distance to perfect

Definition: Minimum number of vertices removed to make the graph into perfect

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	blue■	exclusion	exclusion
bipartite	green■	upper bound	exclusion
bipartite number	blue■	exclusion	exclusion
bisection bandwidth	blue■	exclusion	exclusion
block	green■	upper bound	exclusion
book thickness	blue■	exclusion	exclusion
boolean width	blue■	exclusion	exclusion
bounded components	blue■	exclusion	exclusion
bounded expansion	blue■	exclusion	avoids
boxicity	blue■	exclusion	exclusion
branch width	blue■	exclusion	exclusion
c-closure	blue■	exclusion	exclusion
carving-width	blue■	exclusion	exclusion
chi-bounded	magenta■	exclusion	unknown to HOPS
chordal	green■	upper bound	exclusion
chordality	blue■	exclusion	exclusion
chromatic number	blue■	exclusion	exclusion
clique cover number	blue■	exclusion	exclusion
clique-tree-width	blue■	exclusion	exclusion
clique-width	blue■	exclusion	exclusion
cluster	green■	upper bound	exclusion
co-cluster	green■	upper bound	exclusion
cograph	green■	upper bound	exclusion
complete	green■	upper bound	exclusion
connected	cyan■	unknown to HOPS	avoids
contraction complexity	blue■	exclusion	exclusion
cutwidth	blue■	exclusion	exclusion
cycle	green■	upper bound	exclusion
cycles	blue■	unbounded	exclusion
d-admissibility	magenta■	exclusion	unknown to HOPS
d-path-free	blue■	exclusion	exclusion
degeneracy	blue■	exclusion	exclusion
degree treewidth	blue■	exclusion	exclusion
diameter	blue■	exclusion	exclusion
diameter+max degree	blue■	exclusion	exclusion
distance to bipartite	green■	upper bound	exclusion
distance to block	green■	upper bound	exclusion
distance to bounded components	blue■	exclusion	exclusion
distance to chordal	green■	upper bound	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	green■	upper bound	exclusion
distance to linear forest	green■	upper bound	exclusion
distance to maximum degree	blue■	exclusion	exclusion
distance to outerplanar	blue■	exclusion	exclusion
distance to perfect	yellow■	equal	equal
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	blue■	exclusion	exclusion
edge clique cover number	blue■	exclusion	exclusion
edge connectivity	blue■	exclusion	exclusion
edge-cut width	cyan■	unknown to HOPS	exclusion
edge-treewidth	blue■	exclusion	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	green■	upper bound	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	green■	upper bound	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	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	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	green■	upper bound	exclusion
outerplanar	cyan■	unknown to HOPS	exclusion
overlap treewidth	blue■	exclusion	exclusion
path	green■	upper bound	exclusion
pathwidth	blue■	exclusion	exclusion
pathwidth+maxdegree	blue■	exclusion	exclusion
perfect	green■	upper bound	exclusion
planar	blue■	unbounded	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	blue■	exclusion	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	blue■	exclusion	exclusion
tree	green■	upper bound	exclusion
tree-cut width	blue■	exclusion	exclusion
tree-independence number	magenta■	exclusion	unknown to HOPS
tree-partition-width	blue■	exclusion	exclusion
treebandwidth	blue■	exclusion	exclusion
treedepth	blue■	exclusion	exclusion
treelength	magenta■	exclusion	unknown to HOPS
treespan	blue■	exclusion	exclusion
treewidth	blue■	exclusion	exclusion
twin-cover number	green■	upper bound	exclusion
twin-width	blue■	exclusion	exclusion
vertex connectivity	gray■	unknown to HOPS	unknown to HOPS
vertex cover	green■	upper bound	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 29 : graph classes with bounded edge clique cover number are not bounded distance to perfect – Proposition 4.15. Edge Clique Cover Number is incomparable to Distance to Perfect.
- page 29 : graph classes with bounded distance to perfect are not bounded edge clique cover number – Proposition 4.15. Edge Clique Cover Number is incomparable to Distance to Perfect.
Comparing Graph Parameters by Schröder
- page 15 : graph classes with bounded clique cover number are not bounded distance to perfect – Proposition 3.6
- page 19 : graph classes with bounded distance to perfect are not bounded chordality – Proposition 3.11
- page 26 : graph classes with bounded distance to outerplanar are not bounded distance to perfect – Proposition 3.23
- page 26 : graph classes with bounded bandwidth are not bounded distance to perfect – Proposition 3.24
- page 26 : graph classes with bounded genus are not bounded distance to perfect – Proposition 3.24
- page 26 : graph classes with bounded treedepth are not bounded distance to perfect – Proposition 3.24
assumed
- perfect upper bounds distance to perfect by a constant – by definition
- distance to perfect is equivalent to distance to perfect – assumed
unknown source
- graph class planar is not constant distance to perfect
- graph class cycles is not constant distance to perfect
- graph classes with bounded bounded components are not bounded distance to perfect – By a disjoint union of small components with distance to perfect at least 1.