c-closure

Definition: Minimum $c$ such that if vertices share at least $c$ neighbors, then they are adjacent.

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	cyan■	unknown to HOPS	exclusion
bipartite number	blue■	exclusion	exclusion
bisection bandwidth	blue■	exclusion	exclusion
block	cyan■	unknown to HOPS	exclusion
book thickness	blue■	exclusion	exclusion
boolean width	blue■	exclusion	exclusion
bounded components	green■	upper bound	exclusion
bounded expansion	blue■	exclusion	avoids
boxicity	blue■	exclusion	exclusion
branch width	blue■	exclusion	exclusion
c-closure	yellow■	equal	equal
carving-width	green■	upper bound	exclusion
chi-bounded	magenta■	exclusion	unknown to HOPS
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	blue■	exclusion	exclusion
clique-width	blue■	exclusion	exclusion
cluster	cyan■	unknown to HOPS	exclusion
co-cluster	cyan■	unknown to HOPS	exclusion
cograph	cyan■	unknown to HOPS	exclusion
complete	cyan■	unknown to HOPS	exclusion
connected	cyan■	unknown to HOPS	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	blue■	exclusion	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	blue■	exclusion	exclusion
distance to bounded components	blue■	exclusion	exclusion
distance to chordal	blue■	exclusion	exclusion
distance to cluster	blue■	exclusion	exclusion
distance to co-cluster	blue■	exclusion	exclusion
distance to cograph	blue■	exclusion	exclusion
distance to complete	blue■	exclusion	exclusion
distance to edgeless	blue■	exclusion	exclusion
distance to forest	blue■	exclusion	exclusion
distance to interval	blue■	exclusion	exclusion
distance to linear forest	blue■	exclusion	exclusion
distance to maximum degree	blue■	exclusion	exclusion
distance to outerplanar	blue■	exclusion	exclusion
distance to perfect	blue■	exclusion	exclusion
distance to planar	blue■	exclusion	exclusion
distance to stars	blue■	exclusion	exclusion
domatic number	blue■	exclusion	exclusion
domination number	blue■	exclusion	exclusion
domino treewidth	green■	upper bound	exclusion
edge clique cover number	blue■	exclusion	exclusion
edge connectivity	magenta■	exclusion	unknown to HOPS
edge-cut width	cyan■	unknown to HOPS	exclusion
edge-treewidth	cyan■	unknown to HOPS	exclusion
edgeless	green■	upper bound	avoids
excluded minor	blue■	exclusion	avoids
excluded planar minor	cyan■	unknown to HOPS	avoids
excluded top-minor	blue■	exclusion	avoids
feedback edge set	green■	upper bound	exclusion
feedback vertex set	blue■	exclusion	exclusion
flip-width	magenta■	exclusion	unknown to HOPS
forest	green■	upper bound	exclusion
genus	blue■	exclusion	exclusion
grid	green■	upper bound	exclusion
h-index	blue■	exclusion	exclusion
interval	cyan■	unknown to HOPS	exclusion
iterated type partitions	blue■	exclusion	exclusion
linear clique-width	blue■	exclusion	exclusion
linear forest	green■	upper bound	exclusion
linear NLC-width	blue■	exclusion	exclusion
linear rank-width	blue■	exclusion	exclusion
maximum clique	blue■	exclusion	exclusion
maximum degree	green■	upper bound	exclusion
maximum independent set	blue■	exclusion	exclusion
maximum induced matching	blue■	exclusion	exclusion
maximum leaf number	green■	upper bound	exclusion
maximum matching	blue■	exclusion	exclusion
maximum matching on bipartite graphs	cyan■	unknown to HOPS	exclusion
merge-width	magenta■	exclusion	unknown to HOPS
mim-width	magenta■	exclusion	unknown to HOPS
minimum degree	blue■	exclusion	exclusion
mm-width	blue■	exclusion	exclusion
modular-width	blue■	exclusion	exclusion
module-width	blue■	exclusion	exclusion
monadically dependent	magenta■	exclusion	unknown to HOPS
monadically stable	magenta■	exclusion	unknown to HOPS
neighborhood diversity	blue■	exclusion	exclusion
NLC-width	blue■	exclusion	exclusion
NLCT-width	blue■	exclusion	exclusion
nowhere dense	magenta■	exclusion	unknown to HOPS
odd cycle transversal	blue■	exclusion	exclusion
outerplanar	cyan■	unknown to HOPS	exclusion
overlap treewidth	cyan■	unknown to HOPS	exclusion
path	green■	upper bound	exclusion
pathwidth	blue■	exclusion	exclusion
pathwidth+maxdegree	green■	upper bound	exclusion
perfect	cyan■	unknown to HOPS	exclusion
planar	cyan■	unknown to HOPS	exclusion
radius-inf flip-width	blue■	exclusion	exclusion
radius-r flip-width	magenta■	exclusion	unknown to HOPS
rank-width	blue■	exclusion	exclusion
series-parallel	gray■	unknown to HOPS	unknown to HOPS
shrub-depth	blue■	exclusion	exclusion
sim-width	magenta■	exclusion	unknown to HOPS
size	green■	upper bound	exclusion
slim tree-cut width	cyan■	unknown to HOPS	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	blue■	exclusion	exclusion
topological bandwidth	cyan■	unknown to HOPS	exclusion
tree	green■	upper bound	exclusion
tree-cut width	cyan■	unknown to HOPS	exclusion
tree-independence number	magenta■	exclusion	unknown to HOPS
tree-partition-width	cyan■	unknown to HOPS	exclusion
treebandwidth	cyan■	unknown to HOPS	exclusion
treedepth	blue■	exclusion	exclusion
treelength	magenta■	exclusion	unknown to HOPS
treespan	green■	upper bound	exclusion
treewidth	blue■	exclusion	exclusion
twin-cover number	blue■	exclusion	exclusion
twin-width	blue■	exclusion	exclusion
vertex connectivity	blue■	exclusion	exclusion
vertex cover	blue■	exclusion	exclusion
vertex integrity	blue■	exclusion	exclusion
weak coloring number	blue■	exclusion	exclusion
weak d-coloring number	magenta■	exclusion	unknown to HOPS
weak inf-coloring number	blue■	exclusion	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 14 : c-closure – The c-closure $\mathrm{cc}(G)$ of a graph $G$ is the smallest number $c$ such that any pair of vertices $v,w \in V(G)$ with $|N_G(v) \cap N_G(w)| \ge c$ is adjacent. …
- page 32 : maximum degree upper bounds c-closure by a computable function – Proposition 5.1. Maximum Degree strictly upper bounds $c$-Closure.
- page 32 : graph classes with bounded c-closure are not bounded maximum degree – Proposition 5.1. Maximum Degree strictly upper bounds $c$-Closure.
- page 32 : feedback edge set upper bounds c-closure by a computable function – Theorem 5.2. Feedback Edge Number strictly upper bounds $c$-Closure.
- page 32 : graph classes with bounded c-closure are not bounded feedback edge set – Theorem 5.2. Feedback Edge Number strictly upper bounds $c$-Closure.
- page 34 : graph classes with bounded c-closure are not bounded vertex cover – Proposition 5.3. $c$-Closure is incomparable to Vertex Cover Number.
- page 34 : graph classes with bounded vertex cover are not bounded c-closure – Proposition 5.3. $c$-Closure is incomparable to Vertex Cover Number.
- page 34 : graph classes with bounded c-closure are not bounded distance to complete – Proposition 5.4. $c$-Closure is incomparable to Distance to Clique.
- page 34 : graph classes with bounded distance to complete are not bounded c-closure – Proposition 5.4. $c$-Closure is incomparable to Distance to Clique.
- page 34 : graph classes with bounded c-closure are not bounded bisection bandwidth – Proposition 5.5. $c$-Closure is incomparable to Bisection Width.
- page 34 : graph classes with bounded bisection bandwidth are not bounded c-closure – Proposition 5.5. $c$-Closure is incomparable to Bisection Width.
- page 34 : graph classes with bounded c-closure are not bounded genus – Proposition 5.6. $c$-Closure is incomparable to Genus.
- page 34 : graph classes with bounded genus are not bounded c-closure – Proposition 5.6. $c$-Closure is incomparable to Genus.
- page 34 : graph classes with bounded c-closure are not bounded vertex connectivity – Observation 5.7. $c$-Closure is incomparable to Distance to Disconnected, Domatic Number and Maximum Clique.
- page 34 : graph classes with bounded vertex connectivity are not bounded c-closure – Observation 5.7. $c$-Closure is incomparable to Distance to Disconnected, Domatic Number and Maximum Clique.
- page 34 : graph classes with bounded c-closure are not bounded domatic number – Observation 5.7. $c$-Closure is incomparable to Distance to Disconnected, Domatic Number and Maximum Clique.
- page 34 : graph classes with bounded domatic number are not bounded c-closure – Observation 5.7. $c$-Closure is incomparable to Distance to Disconnected, Domatic Number and Maximum Clique.
- page 34 : graph classes with bounded c-closure are not bounded maximum clique – Observation 5.7. $c$-Closure is incomparable to Distance to Disconnected, Domatic Number and Maximum Clique.
- page 34 : graph classes with bounded maximum clique are not bounded c-closure – Observation 5.7. $c$-Closure is incomparable to Distance to Disconnected, Domatic Number and Maximum Clique.
- page 35 : graph classes with bounded c-closure are not bounded boxicity – Proposition 5.8. $c$-Closure is incomparable to Boxicity.
- page 35 : graph classes with bounded boxicity are not bounded c-closure – Proposition 5.8. $c$-Closure is incomparable to Boxicity.
assumed
- c-closure is equivalent to c-closure – assumed
unknown source
- maximum degree upper bounds c-closure by a computable function
- feedback edge set upper bounds c-closure by a computable function