perfect

providers: ISGCI

Definition: Has maximum clique equal to chromatic number.

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	blue■	exclusion	unbounded
bandwidth	blue■	exclusion	unbounded
bipartite	green■	inclusion	exclusion
bipartite number	blue■	exclusion	unbounded
bisection bandwidth	blue■	exclusion	unbounded
block	green■	upper bound	exclusion
book thickness	blue■	exclusion	unbounded
boolean width	blue■	exclusion	unbounded
bounded components	blue■	exclusion	unbounded
bounded expansion	blue■	exclusion	avoids
boxicity	magenta■	exclusion	unknown to HOPS
branch width	blue■	exclusion	unbounded
c-closure	magenta■	exclusion	unknown to HOPS
carving-width	blue■	exclusion	unbounded
chi-bounded	red■	exclusion	upper bound
chordal	green■	inclusion	exclusion
chordality	magenta■	exclusion	unknown to HOPS
chromatic number	blue■	exclusion	unbounded
clique cover number	blue■	exclusion	unbounded
clique-tree-width	blue■	exclusion	unbounded
clique-width	blue■	exclusion	unbounded
cluster	green■	upper bound	exclusion
co-cluster	green■	upper bound	exclusion
cograph	green■	inclusion	exclusion
complete	green■	upper bound	exclusion
connected	cyan■	unknown to HOPS	avoids
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	blue■	exclusion	unbounded
diameter+max degree	blue■	exclusion	unbounded
distance to bipartite	blue■	exclusion	unbounded
distance to block	blue■	exclusion	unbounded
distance to bounded components	blue■	exclusion	unbounded
distance to chordal	blue■	exclusion	unbounded
distance to cluster	blue■	exclusion	unbounded
distance to co-cluster	blue■	exclusion	unbounded
distance to cograph	blue■	exclusion	unbounded
distance to complete	blue■	exclusion	unbounded
distance to edgeless	blue■	exclusion	unbounded
distance to forest	blue■	exclusion	unbounded
distance to interval	blue■	exclusion	unbounded
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	blue■	exclusion	unbounded
domino treewidth	blue■	exclusion	unbounded
edge clique cover number	blue■	exclusion	unbounded
edge connectivity	blue■	exclusion	unbounded
edge-cut width	blue■	exclusion	unbounded
edge-treewidth	blue■	exclusion	unbounded
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	blue■	exclusion	unbounded
feedback vertex set	blue■	exclusion	unbounded
flip-width	magenta■	exclusion	unknown to HOPS
forest	green■	upper bound	exclusion
genus	blue■	exclusion	unbounded
grid	green■	upper bound	exclusion
h-index	blue■	exclusion	unbounded
interval	green■	upper bound	exclusion
iterated type partitions	blue■	exclusion	unbounded
linear clique-width	blue■	exclusion	unbounded
linear forest	green■	upper bound	exclusion
linear NLC-width	blue■	exclusion	unbounded
linear rank-width	blue■	exclusion	unbounded
maximum clique	blue■	exclusion	unbounded
maximum degree	blue■	exclusion	unbounded
maximum independent set	blue■	exclusion	unbounded
maximum induced matching	blue■	exclusion	unbounded
maximum leaf number	blue■	exclusion	unbounded
maximum matching	blue■	exclusion	unbounded
maximum matching on bipartite graphs	green■	upper bound	unbounded
merge-width	magenta■	exclusion	unknown to HOPS
mim-width	magenta■	exclusion	unknown to HOPS
minimum degree	blue■	exclusion	unbounded
mm-width	blue■	exclusion	unbounded
modular-width	blue■	exclusion	unbounded
module-width	blue■	exclusion	unbounded
monadically dependent	magenta■	exclusion	unknown to HOPS
monadically stable	magenta■	exclusion	unknown to HOPS
neighborhood diversity	blue■	exclusion	unbounded
NLC-width	blue■	exclusion	unbounded
NLCT-width	blue■	exclusion	unbounded
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	green■	upper bound	exclusion
pathwidth	blue■	exclusion	unbounded
pathwidth+maxdegree	blue■	exclusion	unbounded
perfect	yellow■	equal	equal
planar	blue■	exclusion	exclusion
radius-inf flip-width	blue■	exclusion	unbounded
radius-r flip-width	magenta■	exclusion	unknown to HOPS
rank-width	blue■	exclusion	unbounded
series-parallel	gray■	unknown to HOPS	unknown to HOPS
shrub-depth	blue■	exclusion	unbounded
sim-width	magenta■	exclusion	unknown to HOPS
size	blue■	exclusion	unbounded
slim tree-cut width	blue■	exclusion	unbounded
sparse twin-width	blue■	exclusion	unbounded
star	green■	upper bound	exclusion
stars	green■	upper bound	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	green■	upper bound	exclusion
tree-cut width	blue■	exclusion	unbounded
tree-independence number	magenta■	exclusion	unknown to HOPS
tree-partition-width	blue■	exclusion	unbounded
treebandwidth	blue■	exclusion	unbounded
treedepth	blue■	exclusion	unbounded
treelength	magenta■	exclusion	unknown to HOPS
treespan	blue■	exclusion	unbounded
treewidth	blue■	exclusion	unbounded
twin-cover number	blue■	exclusion	unbounded
twin-width	magenta■	exclusion	unknown to HOPS
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

2017 Graph Theory by Diestel
- page 135 : perfect – A graph is called perfect if every induced subgraph $H \subseteq G$ has chromatic number $\chi(H)=\omega(H)$, i.e. if the trivial lower bound of $\omega(H)$ colours always suffices to colour the vertices of $H$.
assumed
- graph class chordal has contained in perfect
- graph class perfect does not have contained in chordal
- graph class cograph has contained in perfect
- graph class perfect does not have contained in cograph
- graph class bipartite has contained in perfect
- graph class perfect does not have contained in bipartite
- perfect upper bounds distance to perfect by a constant – by definition
- graph classes with bounded size do not have contained in perfect – By definition
- perfect is equivalent to perfect – assumed
unknown source
- perfect upper bounds chi-bounded by a constant