linear rank-width

equivalent to: linear NLC-width, linear clique-width

Relations

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

Results

2015 Linear rank-width and linear clique-width of trees by Adler, Kanté
- page 1 : linear rank-width upper bounds linear clique-width by a computable function – Linear rank-width is equivalent to linear clique-width in the sense that any graph class has bounded linear clique-width if and only if it has bounded linear rank-width.
- page 1 : linear clique-width upper bounds linear rank-width by a computable function – Linear rank-width is equivalent to linear clique-width in the sense that any graph class has bounded linear clique-width if and only if it has bounded linear rank-width.
- page 3 : pathwidth upper bounds linear rank-width by a computable function – Lemma 5. Any graph $G$ satisfies $\mathrm{lrw}(G) \le \mathrm{pw}(G)$.
2012 Twin-Cover: Beyond Vertex Cover in Parameterized Algorithmics by Ganian
- page 263 : twin-cover number upper bounds linear rank-width by a constant – The rank-width and linaer rank-width of graph of twin-cover $k$ are at most $k+1$.
unknown source
- linear rank-width upper bounds rank-width by a computable function
- pathwidth upper bounds linear rank-width by a computable function
assumed
- linear rank-width is equivalent to linear rank-width – assumed