bandwidth

providers: ISGCI

Definition: Minimum over vertex labeling by distinct integers, maximum over edges, absolute difference of labels of edge’s endpoints.

Relations

Other		Relation from	Relation to
acyclic chromatic number	red■	exclusion	upper bound
admissibility	red■	exclusion	upper bound
arboricity	red■	exclusion	upper bound
average degree	red■	exclusion	upper bound
average distance	magenta■	exclusion	unknown to HOPS
bandwidth	yellow■	equal	equal
bipartite	blue■	unbounded	exclusion
bipartite number	magenta■	exclusion	unknown to HOPS
bisection bandwidth	red■	exclusion	upper bound
block	blue■	unbounded	exclusion
book thickness	red■	exclusion	upper bound
boolean width	red■	exclusion	upper bound
bounded components	gray■	unknown to HOPS	unknown to HOPS
bounded expansion	red■	exclusion	upper bound
boxicity	red■	exclusion	upper bound
branch width	red■	exclusion	upper bound
c-closure	red■	exclusion	upper bound
carving-width	orange■	unknown to HOPS	upper bound
chi-bounded	red■	exclusion	upper bound
chordal	blue■	unbounded	exclusion
chordality	red■	exclusion	upper bound
chromatic number	red■	exclusion	upper bound
clique cover number	magenta■	exclusion	unknown to HOPS
clique-tree-width	red■	exclusion	upper bound
clique-width	red■	exclusion	upper bound
cluster	blue■	unbounded	exclusion
co-cluster	blue■	unbounded	exclusion
cograph	blue■	unbounded	exclusion
complete	blue■	unbounded	exclusion
connected	blue■	exclusion	avoids
contraction complexity	orange■	unknown to HOPS	upper bound
cutwidth	orange■	unknown to HOPS	non-tight bounds
cycle	green■	upper bound	exclusion
cycles	gray■	unknown to HOPS	unknown to HOPS
d-admissibility	red■	exclusion	upper bound
d-path-free	magenta■	exclusion	unknown to HOPS
degeneracy	red■	exclusion	upper bound
degree treewidth	orange■	unknown to HOPS	upper bound
diameter	magenta■	exclusion	unknown to HOPS
diameter+max degree	gray■	unknown to HOPS	unknown to HOPS
distance to bipartite	blue■	exclusion	exclusion
distance to block	blue■	exclusion	exclusion
distance to bounded components	magenta■	exclusion	unknown to HOPS
distance to chordal	blue■	exclusion	exclusion
distance to cluster	blue■	exclusion	exclusion
distance to co-cluster	blue■	exclusion	exclusion
distance to cograph	blue■	exclusion	exclusion
distance to complete	blue■	exclusion	exclusion
distance to edgeless	blue■	exclusion	exclusion
distance to forest	blue■	exclusion	exclusion
distance to interval	blue■	exclusion	exclusion
distance to linear forest	blue■	exclusion	exclusion
distance to maximum degree	red■	exclusion	upper bound
distance to outerplanar	blue■	exclusion	exclusion
distance to perfect	blue■	exclusion	exclusion
distance to planar	blue■	exclusion	exclusion
distance to stars	blue■	exclusion	exclusion
domatic number	red■	exclusion	upper bound
domination number	magenta■	exclusion	unknown to HOPS
domino treewidth	orange■	unknown to HOPS	upper bound
edge clique cover number	magenta■	exclusion	unknown to HOPS
edge connectivity	red■	exclusion	upper bound
edge-cut width	magenta■	exclusion	unknown to HOPS
edge-treewidth	red■	exclusion	upper bound
edgeless	cyan■	unknown to HOPS	avoids
excluded minor	magenta■	exclusion	unknown to HOPS
excluded planar minor	gray■	unknown to HOPS	unknown to HOPS
excluded top-minor	red■	exclusion	upper bound
feedback edge set	blue■	exclusion	exclusion
feedback vertex set	blue■	exclusion	exclusion
flip-width	red■	exclusion	upper bound
forest	blue■	unbounded	exclusion
genus	blue■	exclusion	exclusion
grid	blue■	unbounded	exclusion
h-index	red■	exclusion	upper bound
interval	blue■	unbounded	exclusion
iterated type partitions	magenta■	exclusion	unknown to HOPS
linear clique-width	red■	exclusion	upper bound
linear forest	cyan■	unknown to HOPS	exclusion
linear NLC-width	red■	exclusion	upper bound
linear rank-width	red■	exclusion	upper bound
maximum clique	red■	exclusion	upper bound
maximum degree	red■	exclusion	upper bound
maximum independent set	magenta■	exclusion	unknown to HOPS
maximum induced matching	magenta■	exclusion	unknown to HOPS
maximum leaf number	green■	upper bound	exclusion
maximum matching	blue■	exclusion	exclusion
maximum matching on bipartite graphs	blue■	exclusion	exclusion
merge-width	red■	exclusion	upper bound
mim-width	red■	exclusion	upper bound
minimum degree	red■	exclusion	upper bound
mm-width	red■	exclusion	upper bound
modular-width	magenta■	exclusion	unknown to HOPS
module-width	red■	exclusion	upper bound
monadically dependent	red■	exclusion	upper bound
monadically stable	red■	exclusion	upper bound
neighborhood diversity	magenta■	exclusion	unknown to HOPS
NLC-width	red■	exclusion	upper bound
NLCT-width	red■	exclusion	upper bound
nowhere dense	red■	exclusion	upper bound
odd cycle transversal	blue■	exclusion	exclusion
outerplanar	cyan■	unknown to HOPS	exclusion
overlap treewidth	red■	exclusion	upper bound
path	cyan■	unknown to HOPS	exclusion
pathwidth	red■	exclusion	upper bound
pathwidth+maxdegree	orange■	unknown to HOPS	upper bound
perfect	blue■	unbounded	exclusion
planar	blue■	unbounded	exclusion
radius-inf flip-width	red■	exclusion	upper bound
radius-r flip-width	red■	exclusion	upper bound
rank-width	red■	exclusion	upper bound
series-parallel	gray■	unknown to HOPS	unknown to HOPS
shrub-depth	magenta■	exclusion	unknown to HOPS
sim-width	red■	exclusion	upper bound
size	green■	upper bound	exclusion
slim tree-cut width	red■	exclusion	upper bound
sparse twin-width	red■	exclusion	upper bound
star	blue■	unbounded	exclusion
stars	blue■	unbounded	exclusion
strong coloring number	red■	exclusion	upper bound
strong d-coloring number	red■	exclusion	upper bound
strong inf-coloring number	red■	exclusion	upper bound
topological bandwidth	orange■	unknown to HOPS	upper bound
tree	blue■	unbounded	exclusion
tree-cut width	red■	exclusion	upper bound
tree-independence number	red■	exclusion	upper bound
tree-partition-width	red■	exclusion	upper bound
treebandwidth	red■	exclusion	upper bound
treedepth	magenta■	exclusion	unknown to HOPS
treelength	magenta■	exclusion	unknown to HOPS
treespan	orange■	unknown to HOPS	upper bound
treewidth	red■	exclusion	upper bound
twin-cover number	blue■	exclusion	exclusion
twin-width	red■	exclusion	upper bound
vertex connectivity	magenta■	exclusion	unknown to HOPS
vertex cover	blue■	exclusion	exclusion
vertex integrity	magenta■	exclusion	unknown to HOPS
weak coloring number	red■	exclusion	upper bound
weak d-coloring number	red■	exclusion	upper bound
weak inf-coloring number	magenta■	exclusion	unknown to HOPS
weakly sparse	red■	exclusion	upper bound
weakly sparse and merge width	red■	exclusion	upper bound

Results

2019 The Graph Parameter Hierarchy by Sorge
- page 10 : maximum leaf number upper bounds bandwidth by a linear function – Lemma 4.25. The max leaf number $\mathrm{ml}$ strictly upper bounds the bandwidth $\mathrm{bw}$.
1998 A partial $k$-arboretum of graphs with bounded treewidth by Bodlaender
- page 22 : bandwidth – Let $G=(V,E)$ be a graph, and let $f\colon V\to {1,2,\dots,n}$ be a linear ordering of $G$. 1. The \emph{bandwidth} of $f$ is $\max{|f(v)-f(w)| \mid (v,w) \in E}$. … The bandwidth … is the minimum bandwidth … over all possible linear orderings of $G$.
- page 23 : bandwidth upper bounds pathwidth by a linear function – Theorem 44. For every graph $G$, the pathwidth of $G$ is at most the bandwidth of $G$, … Proof. Let $f \colon V\to {1,\dots,n}$ be a linear ordering of $G$ with bandwidth $k$. Then $(X_1,\dots,X_{n-k})$ with $X_i={f^{-1}(i), f^{-1}(i+1), \dots, f^{-1}(i+k)}$ is a path decomposition of $G$ with pathwidth $k$. …
Comparing Graph Parameters by Schröder
- page 21 : graph classes with bounded bandwidth are not bounded distance to planar – Proposition 3.13
- page 26 : graph classes with bounded bandwidth are not bounded distance to perfect – Proposition 3.24
- page 30 : graph classes with bounded bandwidth are not bounded genus – Proposition 3.27
assumed
- bandwidth upper bounds topological bandwidth by a linear function – By definition
- bandwidth is equivalent to bandwidth – assumed
unknown source
- bandwidth upper bounds maximum degree by a linear function – Each vertex has an integer $i$ and may be connected only to vertices whose difference from $i$ is at most $k$. There are at most $k$ bigger and $k$ smaller such neighbors.
- bandwidth upper bounds cutwidth by a linear function and lower bounds it by a polynomial function – Any bandwidth bound cutwidth quadratically. An example where this happens is $(P_n)^k$ which has bandwidth $k$ and cutwidth $O(k^2)$; both seem to be optimal.