rank-width

functionally equivalent to: clique-width, boolean width, NLC-width, module-width, radius-inf flip-width

providers: ISGCI

Relations

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

Results

2023 Flip-width: Cops and Robber on dense graphs by Toruńczyk
- radius-inf flip-width upper bounds rank-width by a linear function – For every graph $G$, we have $\mathrm{rankwidth}(G) \le 3 \mathrm{fw}_\infty(G)+1$ …
- rank-width upper bounds radius-inf flip-width by an exponential function – For every graph $G$, we have … $3 \mathrm{fw}_\infty(G)+1 \le O(2^{\mathrm{rankwidth(G)}})$.
2012 Twin-Cover: Beyond Vertex Cover in Parameterized Algorithmics by Ganian
- page 263 : twin-cover number upper bounds rank-width by a constant – The rank-width and linaer rank-width of graph of twin-cover $k$ are at most $k+1$.
2012 Classes of graphs with small rank decompositions are χ-bounded by Dvořák, Král’
- page 2 : rank-width upper bounds chi-bounded by a constant – Theorem 1. For any $k$, the class of graphs with rank-width at most $k$ is $\chi$-bounded.
2011 Boolean-width of graphs by Bui-Xuan, Telle, Vatshelle
- boolean width upper bounds rank-width by an exponential function – \textbf{Corollary 1.} For any graph $G$ and decomposition tree $(T,\gamma)$ of $G$ it holds that … $\log_2 rw(G) \le boolw(G)$ …
- rank-width upper bounds boolean width by a polynomial function – \textbf{Corollary 1.} For any graph $G$ and decomposition tree $(T,\gamma)$ of $G$ it holds that … $boolw(G) \le \frac 14 rw^2(G)+O(rw(G))$.
2010 The rank-width of the square grid by Jelínek
- page 2 : graph class grid is not constant rank-width – The grid $G_{n,n}$ has rank-width equal to $n-1$.
2006 Approximating clique-width and branch-width by Oum, Seymour
- page 9 : rank-width – … and the \emph{rank-width} $\mathrm{rwd}(G)$ of $G$ is the branch-width of $\mathrm{cutrk}_G$.
- page 9 : rank-width upper and lower bounds clique-width by an exponential function – Proposition 6.3
- page 9 : clique-width upper bounds rank-width by a linear function – Proposition 6.3
assumed
- rank-width is equivalent to rank-width – assumed
unknown source
- linear rank-width upper bounds rank-width by a computable function
- branch width upper bounds rank-width by a linear function