A graph in which every connected induced subgraph has a disconnected complement is called a cograph. Such graphs are precisely the graphs that do not have the 4-vertex path as an induced subgraph. We define a $2$-cograph to be a graph in which the complement of every $2$-connected induced subgraph is not $2$-connected. We show that, like cographs, $2$-cographs can be recursively defined and are closed under induced minors. We characterize the class of non-$2$-cographs for which every proper induced minor is a $2$-cograph. We further find the finitely many members of this class whose complements are also induced-minor-minimal non-$2$-cographs.
@article{10_37236_10272,
author = {James Oxley and Jagdeep Singh},
title = {Generalizing cographs to 2-cographs},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {1},
doi = {10.37236/10272},
zbl = {1512.05242},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10272/}
}
TY - JOUR
AU - James Oxley
AU - Jagdeep Singh
TI - Generalizing cographs to 2-cographs
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/10272/
DO - 10.37236/10272
ID - 10_37236_10272
ER -
%0 Journal Article
%A James Oxley
%A Jagdeep Singh
%T Generalizing cographs to 2-cographs
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/10272/
%R 10.37236/10272
%F 10_37236_10272
James Oxley; Jagdeep Singh. Generalizing cographs to 2-cographs. The electronic journal of combinatorics, Tome 30 (2023) no. 1. doi: 10.37236/10272
