linear forest

Definition: A disjoint union of paths.

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	blue■	exclusion	unbounded
bandwidth	magenta■	exclusion	unknown to HOPS
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	red■	exclusion	upper bound
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	blue■	exclusion	avoids
contraction complexity	red■	exclusion	upper bound
cutwidth	red■	exclusion	upper bound
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	red■	exclusion	upper bound
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	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
domino treewidth	red■	exclusion	upper bound
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	green■	inclusion	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	inclusion
genus	red■	exclusion	upper bound
grid	blue■	exclusion	exclusion
h-index	red■	exclusion	upper bound
interval	red■	exclusion	inclusion
iterated type partitions	blue■	exclusion	unbounded
linear clique-width	red■	exclusion	upper bound
linear forest	yellow■	equal	equal
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	magenta■	exclusion	unknown to HOPS
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■	upper bound	exclusion
pathwidth	red■	exclusion	upper bound
pathwidth+maxdegree	red■	exclusion	upper bound
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	blue■	exclusion	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	magenta■	exclusion	unknown to HOPS
tree	blue■	exclusion	exclusion
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	red■	exclusion	upper bound
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

unknown source
- graph class linear forest is not constant average distance
assumed
- path upper bounds linear forest by a constant – by definition
- graph class linear forest has contained in interval
- graph class interval does not have contained in linear forest
- graph class linear forest has contained in forest
- graph class forest does not have contained in linear forest
- graph classes that are edgeless have contained in linear forest
- graph class linear forest is not edgeless
- linear forest upper bounds distance to linear forest by a constant – by definition
- linear forest upper bounds maximum degree by a constant – By definition
- linear forest is equivalent to linear forest – assumed