected graphs have unique 4-tangles, and that every graph with a 4-tangle $\tau$ has an internally 4-connected minor whose unique 4-tangle lifts to $\tau$.
@article{10_37236_12367,
author = {Johannes Carmesin and Jan Kurkofka},
title = {Characterising 4-tangles through a connectivity property},
journal = {The electronic journal of combinatorics},
year = {2025},
volume = {32},
number = {3},
doi = {10.37236/12367},
zbl = {8097654},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12367/}
}
TY - JOUR
AU - Johannes Carmesin
AU - Jan Kurkofka
TI - Characterising 4-tangles through a connectivity property
JO - The electronic journal of combinatorics
PY - 2025
VL - 32
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/12367/
DO - 10.37236/12367
ID - 10_37236_12367
ER -
%0 Journal Article
%A Johannes Carmesin
%A Jan Kurkofka
%T Characterising 4-tangles through a connectivity property
%J The electronic journal of combinatorics
%D 2025
%V 32
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/12367/
%R 10.37236/12367
%F 10_37236_12367
Johannes Carmesin; Jan Kurkofka. Characterising 4-tangles through a connectivity property. The electronic journal of combinatorics, Tome 32 (2025) no. 3. doi: 10.37236/12367
