chordality

Relations

Other		Relation from	Relation to
acyclic chromatic number	green■	upper bound	exclusion
admissibility	green■	upper bound	exclusion
arboricity	green■	upper bound	exclusion
average degree	blue■	exclusion	exclusion
average distance	blue■	exclusion	exclusion
bandwidth	green■	upper bound	exclusion
bipartite	green■	upper bound	exclusion
bipartite number	blue■	exclusion	exclusion
bisection bandwidth	blue■	exclusion	exclusion
block	green■	upper bound	exclusion
book thickness	green■	upper bound	exclusion
boolean width	blue■	exclusion	exclusion
bounded components	green■	upper bound	exclusion
bounded expansion	green■	upper bound	avoids
boxicity	green■	upper bound	exclusion
branch width	green■	upper bound	exclusion
c-closure	cyan■	unknown to HOPS	exclusion
carving-width	green■	upper bound	exclusion
chi-bounded	magenta■	exclusion	unknown to HOPS
chordal	green■	upper bound	exclusion
chordality	yellow■	equal	equal
chromatic number	green■	upper bound	exclusion
clique cover number	blue■	exclusion	exclusion
clique-tree-width	cyan■	unknown to HOPS	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	green■	upper bound	exclusion
cutwidth	green■	upper bound	exclusion
cycle	green■	upper bound	exclusion
cycles	green■	upper bound	exclusion
d-admissibility	gray■	unknown to HOPS	unknown to HOPS
d-path-free	green■	upper bound	exclusion
degeneracy	green■	upper bound	exclusion
degree treewidth	green■	upper bound	exclusion
diameter	blue■	exclusion	exclusion
diameter+max degree	green■	upper bound	exclusion
distance to bipartite	green■	upper bound	exclusion
distance to block	green■	upper bound	exclusion
distance to bounded components	green■	upper bound	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	green■	upper bound	exclusion
distance to outerplanar	green■	upper bound	exclusion
distance to perfect	blue■	exclusion	exclusion
distance to planar	green■	upper bound	exclusion
distance to stars	green■	upper bound	exclusion
domatic number	blue■	exclusion	exclusion
domination number	blue■	exclusion	exclusion
domino treewidth	green■	upper bound	exclusion
edge clique cover number	green■	upper bound	exclusion
edge connectivity	blue■	exclusion	exclusion
edge-cut width	green■	upper bound	exclusion
edge-treewidth	green■	upper bound	exclusion
edgeless	green■	upper bound	avoids
excluded minor	green■	upper bound	avoids
excluded planar minor	green■	upper bound	avoids
excluded top-minor	green■	upper bound	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	green■	upper bound	exclusion
grid	green■	upper bound	exclusion
h-index	green■	upper bound	exclusion
interval	green■	upper bound	exclusion
iterated type partitions	cyan■	unknown to HOPS	exclusion
linear clique-width	cyan■	unknown to HOPS	exclusion
linear forest	green■	upper bound	exclusion
linear NLC-width	cyan■	unknown to HOPS	exclusion
linear rank-width	cyan■	unknown to HOPS	exclusion
maximum clique	cyan■	unknown to HOPS	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	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	green■	upper bound	exclusion
modular-width	blue■	exclusion	exclusion
module-width	blue■	exclusion	exclusion
monadically dependent	magenta■	exclusion	unknown to HOPS
monadically stable	gray■	unknown to HOPS	unknown to HOPS
neighborhood diversity	green■	upper bound	exclusion
NLC-width	blue■	exclusion	exclusion
NLCT-width	cyan■	unknown to HOPS	exclusion
nowhere dense	gray■	unknown to HOPS	unknown to HOPS
odd cycle transversal	green■	upper bound	exclusion
outerplanar	green■	upper bound	exclusion
overlap treewidth	green■	upper bound	exclusion
path	green■	upper bound	exclusion
pathwidth	green■	upper bound	exclusion
pathwidth+maxdegree	green■	upper bound	exclusion
perfect	cyan■	unknown to HOPS	exclusion
planar	green■	upper bound	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	cyan■	unknown to HOPS	exclusion
sim-width	magenta■	exclusion	unknown to HOPS
size	green■	upper bound	exclusion
slim tree-cut width	green■	upper bound	exclusion
sparse twin-width	green■	upper bound	exclusion
star	green■	upper bound	exclusion
stars	green■	upper bound	exclusion
strong coloring number	green■	upper bound	exclusion
strong d-coloring number	gray■	unknown to HOPS	unknown to HOPS
strong inf-coloring number	green■	upper bound	exclusion
topological bandwidth	green■	upper bound	exclusion
tree	green■	upper bound	exclusion
tree-cut width	green■	upper bound	exclusion
tree-independence number	gray■	unknown to HOPS	unknown to HOPS
tree-partition-width	green■	upper bound	exclusion
treebandwidth	green■	upper bound	exclusion
treedepth	green■	upper bound	exclusion
treelength	magenta■	exclusion	unknown to HOPS
treespan	green■	upper bound	exclusion
treewidth	green■	upper bound	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	green■	upper bound	exclusion
weak coloring number	green■	upper bound	exclusion
weak d-coloring number	gray■	unknown to HOPS	unknown to HOPS
weak inf-coloring number	green■	upper bound	exclusion
weakly sparse	gray■	unknown to HOPS	unknown to HOPS
weakly sparse and merge width	green■	upper bound	exclusion

Results

2022 Expanding the Graph Parameter Hierarchy by Tran
- page 16 : chordality – The chordality of a graph $G$ is the minimum amount of chordal graphs required, such that their intersection (ed: fixed typo) results in $G$.
- page 30 : graph classes with bounded modular-width are not bounded chordality – Theorem 4.16. Modular-width is incomparable to Chordality.
- page 30 : graph classes with bounded chordality are not bounded modular-width – Theorem 4.16. Modular-width is incomparable to Chordality.
2019 The Graph Parameter Hierarchy by Sorge
- page 9 : boxicity upper bounds chordality by a linear function – Lemma 4.15 ([8,11]). The boxicity $b$ upper-bounds the chordality $c$. We have $c \le b$.
- page 9 : distance to chordal upper bounds chordality by a linear function – (ed: apparently goes as the lemma for ddist to interval and boxicity) Lemma 4.16. The distance $i$ to an interval graph upper bounds the boxicity $b$. We have $b \le i+1$.
1993 On the chordality of a graph by McKee, Scheinerman
- page 1 : chordality – The \emph{chordality} of a graph $G=(V,E)$ is defined as the minimum $k$ such that we can write $E=E_1,\cap\dots\cap E_k$ with each $(V,E_i)$ a chordal graph.
- page 2 : size upper bounds chordality by a constant – Corollary 3. For any graph $G$, $\mathrm{Chord}(G) \le |V(G)|/2$.
- page 2 : chromatic number upper bounds chordality by a linear function – Corollary 4. For any graph $G$, $\mathrm{Chord}(G) \le \chi(G)$, the chromatic number of $G$.
- page 5 : treewidth upper bounds chordality by a constant – Theorem 7. For any graph $G$, $\mathrm{Chord}(G) \le \tau(G)$.
unknown source
- distance to cograph upper bounds chordality by a linear function
assumed
- chordality is equivalent to chordality – assumed
Comparing Graph Parameters by Schröder
- page 19 : graph classes with bounded clique cover number are not bounded chordality – Proposition 3.11
- page 19 : graph classes with bounded distance to perfect are not bounded chordality – Proposition 3.11
- page 33 : graph classes with bounded bisection bandwidth are not bounded chordality – Proposition 3.31
- page 36 : graph classes with bounded average degree are not bounded chordality – Proposition 3.36