tree-independence number

Relations

Other		Relation from	Relation to
acyclic chromatic number	cyan■	unknown to HOPS	exclusion
admissibility	cyan■	unknown to HOPS	exclusion
arboricity	cyan■	unknown to HOPS	exclusion
average degree	cyan■	unknown to HOPS	exclusion
average distance	cyan■	unknown to HOPS	exclusion
bandwidth	green■	upper bound	exclusion
bipartite	cyan■	unknown to HOPS	exclusion
bipartite number	cyan■	unknown to HOPS	exclusion
bisection bandwidth	cyan■	unknown to HOPS	exclusion
block	green■	upper bound	exclusion
book thickness	cyan■	unknown to HOPS	exclusion
boolean width	cyan■	unknown to HOPS	exclusion
bounded components	green■	upper bound	exclusion
bounded expansion	cyan■	unknown to HOPS	avoids
boxicity	cyan■	unknown to HOPS	exclusion
branch width	green■	upper bound	exclusion
c-closure	cyan■	unknown to HOPS	exclusion
carving-width	green■	upper bound	exclusion
chi-bounded	gray■	unknown to HOPS	unknown to HOPS
chordal	green■	upper bound	exclusion
chordality	gray■	unknown to HOPS	unknown to HOPS
chromatic number	cyan■	unknown to HOPS	exclusion
clique cover number	cyan■	unknown to HOPS	exclusion
clique-tree-width	cyan■	unknown to HOPS	exclusion
clique-width	cyan■	unknown to HOPS	exclusion
cluster	green■	upper bound	exclusion
co-cluster	cyan■	unknown to HOPS	exclusion
cograph	cyan■	unknown to HOPS	exclusion
complete	green■	upper bound	exclusion
connected	cyan■	unknown to HOPS	avoids
contraction complexity	green■	upper bound	exclusion
cutwidth	green■	upper bound	exclusion
cycle	green■	upper bound	exclusion
cycles	green■	upper bound	exclusion
d-admissibility	gray■	unknown to HOPS	unknown to HOPS
d-path-free	green■	upper bound	exclusion
degeneracy	cyan■	unknown to HOPS	exclusion
degree treewidth	green■	upper bound	exclusion
diameter	cyan■	unknown to HOPS	exclusion
diameter+max degree	green■	upper bound	exclusion
distance to bipartite	cyan■	unknown to HOPS	exclusion
distance to block	green■	upper bound	exclusion
distance to bounded components	green■	upper bound	exclusion
distance to chordal	green■	upper bound	exclusion
distance to cluster	green■	upper bound	exclusion
distance to co-cluster	cyan■	unknown to HOPS	exclusion
distance to cograph	cyan■	unknown to HOPS	exclusion
distance to complete	green■	upper bound	exclusion
distance to edgeless	green■	upper bound	exclusion
distance to forest	green■	upper bound	exclusion
distance to interval	green■	upper bound	exclusion
distance to linear forest	green■	upper bound	exclusion
distance to maximum degree	cyan■	unknown to HOPS	exclusion
distance to outerplanar	green■	upper bound	exclusion
distance to perfect	cyan■	unknown to HOPS	exclusion
distance to planar	cyan■	unknown to HOPS	exclusion
distance to stars	green■	upper bound	exclusion
domatic number	cyan■	unknown to HOPS	exclusion
domination number	cyan■	unknown to HOPS	exclusion
domino treewidth	green■	upper bound	exclusion
edge clique cover number	cyan■	unknown to HOPS	exclusion
edge connectivity	cyan■	unknown to HOPS	exclusion
edge-cut width	green■	upper bound	exclusion
edge-treewidth	green■	upper bound	exclusion
edgeless	green■	upper bound	avoids
excluded minor	cyan■	unknown to HOPS	avoids
excluded planar minor	green■	upper bound	avoids
excluded top-minor	cyan■	unknown to HOPS	avoids
feedback edge set	green■	upper bound	exclusion
feedback vertex set	green■	upper bound	exclusion
flip-width	gray■	unknown to HOPS	unknown to HOPS
forest	green■	upper bound	exclusion
genus	cyan■	unknown to HOPS	exclusion
grid	cyan■	unknown to HOPS	exclusion
h-index	cyan■	unknown to HOPS	exclusion
interval	green■	upper bound	exclusion
iterated type partitions	cyan■	unknown to HOPS	exclusion
linear clique-width	cyan■	unknown to HOPS	exclusion
linear forest	green■	upper bound	exclusion
linear NLC-width	cyan■	unknown to HOPS	exclusion
linear rank-width	cyan■	unknown to HOPS	exclusion
maximum clique	cyan■	unknown to HOPS	exclusion
maximum degree	cyan■	unknown to HOPS	exclusion
maximum independent set	cyan■	unknown to HOPS	exclusion
maximum induced matching	cyan■	unknown to HOPS	exclusion
maximum leaf number	green■	upper bound	exclusion
maximum matching	green■	upper bound	exclusion
maximum matching on bipartite graphs	green■	upper bound	exclusion
merge-width	gray■	unknown to HOPS	unknown to HOPS
mim-width	gray■	unknown to HOPS	unknown to HOPS
minimum degree	cyan■	unknown to HOPS	exclusion
mm-width	green■	upper bound	exclusion
modular-width	cyan■	unknown to HOPS	exclusion
module-width	cyan■	unknown to HOPS	exclusion
monadically dependent	gray■	unknown to HOPS	unknown to HOPS
monadically stable	gray■	unknown to HOPS	unknown to HOPS
neighborhood diversity	cyan■	unknown to HOPS	exclusion
NLC-width	cyan■	unknown to HOPS	exclusion
NLCT-width	cyan■	unknown to HOPS	exclusion
nowhere dense	gray■	unknown to HOPS	unknown to HOPS
odd cycle transversal	cyan■	unknown to HOPS	exclusion
outerplanar	green■	upper bound	exclusion
overlap treewidth	green■	upper bound	exclusion
path	green■	upper bound	exclusion
pathwidth	green■	upper bound	exclusion
pathwidth+maxdegree	green■	upper bound	exclusion
perfect	cyan■	unknown to HOPS	exclusion
planar	cyan■	unknown to HOPS	exclusion
radius-inf flip-width	cyan■	unknown to HOPS	exclusion
radius-r flip-width	gray■	unknown to HOPS	unknown to HOPS
rank-width	cyan■	unknown to HOPS	exclusion
series-parallel	gray■	unknown to HOPS	unknown to HOPS
shrub-depth	cyan■	unknown to HOPS	exclusion
sim-width	orange■	unknown to HOPS	upper bound
size	green■	upper bound	exclusion
slim tree-cut width	green■	upper bound	exclusion
sparse twin-width	cyan■	unknown to HOPS	exclusion
star	green■	upper bound	exclusion
stars	green■	upper bound	exclusion
strong coloring number	cyan■	unknown to HOPS	exclusion
strong d-coloring number	gray■	unknown to HOPS	unknown to HOPS
strong inf-coloring number	green■	upper bound	exclusion
topological bandwidth	green■	upper bound	exclusion
tree	green■	upper bound	exclusion
tree-cut width	green■	upper bound	exclusion
tree-independence number	yellow■	equal	equal
tree-partition-width	green■	upper bound	exclusion
treebandwidth	green■	upper bound	exclusion
treedepth	green■	upper bound	exclusion
treelength	gray■	unknown to HOPS	unknown to HOPS
treespan	green■	upper bound	exclusion
treewidth	green■	upper bound	exclusion
twin-cover number	green■	upper bound	exclusion
twin-width	cyan■	unknown to HOPS	exclusion
vertex connectivity	gray■	unknown to HOPS	unknown to HOPS
vertex cover	green■	upper bound	exclusion
vertex integrity	green■	upper bound	exclusion
weak coloring number	cyan■	unknown to HOPS	exclusion
weak d-coloring number	gray■	unknown to HOPS	unknown to HOPS
weak inf-coloring number	green■	upper bound	exclusion
weakly sparse	gray■	unknown to HOPS	unknown to HOPS
weakly sparse and merge width	cyan■	unknown to HOPS	exclusion

Results

unknown source
- treewidth upper bounds tree-independence number by a computable function
- tree-independence number upper bounds sim-width by a computable function
- distance to chordal upper bounds tree-independence number by a linear function – Put the modulator to every bag of the natural chodal graph tree decomposition which contains a clique in every bag. The biggest independent set can contain the modulator and no more than a single vertex of the clique.
assumed
- tree-independence number is equivalent to tree-independence number – assumed