clique-width

abbr: cw

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

providers: ISGCI

Definition: Minimum number of labels (colors) required to construct the graph using the following operations for constructing labeled graphs: 1) create a new labeled vertex, 2) disjoint union, 3) complete join between two labels, and 4) change all vertices from one to another label.

Clique-width is a generalization of treewidth to dense graphs. There are quite a few parameters that are functionally equivalent to cilque-width; notably rank-width which has a better bound for graphs of bounded treewidth.

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■	equal	equal
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■	tight bounds	upper bound
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

2022 Expanding the Graph Parameter Hierarchy by Tran
- page 25 : modular-width upper bounds clique-width by a computable function – Proposition 4.6. Modular-width strictly upper bounds Clique-width.
- page 25 : graph classes with bounded clique-width are not bounded modular-width – Proposition 4.6. Modular-width strictly upper bounds Clique-width.
- page 36 : clique-width upper bounds twin-width by a tower function – Proposition 6.2. Clique-width strictly upper bounds Twin-width.
- page 36 : graph classes with bounded twin-width are not bounded clique-width – Proposition 6.2. Clique-width strictly upper bounds Twin-width.
2019 The Graph Parameter Hierarchy by Sorge
- page 9 : distance to cograph upper bounds clique-width by an exponential function – Lemma 4.17. The distance $c$ to a cograph upper bounds the cliquewidth $q$. We have $q \le 2^{3+c}-1$.
2012 Twin-Cover: Beyond Vertex Cover in Parameterized Algorithmics by Ganian
- page 263 : twin-cover number upper bounds clique-width by a constant – The clique-width of graphs of twin-cover $k$ is at most $k+2$.
2006 Approximating clique-width and branch-width by Oum, Seymour
- 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
2005 On the relationship between NLC-width and linear NLC-width by Gurski, Wanke
- page 8 : clique-tree-width upper bounds clique-width by a linear function
2000 Upper bounds to the clique width of graphs by Courcelle, Olariu
- page 18 : treewidth upper bounds clique-width by an exponential function – We will prove that for every undirected graph $G$, $cwd(G) \le 2^{twd(G)+1}+1$ …
1998 Clique-decomposition, NLC-decomposition and modular decomposition—relationships and results for random graphs by Johansson
- clique-width upper bounds NLC-width by a linear function
- NLC-width upper bounds clique-width by a linear function
Comparing Graph Parameters by Schröder
- page 16 : graph classes with bounded clique cover number are not bounded clique-width – Proposition 3.9
- page 23 : graph classes with bounded genus are not bounded clique-width – Proposition 3.17
- page 23 : graph classes with bounded distance to planar are not bounded clique-width – Proposition 3.17
- page 28 : graph classes with bounded maximum degree are not bounded clique-width – Proposition 3.26
- page 28 : graph classes with bounded distance to bipartite are not bounded clique-width – Proposition 3.26
- page 33 : graph classes with bounded bisection bandwidth are not bounded clique-width – Proposition 3.32
assumed
- clique-width is equivalent to clique-width – assumed
unknown source
- linear clique-width upper bounds clique-width by a linear function
- clique-width upper bounds boolean width by a linear function
- boolean width upper bounds clique-width by an exponential function
- module-width upper bounds clique-width by a computable function
- clique-width upper bounds module-width by a computable function
- modular-width upper bounds clique-width by a computable function
- clique-width upper bounds mim-width by a linear function
- clique-width upper bounds twin-width by a tower function
- clique-width upper bounds chi-bounded by a constant