nowhere dense

Definition: A graph class $C$ is nowhere dense if for every $r \in \mathbb N$, the family of $r$-shallow minors does not include the family of all cliques.

Relations

Other		Relation from	Relation to
acyclic chromatic number	gray■	unknown to HOPS	unknown to HOPS
admissibility	lime■	upper bound	unknown to HOPS
arboricity	gray■	unknown to HOPS	unknown to HOPS
average degree	gray■	unknown to HOPS	unknown to HOPS
average distance	cyan■	unknown to HOPS	exclusion
bandwidth	green■	upper bound	exclusion
bipartite	cyan■	unknown to HOPS	exclusion
bipartite number	cyan■	unknown to HOPS	exclusion
bisection bandwidth	cyan■	unknown to HOPS	exclusion
block	cyan■	unknown to HOPS	exclusion
book thickness	gray■	unknown to HOPS	unknown to HOPS
boolean width	cyan■	unknown to HOPS	exclusion
bounded components	green■	upper bound	exclusion
bounded expansion	lime■	upper bound	unknown to HOPS
boxicity	gray■	unknown to HOPS	unknown to HOPS
branch width	green■	upper bound	exclusion
c-closure	cyan■	unknown to HOPS	exclusion
carving-width	green■	upper bound	exclusion
chi-bounded	gray■	unknown to HOPS	unknown to HOPS
chordal	cyan■	unknown to HOPS	exclusion
chordality	gray■	unknown to HOPS	unknown to HOPS
chromatic number	gray■	unknown to HOPS	unknown to HOPS
clique cover number	cyan■	unknown to HOPS	exclusion
clique-tree-width	cyan■	unknown to HOPS	exclusion
clique-width	cyan■	unknown to HOPS	exclusion
cluster	cyan■	unknown to HOPS	exclusion
co-cluster	cyan■	unknown to HOPS	exclusion
cograph	cyan■	unknown to HOPS	exclusion
complete	cyan■	unknown to HOPS	exclusion
connected	cyan■	unknown to HOPS	avoids
contraction complexity	green■	upper bound	exclusion
cutwidth	green■	upper bound	exclusion
cycle	green■	upper bound	exclusion
cycles	green■	upper bound	exclusion
d-admissibility	gray■	unknown to HOPS	unknown to HOPS
d-path-free	green■	upper bound	exclusion
degeneracy	gray■	unknown to HOPS	unknown to HOPS
degree treewidth	green■	upper bound	exclusion
diameter	cyan■	unknown to HOPS	exclusion
diameter+max degree	green■	upper bound	exclusion
distance to bipartite	cyan■	unknown to HOPS	exclusion
distance to block	cyan■	unknown to HOPS	exclusion
distance to bounded components	green■	upper bound	exclusion
distance to chordal	cyan■	unknown to HOPS	exclusion
distance to cluster	cyan■	unknown to HOPS	exclusion
distance to co-cluster	cyan■	unknown to HOPS	exclusion
distance to cograph	cyan■	unknown to HOPS	exclusion
distance to complete	cyan■	unknown to HOPS	exclusion
distance to edgeless	green■	upper bound	exclusion
distance to forest	green■	upper bound	exclusion
distance to interval	cyan■	unknown to HOPS	exclusion
distance to linear forest	green■	upper bound	exclusion
distance to maximum degree	cyan■	unknown to HOPS	exclusion
distance to outerplanar	green■	upper bound	exclusion
distance to perfect	cyan■	unknown to HOPS	exclusion
distance to planar	green■	upper bound	exclusion
distance to stars	green■	upper bound	exclusion
domatic number	gray■	unknown to HOPS	unknown to HOPS
domination number	cyan■	unknown to HOPS	exclusion
domino treewidth	green■	upper bound	exclusion
edge clique cover number	cyan■	unknown to HOPS	exclusion
edge connectivity	gray■	unknown to HOPS	unknown to HOPS
edge-cut width	green■	upper bound	exclusion
edge-treewidth	green■	upper bound	exclusion
edgeless	green■	upper bound	avoids
excluded minor	lime■	upper bound	unknown to HOPS
excluded planar minor	green■	upper bound	avoids
excluded top-minor	lime■	upper bound	unknown to HOPS
feedback edge set	green■	upper bound	exclusion
feedback vertex set	green■	upper bound	exclusion
flip-width	gray■	unknown to HOPS	unknown to HOPS
forest	green■	upper bound	exclusion
genus	green■	upper bound	exclusion
grid	green■	upper bound	exclusion
h-index	cyan■	unknown to HOPS	exclusion
interval	cyan■	unknown to HOPS	exclusion
iterated type partitions	cyan■	unknown to HOPS	exclusion
linear clique-width	cyan■	unknown to HOPS	exclusion
linear forest	green■	upper bound	exclusion
linear NLC-width	cyan■	unknown to HOPS	exclusion
linear rank-width	cyan■	unknown to HOPS	exclusion
maximum clique	gray■	unknown to HOPS	unknown to HOPS
maximum degree	green■	upper bound	exclusion
maximum independent set	cyan■	unknown to HOPS	exclusion
maximum induced matching	cyan■	unknown to HOPS	exclusion
maximum leaf number	green■	upper bound	exclusion
maximum matching	green■	upper bound	exclusion
maximum matching on bipartite graphs	green■	upper bound	exclusion
merge-width	gray■	unknown to HOPS	unknown to HOPS
mim-width	gray■	unknown to HOPS	unknown to HOPS
minimum degree	gray■	unknown to HOPS	unknown to HOPS
mm-width	green■	upper bound	exclusion
modular-width	cyan■	unknown to HOPS	exclusion
module-width	cyan■	unknown to HOPS	exclusion
monadically dependent	orange■	unknown to HOPS	upper bound
monadically stable	orange■	unknown to HOPS	upper bound
neighborhood diversity	cyan■	unknown to HOPS	exclusion
NLC-width	cyan■	unknown to HOPS	exclusion
NLCT-width	cyan■	unknown to HOPS	exclusion
nowhere dense	yellow■	equal	equal
odd cycle transversal	cyan■	unknown to HOPS	exclusion
outerplanar	green■	upper bound	exclusion
overlap treewidth	green■	upper bound	exclusion
path	green■	upper bound	exclusion
pathwidth	green■	upper bound	exclusion
pathwidth+maxdegree	green■	upper bound	exclusion
perfect	cyan■	unknown to HOPS	exclusion
planar	green■	upper bound	exclusion
radius-inf flip-width	cyan■	unknown to HOPS	exclusion
radius-r flip-width	gray■	unknown to HOPS	unknown to HOPS
rank-width	cyan■	unknown to HOPS	exclusion
series-parallel	gray■	unknown to HOPS	unknown to HOPS
shrub-depth	cyan■	unknown to HOPS	exclusion
sim-width	gray■	unknown to HOPS	unknown to HOPS
size	green■	upper bound	exclusion
slim tree-cut width	green■	upper bound	exclusion
sparse twin-width	green■	upper bound	exclusion
star	green■	upper bound	exclusion
stars	green■	upper bound	exclusion
strong coloring number	lime■	upper bound	unknown to HOPS
strong d-coloring number	gray■	unknown to HOPS	unknown to HOPS
strong inf-coloring number	green■	upper bound	exclusion
topological bandwidth	green■	upper bound	exclusion
tree	green■	upper bound	exclusion
tree-cut width	green■	upper bound	exclusion
tree-independence number	gray■	unknown to HOPS	unknown to HOPS
tree-partition-width	green■	upper bound	exclusion
treebandwidth	green■	upper bound	exclusion
treedepth	green■	upper bound	exclusion
treelength	gray■	unknown to HOPS	unknown to HOPS
treespan	green■	upper bound	exclusion
treewidth	green■	upper bound	exclusion
twin-cover number	cyan■	unknown to HOPS	exclusion
twin-width	cyan■	unknown to HOPS	exclusion
vertex connectivity	gray■	unknown to HOPS	unknown to HOPS
vertex cover	green■	upper bound	exclusion
vertex integrity	green■	upper bound	exclusion
weak coloring number	lime■	upper bound	unknown to HOPS
weak d-coloring number	gray■	unknown to HOPS	unknown to HOPS
weak inf-coloring number	green■	upper bound	exclusion
weakly sparse	orange■	unknown to HOPS	upper bound
weakly sparse and merge width	lime■	upper bound	unknown to HOPS

Results

assumed
- nowhere dense is equivalent to nowhere dense – assumed
unknown source
- bounded expansion upper bounds nowhere dense by a constant – By definition
- nowhere dense upper bounds weakly sparse by a constant
- nowhere dense upper bounds monadically stable by a constant