bounded components

functionally equivalent to: diameter+max degree

Relations

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

Results

unknown source
- bounded components upper bounds cutwidth by a linear function – By greedily placing one component after another.
- graph classes with bounded bounded components are not bounded distance to perfect – By a disjoint union of small components with distance to perfect at least 1.
- graph classes with bounded bounded components are not bounded distance to planar – By a disjoint union of many $K_5$ graphs.
assumed
- bounded components upper bounds distance to bounded components by a linear function – by definition
- edgeless upper bounds bounded components by a constant – By definition
- bounded components upper bounds maximum degree by a linear function – By definition
- diameter+max degree upper bounds bounded components by an exponential function – folklore
- size upper bounds bounded components by a linear function – By definition
- bounded components is equivalent to bounded components – assumed