girth

Relations

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

Results

assumed
- girth is equivalent to girth – assumed
unknown source
- average distance upper bounds girth by a computable function – Small average distance implies a small cycle while adding a triangle makes the girth constant and minimally changes the average distance.
- bounded girth does not imply bounded average distance – Small average distance implies a small cycle while adding a triangle makes the girth constant and minimally changes the average distance.
- graph class bipartite has unbounded girth
- graph class forest has unbounded girth
- graph class disjoint cycles has unbounded girth
- graph class planar has unbounded girth
- graph class cycle has unbounded girth
Comparing Graph Parameters by Schröder
- page 21 : bounded maximum leaf number does not imply bounded girth – Proposition 3.14