tree

Definitions:

an intersetcion of connected, and forest
A connected graph without cycles.
A connected graph with $n$ vertices and $n-1$ edges.

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	magenta■	exclusion	unknown to HOPS
bandwidth	blue■	exclusion	unbounded
bipartite	red■	exclusion	upper bound
bipartite number	blue■	exclusion	unbounded
bisection bandwidth	magenta■	exclusion	unknown to HOPS
block	red■	exclusion	upper bound
book thickness	red■	exclusion	upper bound
boolean width	red■	exclusion	upper bound
bounded components	blue■	exclusion	unbounded
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	blue■	exclusion	unbounded
chi-bounded	red■	exclusion	upper bound
chordal	red■	exclusion	upper bound
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	blue■	exclusion	unbounded
cutwidth	blue■	exclusion	unbounded
cycle	cyan■	unknown to HOPS	exclusion
cycles	blue■	exclusion	exclusion
d-admissibility	red■	exclusion	upper bound
d-path-free	blue■	exclusion	unbounded
degeneracy	red■	exclusion	upper bound
degree treewidth	blue■	exclusion	unbounded
diameter	blue■	exclusion	unbounded
diameter+max degree	blue■	exclusion	unbounded
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 edgeless	blue■	exclusion	unbounded
distance to forest	red■	exclusion	upper bound
distance to interval	magenta■	exclusion	unknown to HOPS
distance to linear forest	blue■	exclusion	unbounded
distance to maximum degree	blue■	exclusion	unbounded
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
domino treewidth	blue■	exclusion	unbounded
edge clique cover number	blue■	exclusion	unbounded
edge connectivity	red■	exclusion	upper bound
edge-cut width	red■	exclusion	upper bound
edge-treewidth	red■	exclusion	upper bound
edgeless	blue■	exclusion	avoids
excluded minor	red■	exclusion	upper bound
excluded planar minor	gray■	unknown to HOPS	unknown to HOPS
excluded top-minor	red■	exclusion	upper bound
feedback edge set	red■	exclusion	upper bound
feedback vertex set	red■	exclusion	upper bound
flip-width	red■	exclusion	upper bound
forest	red■	exclusion	upper bound
genus	red■	exclusion	upper bound
grid	blue■	exclusion	exclusion
h-index	blue■	exclusion	unbounded
interval	magenta■	exclusion	unknown to HOPS
iterated type partitions	blue■	exclusion	unbounded
linear clique-width	magenta■	exclusion	unknown to HOPS
linear forest	blue■	exclusion	exclusion
linear NLC-width	magenta■	exclusion	unknown to HOPS
linear rank-width	magenta■	exclusion	unknown to HOPS
maximum clique	red■	exclusion	upper bound
maximum degree	blue■	exclusion	unbounded
maximum independent set	blue■	exclusion	unbounded
maximum induced matching	blue■	exclusion	unbounded
maximum leaf number	blue■	exclusion	unbounded
maximum matching	blue■	exclusion	unbounded
maximum matching on bipartite graphs	blue■	exclusion	unbounded
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	blue■	exclusion	unbounded
module-width	red■	exclusion	upper bound
monadically dependent	red■	exclusion	upper bound
monadically stable	red■	exclusion	upper bound
neighborhood diversity	blue■	exclusion	unbounded
NLC-width	red■	exclusion	upper bound
NLCT-width	red■	exclusion	upper bound
nowhere dense	red■	exclusion	upper bound
odd cycle transversal	red■	exclusion	upper bound
outerplanar	gray■	unknown to HOPS	unknown to HOPS
overlap treewidth	red■	exclusion	upper bound
path	green■	inclusion	exclusion
pathwidth	blue■	exclusion	unbounded
pathwidth+maxdegree	blue■	exclusion	unbounded
perfect	red■	exclusion	upper bound
planar	magenta■	exclusion	unknown to HOPS
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	magenta■	exclusion	unknown to HOPS
sim-width	red■	exclusion	upper bound
size	blue■	exclusion	unbounded
slim tree-cut width	red■	exclusion	upper bound
sparse twin-width	red■	exclusion	upper bound
star	green■	inclusion	exclusion
stars	blue■	exclusion	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	blue■	exclusion	unbounded
tree	yellow■	equal	equal
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	blue■	exclusion	unbounded
treelength	magenta■	exclusion	unknown to HOPS
treespan	blue■	exclusion	unbounded
treewidth	red■	exclusion	upper bound
twin-cover number	blue■	exclusion	unbounded
twin-width	red■	exclusion	upper bound
vertex connectivity	magenta■	exclusion	unknown to HOPS
vertex cover	blue■	exclusion	unbounded
vertex integrity	blue■	exclusion	unbounded
weak coloring number	red■	exclusion	upper bound
weak d-coloring number	red■	exclusion	upper bound
weak inf-coloring number	blue■	exclusion	unbounded
weakly sparse	red■	exclusion	upper bound
weakly sparse and merge width	red■	exclusion	upper bound

Results

2010 Comparing 17 graph parameters by Sasák
- page 16 : graph class tree is not constant pathwidth – Theorem 2.1
assumed
- tree upper bounds connected by a constant – by definition
- tree upper bounds forest by a constant – by definition
- graph class star has contained in tree
- graph class tree does not have contained in star
- graph class path has contained in tree
- graph class tree does not have contained in path
- tree is equivalent to tree – assumed
unknown source
- graph class tree is not constant h-index – trivially