bounded expansion

functionally equivalent to: weak coloring number, weakly sparse and merge width, admissibility, strong coloring number

Definition: A graph class $C$ has bounded expansion if for every $r \in \mathbb N$, the family of $r$-shallow minors does not include the family of graphs with unbounded density ($|E(G)|/|V(G)|$).

Relations

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

Results

2024 Merge-width and First-Order Model Checking by Dreier, Toruńczyk
- page 7 : bounded expansion upper bounds merge-width by a computable function – Theorem 1.6. Graph classes of bounded expansion have bounded merge-width.
- page 7 : weakly sparse and merge width upper bounds bounded expansion by a computable function – Corollary 1.8. A graph class has bounded expansion if and only if it has bounded merge-width, and is weakly sparse (exclydes some biclique $K_{t,t}$ as a subgraph).
- page 7 : bounded expansion upper bounds weakly sparse and merge width by a computable function – Corollary 1.8. A graph class has bounded expansion if and only if it has bounded merge-width, and is weakly sparse (exclydes some biclique $K_{t,t}$ as a subgraph).
unknown source
- bounded expansion upper bounds degeneracy by a constant – $wcol_1-1 = col_1-1=adm_1=degeneracy$
- bounded expansion upper bounds weak coloring number by a computable function
- weak coloring number upper bounds bounded expansion by a computable function
- bounded expansion upper bounds strong coloring number by a computable function
- strong coloring number upper bounds bounded expansion by a computable function
- bounded expansion upper bounds admissibility by a computable function
- admissibility upper bounds bounded expansion by a computable function
- bounded expansion upper bounds nowhere dense by a constant – By definition
- excluded top-minor upper bounds bounded expansion by a constant
- sparse twin-width upper bounds bounded expansion by a linear function
assumed
- sparse twin-width upper bounds bounded expansion by a linear function – by definition
- bounded expansion is equivalent to bounded expansion – assumed