bisection bandwidth

Definition: Minimum over bi-partitions into (almost) equal-sized parts, the number of edges between parts.

Relations

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

Results

2022 Expanding the Graph Parameter Hierarchy by Tran
- page 34 : graph classes with bounded c-closure are not bounded bisection bandwidth – Proposition 5.5. $c$-Closure is incomparable to Bisection Width.
- page 34 : graph classes with bounded bisection bandwidth are not bounded c-closure – Proposition 5.5. $c$-Closure is incomparable to Bisection Width.
- page 40 : graph classes with bounded twin-width are not bounded bisection bandwidth – Observation 6.9. Twin-width is incomparable to Bisection Width.
- page 40 : graph classes with bounded bisection bandwidth are not bounded twin-width – Observation 6.9. Twin-width is incomparable to Bisection Width.
Comparing Graph Parameters by Schröder
- page 24 : graph classes with bounded vertex cover are not bounded bisection bandwidth – Proposition 3.20
- page 28 : graph classes with bounded maximum degree are not bounded bisection bandwidth – Proposition 3.26
- page 28 : graph classes with bounded distance to bipartite are not bounded bisection bandwidth – Proposition 3.26
- page 30 : graph classes with bounded bisection bandwidth are not bounded domatic number – Proposition 3.28
- page 30 : graph classes with bounded feedback edge set are not bounded bisection bandwidth – Proposition 3.29
- page 33 : graph classes with bounded bisection bandwidth are not bounded chordality – Proposition 3.31
- page 33 : graph classes with bounded bisection bandwidth are not bounded clique-width – Proposition 3.32
- page 33 : graph classes with bounded bisection bandwidth are not bounded maximum clique – Proposition 3.33
assumed
- bisection bandwidth upper bounds edge connectivity by a linear function – By definition
- bisection bandwidth is equivalent to bisection bandwidth – assumed
unknown source
- topological bandwidth upper bounds bisection bandwidth by a linear function – Order vertices by their bandwidth integer. We split the graph in the middle of this ordering. There are at most roughly $k^2/2$ edges over this split.
- graph class grid is not constant bisection bandwidth
- outerplanar upper bounds bisection bandwidth by a constant