boxicity

Relations

Other		Relation from	Relation to
acyclic chromatic number	green■	upper bound	exclusion
admissibility	cyan■	unknown to HOPS	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	green■	upper bound	exclusion
boolean width	blue■	exclusion	exclusion
bounded components	green■	upper bound	exclusion
bounded expansion	cyan■	unknown to HOPS	avoids
boxicity	yellow■	equal	equal
branch width	green■	upper bound	exclusion
c-closure	blue■	exclusion	exclusion
carving-width	green■	upper bound	exclusion
chi-bounded	magenta■	exclusion	unknown to HOPS
chordal	cyan■	unknown to HOPS	exclusion
chordality	red■	exclusion	upper bound
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	cyan■	unknown to HOPS	exclusion
cograph	cyan■	unknown to HOPS	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	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	cyan■	unknown to HOPS	exclusion
distance to bounded components	green■	upper bound	exclusion
distance to chordal	blue■	exclusion	exclusion
distance to cluster	green■	upper bound	exclusion
distance to co-cluster	blue■	exclusion	exclusion
distance to cograph	blue■	exclusion	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	cyan■	unknown to HOPS	avoids
excluded planar minor	green■	upper bound	avoids
excluded top-minor	cyan■	unknown to HOPS	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	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	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	blue■	exclusion	exclusion
nowhere dense	gray■	unknown to HOPS	unknown to HOPS
odd cycle transversal	blue■	exclusion	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	blue■	exclusion	exclusion
sim-width	magenta■	exclusion	unknown to HOPS
size	green■	upper bound	exclusion
slim tree-cut width	green■	upper bound	exclusion
sparse twin-width	cyan■	unknown to HOPS	exclusion
star	green■	upper bound	exclusion
stars	green■	upper bound	exclusion
strong coloring number	cyan■	unknown to HOPS	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	magenta■	exclusion	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	cyan■	unknown to HOPS	exclusion
weak d-coloring number	gray■	unknown to HOPS	unknown to HOPS
weak inf-coloring number	green■	upper bound	exclusion
weakly sparse	magenta■	exclusion	unknown to HOPS
weakly sparse and merge width	cyan■	unknown to HOPS	exclusion

Results

2022 Expanding the Graph Parameter Hierarchy by Tran
- page 16 : boxicity – The boxicity of a graph $G$ is the minimum amount of interval graphs required, such that their intersection (ed: fixed typo) results in $G$.
- page 26 : neighborhood diversity upper bounds boxicity by a polynomial function – Note that given a path of length $n > 3$. The boxicity is 1 while … neighborhood diversity is $n$. … a graph … with neighborhood diversity of $k$, its boxicity is at most $k+k^2$.
- page 26 : graph classes with bounded boxicity are not bounded neighborhood diversity – Note that given a path of length $n > 3$. The boxicity is 1 while … neighborhood diversity is $n$. … a graph … with neighborhood diversity of $k$, its boxicity is at most $k+k^2$.
- 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.
- page 42 : graph classes with bounded degeneracy are not bounded boxicity – Proposition 7.1. Degeneracy is incomparable to Boxicity.
- page 42 : graph classes with bounded boxicity are not bounded degeneracy – Proposition 7.1. Degeneracy is incomparable to Boxicity.
2019 The Graph Parameter Hierarchy by Sorge
- page 8 : acyclic chromatic number upper bounds boxicity by a computable function – Lemma 4.9. The boxicity $b$ is upper bounded by the acyclic chromatic number $\chi_a$ (for every graph with $\chi_a>1$). We have $b \le \chi_a(\chi_a-1)$.
- 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 interval upper bounds boxicity by a linear function – Lemma 4.16. The distance $i$ to an interval graph upper bounds the boxicity $b$. We have $b \le i+1$.
2011 Chordal Bipartite Graphs with High Boxicity by Chandran, Francis, Mathew
- page 9 : graph classes with bounded distance to bipartite are not bounded boxicity – Theorem 2 For any $b \in \mathbb N^+$, there exists a chordal bipartite graph $G$ (ed: i.e. bipartite graph with no induced cycle on more than 4 vertices) with $\mathrm{box}(G) > b$.
Comparing Graph Parameters by Schröder
- page 14 : graph classes with bounded distance to co-cluster are not bounded boxicity – Proposition 3.4
- page 27 : graph classes with bounded distance to chordal are not bounded boxicity – Proposition 3.25
assumed
- distance to interval upper bounds boxicity by a linear function – By definition
- boxicity is equivalent to boxicity – assumed
unknown source
- acyclic chromatic number upper bounds boxicity by a computable function