grid
providers: ISGCI
Definition: Cartesian product of two paths.
Relations
Other | Relation from | Relation to | |
---|---|---|---|
acyclic chromatic number | ■ | exclusion | upper bound |
admissibility | ■ | exclusion | upper bound |
arboricity | ■ | exclusion | upper bound |
average degree | ■ | exclusion | upper bound |
average distance | ■ | exclusion | unbounded |
bandwidth | ■ | exclusion | unbounded |
bipartite | ■ | exclusion | inclusion |
bipartite number | ■ | exclusion | unbounded |
bisection bandwidth | ■ | exclusion | unbounded |
block | ■ | exclusion | exclusion |
book thickness | ■ | exclusion | upper bound |
boolean width | ■ | exclusion | unbounded |
bounded components | ■ | exclusion | unbounded |
bounded expansion | ■ | exclusion | upper bound |
boxicity | ■ | exclusion | upper bound |
branch width | ■ | exclusion | unbounded |
c-closure | ■ | exclusion | upper bound |
carving-width | ■ | exclusion | unbounded |
chi-bounded | ■ | exclusion | upper bound |
chordal | ■ | exclusion | exclusion |
chordality | ■ | exclusion | upper bound |
chromatic number | ■ | exclusion | upper bound |
clique cover number | ■ | exclusion | unbounded |
clique-tree-width | ■ | exclusion | unbounded |
clique-width | ■ | exclusion | unbounded |
cluster | ■ | exclusion | exclusion |
co-cluster | ■ | exclusion | exclusion |
cograph | ■ | exclusion | exclusion |
complete | ■ | exclusion | exclusion |
connected | ■ | exclusion | has |
contraction complexity | ■ | exclusion | unbounded |
cutwidth | ■ | exclusion | unbounded |
cycle | ■ | unknown to HOPS | exclusion |
cycles | ■ | exclusion | exclusion |
d-admissibility | ■ | exclusion | upper bound |
d-path-free | ■ | exclusion | unbounded |
degeneracy | ■ | exclusion | upper bound |
degree treewidth | ■ | exclusion | unbounded |
diameter | ■ | exclusion | unbounded |
diameter+max degree | ■ | exclusion | unbounded |
distance to bipartite | ■ | exclusion | upper bound |
distance to block | ■ | exclusion | unbounded |
distance to bounded components | ■ | exclusion | unbounded |
distance to chordal | ■ | exclusion | unbounded |
distance to cluster | ■ | exclusion | unbounded |
distance to co-cluster | ■ | exclusion | unbounded |
distance to cograph | ■ | exclusion | unbounded |
distance to complete | ■ | exclusion | unbounded |
distance to edgeless | ■ | exclusion | unbounded |
distance to forest | ■ | exclusion | unbounded |
distance to interval | ■ | exclusion | unbounded |
distance to linear forest | ■ | exclusion | unbounded |
distance to maximum degree | ■ | exclusion | upper bound |
distance to outerplanar | ■ | exclusion | unbounded |
distance to perfect | ■ | exclusion | upper bound |
distance to planar | ■ | exclusion | upper bound |
distance to stars | ■ | exclusion | unbounded |
domatic number | ■ | exclusion | upper bound |
domination number | ■ | exclusion | unbounded |
domino treewidth | ■ | exclusion | unbounded |
edge clique cover number | ■ | exclusion | unbounded |
edge connectivity | ■ | exclusion | upper bound |
edge-cut width | ■ | exclusion | unbounded |
edge-treewidth | ■ | exclusion | unbounded |
edgeless | ■ | exclusion | avoids |
excluded minor | ■ | exclusion | upper bound |
excluded planar minor | ■ | unknown to HOPS | avoids |
excluded top-minor | ■ | exclusion | upper bound |
feedback edge set | ■ | exclusion | unbounded |
feedback vertex set | ■ | exclusion | unbounded |
flip-width | ■ | exclusion | upper bound |
forest | ■ | exclusion | exclusion |
genus | ■ | exclusion | upper bound |
grid | ■ | equal | equal |
h-index | ■ | exclusion | upper bound |
interval | ■ | exclusion | exclusion |
iterated type partitions | ■ | exclusion | unbounded |
linear clique-width | ■ | exclusion | unbounded |
linear forest | ■ | exclusion | exclusion |
linear NLC-width | ■ | exclusion | unbounded |
linear rank-width | ■ | exclusion | unbounded |
maximum clique | ■ | exclusion | upper bound |
maximum degree | ■ | exclusion | upper bound |
maximum independent set | ■ | exclusion | unbounded |
maximum induced matching | ■ | exclusion | unbounded |
maximum leaf number | ■ | exclusion | unbounded |
maximum matching | ■ | exclusion | unbounded |
maximum matching on bipartite graphs | ■ | exclusion | unbounded |
merge-width | ■ | exclusion | upper bound |
mim-width | ■ | exclusion | unknown to HOPS |
minimum degree | ■ | exclusion | upper bound |
mm-width | ■ | exclusion | unbounded |
modular-width | ■ | exclusion | unbounded |
module-width | ■ | exclusion | unbounded |
monadically dependent | ■ | exclusion | upper bound |
monadically stable | ■ | exclusion | upper bound |
neighborhood diversity | ■ | exclusion | unbounded |
NLC-width | ■ | exclusion | unbounded |
NLCT-width | ■ | exclusion | unbounded |
nowhere dense | ■ | exclusion | upper bound |
odd cycle transversal | ■ | exclusion | upper bound |
outerplanar | ■ | unknown to HOPS | exclusion |
overlap treewidth | ■ | exclusion | unbounded |
path | ■ | inclusion | exclusion |
pathwidth | ■ | exclusion | unbounded |
pathwidth+maxdegree | ■ | exclusion | unbounded |
perfect | ■ | exclusion | upper bound |
planar | ■ | exclusion | inclusion |
radius-inf flip-width | ■ | exclusion | unbounded |
radius-r flip-width | ■ | exclusion | upper bound |
rank-width | ■ | exclusion | unbounded |
series-parallel | ■ | unknown to HOPS | unknown to HOPS |
shrub-depth | ■ | exclusion | unbounded |
sim-width | ■ | exclusion | unknown to HOPS |
size | ■ | exclusion | unbounded |
slim tree-cut width | ■ | exclusion | unbounded |
sparse twin-width | ■ | exclusion | upper bound |
star | ■ | exclusion | exclusion |
stars | ■ | exclusion | exclusion |
strong coloring number | ■ | exclusion | upper bound |
strong d-coloring number | ■ | exclusion | upper bound |
strong inf-coloring number | ■ | exclusion | unbounded |
topological bandwidth | ■ | exclusion | unbounded |
tree | ■ | exclusion | exclusion |
tree-cut width | ■ | exclusion | unbounded |
tree-independence number | ■ | exclusion | unknown to HOPS |
tree-partition-width | ■ | exclusion | unbounded |
treebandwidth | ■ | exclusion | unbounded |
treedepth | ■ | exclusion | unbounded |
treelength | ■ | exclusion | unknown to HOPS |
treespan | ■ | exclusion | unbounded |
treewidth | ■ | exclusion | unbounded |
twin-cover number | ■ | exclusion | unbounded |
twin-width | ■ | exclusion | upper bound |
vertex connectivity | ■ | exclusion | unknown to HOPS |
vertex cover | ■ | exclusion | unbounded |
vertex integrity | ■ | exclusion | unbounded |
weak coloring number | ■ | exclusion | upper bound |
weak d-coloring number | ■ | exclusion | upper bound |
weak inf-coloring number | ■ | exclusion | unbounded |
weakly sparse | ■ | exclusion | upper bound |
weakly sparse and merge width | ■ | exclusion | upper bound |
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 Comparing 17 graph parameters by Sasák
- 2010 The rank-width of the square grid by Jelínek
- page 2 : graph class grid is not constant rank-width – The grid $G_{n,n}$ has rank-width equal to $n-1$.
- 1998 A partial $k$-arboretum of graphs with bounded treewidth by Bodlaender
- unknown source
- graph class grid is not constant distance to chordal
- graph class grid is not constant average distance
- graph class grid is not constant bisection bandwidth
- grid upper bounds maximum degree by a constant
- assumed
- graph class grid has contained in planar
- graph class planar does not have contained in grid
- graph class grid has contained in bipartite
- graph class bipartite does not have contained in grid
- graph class grid is connected
- graph classes that are connected do not have contained in grid
- graph class path has contained in grid
- graph class grid does not have contained in path
- grid upper bounds maximum degree by a constant – By definition
- grid is equivalent to grid – assumed