forest

providers: ISGCI

Definition: A graph with no cycle.

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	blue■	exclusion	unbounded
bipartite	red■	exclusion	inclusion
bipartite number	magenta■	exclusion	unknown to HOPS
bisection bandwidth	red■	exclusion	upper bound
block	red■	exclusion	inclusion
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	blue■	exclusion	unbounded
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	blue■	exclusion	exclusion
contraction complexity	blue■	exclusion	unbounded
cutwidth	blue■	exclusion	unbounded
cycle	cyan■	unknown to HOPS	exclusion
cycles	blue■	exclusion	exclusion
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
disconnected	gray■	unknown to HOPS	unknown to HOPS
disjoint cycles	red■	exclusion	inclusion
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	blue■	exclusion	unbounded
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
edge clique cover number	blue■	exclusion	unbounded
edge connectivity	red■	exclusion	upper bound
edgeless	green■	upper bound	exclusion
feedback edge set	red■	exclusion	upper bound
feedback vertex set	red■	exclusion	upper bound
forest	yellow■	equal	equal
genus	red■	exclusion	upper bound
girth	blue■	exclusion	unbounded
grid	blue■	exclusion	exclusion
h-index	blue■	exclusion	unbounded
inf-flip-width	red■	exclusion	upper bound
interval	blue■	exclusion	exclusion
iterated type partitions	blue■	exclusion	unbounded
linear clique-width	magenta■	exclusion	unknown to HOPS
linear forest	green■	inclusion	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	cyan■	unknown to HOPS	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	green■	upper bound	exclusion
pathwidth	blue■	exclusion	unbounded
pathwidth+maxdegree	blue■	exclusion	unbounded
perfect	red■	exclusion	upper bound
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	green■	upper bound	exclusion
stars	green■	inclusion	exclusion
topological bandwidth	blue■	exclusion	unbounded
tree	green■	upper bound	exclusion
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

2017 Graph Theory by Diestel
- page 13 : forest – An acyclic graph, one not containing any cycles, is called a \emph{forest}.
assumed
- tree upper bounds forest by a constant – by definition
- graph class forest is included in graph class bipartite
- graph class bipartite is not included in graph class forest
- graph class linear forest is included in graph class forest
- graph class forest is not included in graph class linear forest
- graph class forest is included in graph class disjoint cycles
- graph class disjoint cycles is not included in graph class forest
- graph class forest is included in graph class block
- graph class block is not included in graph class forest
- graph class stars is included in graph class forest
- graph class forest is not included in graph class stars
- forest upper bounds distance to forest by a constant – by definition
- forest is equivalent to forest – assumed
unknown source
- forest upper bounds feedback edge set by a constant
- graph class forest has unbounded girth
- graph class forest has unbounded distance to interval