chordal

tags: vertex order, tree decomposition, intersection

providers: ISGCI

Definitions:

Contains no induced cycle on 4 or more vertices.
Has a vertex order, called perfect elimination scheme, such that future neighborhoods are always cliques.
Has a tree-decomposition such that every bag is a clique.
Is an intersection graph of subtrees of a tree.

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	blue■	exclusion	exclusion
bipartite number	blue■	exclusion	unbounded
bisection bandwidth	blue■	exclusion	unbounded
block	green■	inclusion	exclusion
book thickness	blue■	exclusion	unbounded
boolean width	magenta■	exclusion	unknown to HOPS
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	yellow■	equal	equal
chordality	red■	exclusion	upper bound
chromatic number	blue■	exclusion	unbounded
clique cover number	blue■	exclusion	unbounded
clique-tree-width	magenta■	exclusion	unknown to HOPS
clique-width	magenta■	exclusion	unknown to HOPS
cluster	green■	upper bound	exclusion
co-cluster	blue■	exclusion	exclusion
cograph	blue■	exclusion	exclusion
complete	green■	upper bound	exclusion
connected	blue■	exclusion	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	magenta■	exclusion	unknown to HOPS
distance to bounded components	blue■	exclusion	unbounded
distance to chordal	red■	exclusion	upper bound
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	blue■	exclusion	exclusion
h-index	blue■	exclusion	unbounded
interval	green■	inclusion	exclusion
iterated type partitions	blue■	exclusion	unbounded
linear clique-width	magenta■	exclusion	unknown to HOPS
linear forest	green■	upper bound	exclusion
linear NLC-width	magenta■	exclusion	unknown to HOPS
linear rank-width	magenta■	exclusion	unknown to HOPS
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	cyan■	unknown to HOPS	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	magenta■	exclusion	unknown to HOPS
monadically dependent	magenta■	exclusion	unknown to HOPS
monadically stable	magenta■	exclusion	unknown to HOPS
neighborhood diversity	blue■	exclusion	unbounded
NLC-width	magenta■	exclusion	unknown to HOPS
NLCT-width	magenta■	exclusion	unknown to HOPS
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	red■	exclusion	inclusion
planar	blue■	exclusion	exclusion
radius-inf flip-width	magenta■	exclusion	unknown to HOPS
radius-r flip-width	magenta■	exclusion	unknown to HOPS
rank-width	magenta■	exclusion	unknown to HOPS
series-parallel	gray■	unknown to HOPS	unknown to HOPS
shrub-depth	magenta■	exclusion	unknown to HOPS
sim-width	red■	exclusion	upper bound
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	red■	exclusion	upper bound
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 : chordal – … a graph is chordal (or triangulated) if each of its cycles of length at least $4$ has a chord, i.e. if it contains no induced cycles other than triangles.
assumed
- graph class chordal has contained in perfect
- graph class perfect does not have contained in chordal
- graph class interval has contained in chordal
- graph class chordal does not have contained in interval
- graph class block has contained in chordal
- graph class chordal does not have contained in block
- chordal upper bounds distance to chordal by a constant – by definition
- chordal is equivalent to chordal – assumed