complete

providers: ISGCI

Definition: Contains every edge.

Relations

Other		Relation from	Relation to
acyclic chromatic number	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
boxicity	red■	exclusion	upper bound
branch width	blue■	exclusion	unbounded
c-closure	magenta■	exclusion	unknown to HOPS
carving-width	blue■	exclusion	unbounded
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	blue■	exclusion	exclusion
cycles	blue■	exclusion	exclusion
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
disconnected	gray■	unknown to HOPS	unknown to HOPS
disjoint cycles	blue■	exclusion	exclusion
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 disconnected	magenta■	exclusion	unknown to HOPS
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
edge clique cover number	red■	exclusion	upper bound
edge connectivity	blue■	exclusion	unbounded
edgeless	blue■	exclusion	exclusion
feedback edge set	blue■	exclusion	unbounded
feedback vertex set	blue■	exclusion	unbounded
forest	blue■	exclusion	exclusion
genus	blue■	exclusion	unbounded
girth	red■	exclusion	upper bound
grid	blue■	exclusion	exclusion
h-index	blue■	exclusion	unbounded
inf-flip-width	red■	exclusion	upper bound
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
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
neighborhood diversity	red■	exclusion	upper bound
NLC-width	red■	exclusion	upper bound
NLCT-width	red■	exclusion	upper bound
odd cycle transversal	blue■	exclusion	unbounded
outerplanar	blue■	exclusion	exclusion
path	blue■	exclusion	exclusion
pathwidth	blue■	exclusion	unbounded
pathwidth+maxdegree	blue■	exclusion	unbounded
perfect	red■	exclusion	upper bound
planar	blue■	exclusion	exclusion
radius-r flip-width	red■	exclusion	upper bound
rank-width	red■	exclusion	upper bound
shrub-depth	red■	exclusion	upper bound
sim-width	red■	exclusion	upper bound
size	blue■	exclusion	unbounded
star	cyan■	unknown to HOPS	exclusion
stars	blue■	exclusion	exclusion
topological bandwidth	blue■	exclusion	unbounded
tree	blue■	exclusion	exclusion
tree-independence number	magenta■	exclusion	unknown to HOPS
treedepth	blue■	exclusion	unbounded
treelength	red■	exclusion	upper bound
treewidth	blue■	exclusion	unbounded
twin-cover number	red■	exclusion	upper bound
twin-width	red■	exclusion	upper bound
vertex connectivity	magenta■	exclusion	unknown to HOPS
vertex cover	blue■	exclusion	unbounded
vertex integrity	blue■	exclusion	unbounded

Results

2022 Expanding the Graph Parameter Hierarchy by Tran
- page 18 : graph class complete has unbounded 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 has unbounded treewidth – Theorem 2.2
1993 The Pathwidth and Treewidth of Cographs by Bodlaender, Möhring
- page 4 : graph class complete has unbounded 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 is included in graph class co-cluster
- graph class co-cluster is not included in graph class complete
- complete upper bounds distance to complete by a constant – by definition
- complete is equivalent to complete – assumed
unknown source
- graph class complete has unbounded maximum clique – Parameter is unbounded for the graph class of cliques.
- graph class complete has unbounded domatic number – Parameter is unbounded for the graph class of cliques.
- graph class complete has unbounded edge connectivity – Parameter is unbounded for the graph class of cliques.
- graph class complete has unbounded maximum matching