grid

providers: ISGCI

Definition: Cartesian product of two paths.

Relations

Other		Relation from	Relation to
acyclic chromatic number	red■	exclusion	upper bound
arboricity	red■	exclusion	upper bound
average degree	red■	exclusion	upper bound
average distance	blue■	exclusion	unbounded
bandwidth	blue■	exclusion	unbounded
bipartite	red■	exclusion	inclusion
bipartite number	magenta■	exclusion	unknown to HOPS
bisection bandwidth	blue■	exclusion	unbounded
block	blue■	exclusion	exclusion
book thickness	red■	exclusion	upper bound
boolean width	blue■	exclusion	unbounded
bounded components	blue■	exclusion	unbounded
boxicity	red■	exclusion	upper bound
branch width	blue■	exclusion	unbounded
c-closure	red■	exclusion	upper bound
carving-width	blue■	exclusion	unbounded
chordal	blue■	exclusion	exclusion
chordality	red■	exclusion	upper bound
chromatic number	red■	exclusion	upper bound
clique cover number	blue■	exclusion	unbounded
clique-tree-width	blue■	exclusion	unbounded
clique-width	blue■	exclusion	unbounded
cluster	blue■	exclusion	exclusion
co-cluster	blue■	exclusion	exclusion
cograph	blue■	exclusion	exclusion
complete	blue■	exclusion	exclusion
connected	red■	exclusion	inclusion
contraction complexity	blue■	exclusion	unbounded
cutwidth	blue■	exclusion	unbounded
cycle	cyan■	unknown to HOPS	exclusion
cycles	blue■	exclusion	exclusion
d-path-free	blue■	exclusion	unbounded
degeneracy	red■	exclusion	upper bound
degree treewidth	blue■	exclusion	unbounded
diameter	blue■	exclusion	unbounded
diameter+max degree	blue■	exclusion	unbounded
disconnected	gray■	unknown to HOPS	unknown to HOPS
disjoint cycles	blue■	exclusion	exclusion
distance to bipartite	red■	exclusion	upper bound
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 disconnected	red■	exclusion	upper bound
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	red■	exclusion	upper bound
distance to outerplanar	blue■	exclusion	unbounded
distance to perfect	red■	exclusion	upper bound
distance to planar	red■	exclusion	upper bound
distance to stars	blue■	exclusion	unbounded
domatic number	red■	exclusion	upper bound
domination number	blue■	exclusion	unbounded
edge clique cover number	blue■	exclusion	unbounded
edge connectivity	red■	exclusion	upper bound
edgeless	blue■	exclusion	exclusion
feedback edge set	blue■	exclusion	unbounded
feedback vertex set	blue■	exclusion	unbounded
forest	blue■	exclusion	exclusion
genus	red■	exclusion	upper bound
girth	magenta■	exclusion	unknown to HOPS
grid	yellow■	equal	equal
h-index	red■	exclusion	upper bound
inf-flip-width	blue■	exclusion	unbounded
interval	blue■	exclusion	exclusion
iterated type partitions	blue■	exclusion	unbounded
linear clique-width	blue■	exclusion	unbounded
linear forest	blue■	exclusion	exclusion
linear NLC-width	blue■	exclusion	unbounded
linear rank-width	blue■	exclusion	unbounded
maximum clique	red■	exclusion	upper bound
maximum degree	red■	exclusion	upper bound
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	blue■	exclusion	unbounded
mim-width	magenta■	exclusion	unknown to HOPS
minimum degree	red■	exclusion	upper bound
mm-width	blue■	exclusion	unbounded
modular-width	blue■	exclusion	unbounded
module-width	blue■	exclusion	unbounded
neighborhood diversity	blue■	exclusion	unbounded
NLC-width	blue■	exclusion	unbounded
NLCT-width	blue■	exclusion	unbounded
odd cycle transversal	red■	exclusion	upper bound
outerplanar	blue■	exclusion	exclusion
path	green■	inclusion	exclusion
pathwidth	blue■	exclusion	unbounded
pathwidth+maxdegree	blue■	exclusion	unbounded
perfect	red■	exclusion	upper bound
planar	red■	exclusion	inclusion
radius-r flip-width	red■	exclusion	upper bound
rank-width	blue■	exclusion	unbounded
shrub-depth	blue■	exclusion	unbounded
sim-width	magenta■	exclusion	unknown to HOPS
size	blue■	exclusion	unbounded
star	blue■	exclusion	exclusion
stars	blue■	exclusion	exclusion
topological bandwidth	blue■	exclusion	unbounded
tree	blue■	exclusion	exclusion
tree-independence number	magenta■	exclusion	unknown to HOPS
treedepth	blue■	exclusion	unbounded
treelength	magenta■	exclusion	unknown to HOPS
treewidth	blue■	exclusion	unbounded
twin-cover number	blue■	exclusion	unbounded
twin-width	red■	exclusion	upper bound
vertex connectivity	red■	exclusion	upper bound
vertex cover	blue■	exclusion	unbounded
vertex integrity	blue■	exclusion	unbounded

Results

2021 Twin-width I: Tractable FO Model Checking by Bonnet, Kim, Thomassé, Watrigant
- page 15 : grid upper bounds twin-width by a constant – Theorem 4.3. For every positive integers $d$ and $n$, the $d$-dimensional $n$-grid has twin-width at most $3d$.
2010 The rank-width of the square grid by Jelínek
- page 2 : graph class grid has unbounded rank-width – The grid $G_{n,n}$ has rank-width equal to $n-1$.
2010 Comparing 17 graph parameters by Sasák
- page 17 : graph class grid has unbounded treewidth – Theorem 2.4
- page 32 : graph class grid has unbounded treedepth – Corollary 2.24 Tree-depth of a grid is at least $\lceil \log_2(n+1)\rceil$.
1998 A partial $k$-arboretum of graphs with bounded treewidth by Bodlaender
- page 37 : graph class grid has unbounded treewidth – Lemma 88. The treewidth of an $n \times n$ grid graph … is at least $n$.
assumed
- graph class grid is included in graph class planar
- graph class planar is not included in graph class grid
- graph class grid is included in graph class bipartite
- graph class bipartite is not included in graph class grid
- graph class grid is included in graph class connected
- graph class connected is not included in graph class grid
- graph class path is included in graph class grid
- graph class grid is not included in graph class path
- grid upper bounds maximum degree by a constant – By definition
- grid is equivalent to grid – assumed
unknown source
- graph class grid has unbounded distance to chordal
- graph class grid has unbounded average distance
- graph class grid has unbounded bisection bandwidth
- grid upper bounds maximum degree by a constant