complete

providers: ISGCI

Definitions:

an intersetcion of connected, and cluster
Contains all the edges.

Relations

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

Results

2022 Expanding the Graph Parameter Hierarchy by Tran
- page 18 : graph class complete is not constant vertex cover – Note that a clique of size $n$ has … a vertex cover number of $n-1$
- page 18 : complete upper bounds twin-cover number by a constant – Note that a clique of size $n$ has a twin cover number of 0 …
2017 Graph Theory by Diestel
- page 3 : complete – If all the vertices of $G$ are pairwise adjacent, then $G$ is \emph{complete}.
2012 Twin-Cover: Beyond Vertex Cover in Parameterized Algorithmics by Ganian
- page 263 : complete upper bounds twin-cover number by a constant – We note that complete graphs indeed have a twin-cover of zero.
2010 Comparing 17 graph parameters by Sasák
- page 17 : graph class complete is not constant treewidth – Theorem 2.2
1993 The Pathwidth and Treewidth of Cographs by Bodlaender, Möhring
- page 4 : graph class complete is not constant treewidth – Lemma 3.1 (“clique containment lemma”). Let $({X_i\mid u\in I},T=(I,F))$ be a tree-decomposition of $G=(V,E)$ and let $W \subseteq V$ be a clique in $G$. Then there exists $i \in I$ with $W \subseteq X_i$.
assumed
- complete upper bounds connected by a constant – by definition
- complete upper bounds cluster by a constant – by definition
- graph class complete has contained in co-cluster
- graph class co-cluster does not have contained in complete
- complete upper bounds distance to complete by a constant – by definition
- complete is equivalent to complete – assumed
unknown source
- graph class complete is not constant maximum clique – Parameter is unbounded for the graph class of cliques.
- graph class complete is not constant domatic number – Parameter is unbounded for the graph class of cliques.
- graph class complete is not constant edge connectivity – Parameter is unbounded for the graph class of cliques.
- graph class complete is not constant maximum matching