cycle

Definitions:

Has a cyclic vertex order such that the graph contains edges for each pair of vertices that are next to each other in the order.
A connected graph with all vertices having degree two.

Relations

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

Results

unknown source
- cycle upper bounds maximum leaf number by a constant
- graph class cycle has unbounded girth
assumed
- cycle upper bounds connected by a constant – by definition
- cycle upper bounds cycles by a constant – by definition
- cycle is equivalent to cycle – assumed