cograph

providers: ISGCI

Definitions:

Can be created from single vertices by disjoint unions and complements.
$P_4$-free

Relations

Other		Relation from	Relation to
acyclic chromatic number	blue■	exclusion	unbounded
arboricity	blue■	exclusion	unbounded
average degree	blue■	exclusion	unbounded
average distance	red■	exclusion	upper bound
bandwidth	blue■	exclusion	unbounded
bipartite	blue■	exclusion	exclusion
bipartite number	red■	exclusion	upper bound
bisection bandwidth	blue■	exclusion	unbounded
block	blue■	exclusion	exclusion
book thickness	blue■	exclusion	unbounded
boolean width	red■	exclusion	upper bound
bounded components	blue■	exclusion	unbounded
boxicity	magenta■	exclusion	unknown to HOPS
branch width	blue■	exclusion	unbounded
c-closure	magenta■	exclusion	unknown to HOPS
carving-width	blue■	exclusion	unbounded
chordal	blue■	exclusion	exclusion
chordality	red■	exclusion	upper bound
chromatic number	blue■	exclusion	unbounded
clique cover number	blue■	exclusion	unbounded
clique-tree-width	magenta■	exclusion	unknown to HOPS
clique-width	red■	exclusion	upper bound
cluster	green■	inclusion	exclusion
co-cluster	green■	inclusion	exclusion
cograph	yellow■	equal	equal
complete	green■	upper bound	exclusion
connected	blue■	exclusion	exclusion
contraction complexity	blue■	exclusion	unbounded
cutwidth	blue■	exclusion	unbounded
cycle	blue■	exclusion	exclusion
cycles	blue■	exclusion	exclusion
d-path-free	blue■	exclusion	unbounded
degeneracy	blue■	exclusion	unbounded
degree treewidth	blue■	exclusion	unbounded
diameter	red■	exclusion	upper bound
diameter+max degree	blue■	exclusion	unbounded
disconnected	gray■	unknown to HOPS	unknown to HOPS
disjoint cycles	blue■	exclusion	exclusion
distance to bipartite	blue■	exclusion	unbounded
distance to block	blue■	exclusion	unbounded
distance to bounded components	blue■	exclusion	unbounded
distance to chordal	blue■	exclusion	unbounded
distance to cluster	blue■	exclusion	unbounded
distance to co-cluster	magenta■	exclusion	unknown to HOPS
distance to cograph	red■	exclusion	upper bound
distance to complete	blue■	exclusion	unbounded
distance to disconnected	magenta■	exclusion	unknown to HOPS
distance to edgeless	blue■	exclusion	unbounded
distance to forest	blue■	exclusion	unbounded
distance to interval	blue■	exclusion	unbounded
distance to linear forest	blue■	exclusion	unbounded
distance to maximum degree	blue■	exclusion	unbounded
distance to outerplanar	blue■	exclusion	unbounded
distance to perfect	red■	exclusion	upper bound
distance to planar	blue■	exclusion	unbounded
distance to stars	blue■	exclusion	unbounded
domatic number	blue■	exclusion	unbounded
domination number	blue■	exclusion	unbounded
edge clique cover number	magenta■	exclusion	unknown to HOPS
edge connectivity	blue■	exclusion	unbounded
edgeless	green■	upper bound	exclusion
feedback edge set	blue■	exclusion	unbounded
feedback vertex set	blue■	exclusion	unbounded
forest	blue■	exclusion	exclusion
genus	blue■	exclusion	unbounded
girth	red■	exclusion	upper bound
grid	blue■	exclusion	exclusion
h-index	blue■	exclusion	unbounded
inf-flip-width	red■	exclusion	upper bound
interval	blue■	exclusion	exclusion
iterated type partitions	magenta■	exclusion	unknown to HOPS
linear clique-width	magenta■	exclusion	unknown to HOPS
linear forest	blue■	exclusion	exclusion
linear NLC-width	magenta■	exclusion	unknown to HOPS
linear rank-width	magenta■	exclusion	unknown to HOPS
maximum clique	blue■	exclusion	unbounded
maximum degree	blue■	exclusion	unbounded
maximum independent set	blue■	exclusion	unbounded
maximum induced matching	magenta■	exclusion	unknown to HOPS
maximum leaf number	blue■	exclusion	unbounded
maximum matching	blue■	exclusion	unbounded
maximum matching on bipartite graphs	cyan■	unknown to HOPS	unbounded
mim-width	red■	exclusion	upper bound
minimum degree	blue■	exclusion	unbounded
mm-width	blue■	exclusion	unbounded
modular-width	magenta■	exclusion	unknown to HOPS
module-width	red■	exclusion	upper bound
neighborhood diversity	magenta■	exclusion	unknown to HOPS
NLC-width	red■	exclusion	upper bound
NLCT-width	magenta■	exclusion	unknown to HOPS
odd cycle transversal	blue■	exclusion	unbounded
outerplanar	blue■	exclusion	exclusion
path	blue■	exclusion	exclusion
pathwidth	blue■	exclusion	unbounded
pathwidth+maxdegree	blue■	exclusion	unbounded
perfect	red■	exclusion	inclusion
planar	blue■	exclusion	exclusion
radius-r flip-width	red■	exclusion	upper bound
rank-width	red■	exclusion	upper bound
shrub-depth	magenta■	exclusion	unknown to HOPS
sim-width	red■	exclusion	upper bound
size	blue■	exclusion	unbounded
star	cyan■	unknown to HOPS	exclusion
stars	cyan■	unknown to HOPS	exclusion
topological bandwidth	blue■	exclusion	unbounded
tree	blue■	exclusion	exclusion
tree-independence number	magenta■	exclusion	unknown to HOPS
treedepth	blue■	exclusion	unbounded
treelength	red■	exclusion	upper bound
treewidth	blue■	exclusion	unbounded
twin-cover number	blue■	exclusion	unbounded
twin-width	red■	exclusion	upper bound
vertex connectivity	magenta■	exclusion	unknown to HOPS
vertex cover	blue■	exclusion	unbounded
vertex integrity	blue■	exclusion	unbounded

Results

1994 k-NLC graphs and polynomial algorithms by Wanke
- page 5 : cograph upper bounds NLC-width by a constant – Fact 2.3. $G$ is a $1$-NLC graph if and only if $unlab(G)$ is a co-graph.
assumed
- graph class cograph is included in graph class perfect
- graph class perfect is not included in graph class cograph
- graph class cluster is included in graph class cograph
- graph class cograph is not included in graph class cluster
- graph class co-cluster is included in graph class cograph
- graph class cograph is not included in graph class co-cluster
- cograph upper bounds distance to cograph by a constant – by definition
- cograph is equivalent to cograph – assumed