connected

Definition: There is a path between any pair of vertices.

Relations

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

Results

assumed
- complete upper bounds connected by a constant – by definition
- tree upper bounds connected by a constant – by definition
- path upper bounds connected by a constant – by definition
- star upper bounds connected by a constant – by definition
- cycle upper bounds connected by a constant – by definition
- graph class grid is connected
- graph classes that are connected do not have contained in grid
- graph classes that are edgeless are not connected – By definition
- graph classes with bounded size are not connected – By definition
- connected is equivalent to connected – assumed