feedback edge set

Definition: In the cardinality of a minimum edge set where every cycle has at least one edge in the edge set.

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	blue■	exclusion	exclusion
bandwidth	blue■	exclusion	exclusion
bipartite	blue■	unbounded	exclusion
bipartite number	blue■	exclusion	exclusion
bisection bandwidth	blue■	exclusion	exclusion
block	blue■	unbounded	exclusion
book thickness	red■	exclusion	upper bound
boolean width	red■	exclusion	upper bound
bounded components	blue■	exclusion	exclusion
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	blue■	exclusion	exclusion
chi-bounded	red■	exclusion	upper bound
chordal	blue■	unbounded	exclusion
chordality	red■	exclusion	upper bound
chromatic number	red■	exclusion	upper bound
clique cover number	blue■	exclusion	exclusion
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	blue■	exclusion	exclusion
cutwidth	blue■	exclusion	exclusion
cycle	green■	upper bound	exclusion
cycles	blue■	unbounded	exclusion
d-admissibility	red■	exclusion	upper bound
d-path-free	blue■	exclusion	exclusion
degeneracy	red■	exclusion	upper bound
degree treewidth	blue■	exclusion	exclusion
diameter	blue■	exclusion	exclusion
diameter+max degree	blue■	exclusion	exclusion
distance to bipartite	red■	exclusion	upper bound
distance to block	red■	exclusion	upper bound
distance to bounded components	blue■	exclusion	exclusion
distance to chordal	red■	exclusion	upper bound
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	red■	exclusion	upper bound
distance to interval	blue■	exclusion	exclusion
distance to linear forest	blue■	exclusion	exclusion
distance to maximum degree	blue■	exclusion	exclusion
distance to outerplanar	red■	exclusion	upper bound
distance to perfect	red■	exclusion	upper bound
distance to planar	red■	exclusion	upper bound
distance to stars	blue■	exclusion	exclusion
domatic number	red■	exclusion	upper bound
domination number	blue■	exclusion	exclusion
domino treewidth	blue■	exclusion	exclusion
edge clique cover number	blue■	exclusion	exclusion
edge connectivity	red■	exclusion	upper bound
edge-cut width	orange■	unknown to HOPS	upper bound
edge-treewidth	red■	exclusion	upper bound
edgeless	green■	upper bound	avoids
excluded minor	red■	exclusion	upper bound
excluded planar minor	gray■	unknown to HOPS	unknown to HOPS
excluded top-minor	red■	exclusion	upper bound
feedback edge set	yellow■	equal	equal
feedback vertex set	red■	exclusion	upper bound
flip-width	red■	exclusion	upper bound
forest	green■	upper bound	exclusion
genus	red■	exclusion	upper bound
grid	blue■	unbounded	exclusion
h-index	blue■	exclusion	exclusion
interval	blue■	unbounded	exclusion
iterated type partitions	blue■	exclusion	exclusion
linear clique-width	magenta■	exclusion	unknown to HOPS
linear forest	green■	upper bound	exclusion
linear NLC-width	magenta■	exclusion	unknown to HOPS
linear rank-width	magenta■	exclusion	unknown to HOPS
maximum clique	red■	exclusion	upper bound
maximum degree	blue■	exclusion	exclusion
maximum independent set	blue■	exclusion	exclusion
maximum induced matching	blue■	exclusion	exclusion
maximum leaf number	green■	upper bound	exclusion
maximum matching	blue■	exclusion	exclusion
maximum matching on bipartite graphs	cyan■	unknown to HOPS	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	blue■	exclusion	exclusion
module-width	red■	exclusion	upper bound
monadically dependent	red■	exclusion	upper bound
monadically stable	red■	exclusion	upper bound
neighborhood diversity	blue■	exclusion	exclusion
NLC-width	red■	exclusion	upper bound
NLCT-width	red■	exclusion	upper bound
nowhere dense	red■	exclusion	upper bound
odd cycle transversal	red■	exclusion	upper bound
outerplanar	cyan■	unknown to HOPS	exclusion
overlap treewidth	red■	exclusion	upper bound
path	green■	upper bound	exclusion
pathwidth	blue■	exclusion	exclusion
pathwidth+maxdegree	blue■	exclusion	exclusion
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	green■	upper bound	exclusion
stars	green■	upper bound	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	blue■	exclusion	exclusion
tree	green■	upper bound	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	blue■	exclusion	exclusion
treelength	magenta■	exclusion	unknown to HOPS
treespan	blue■	exclusion	exclusion
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	blue■	exclusion	exclusion
weak coloring number	red■	exclusion	upper bound
weak d-coloring number	red■	exclusion	upper bound
weak inf-coloring number	blue■	exclusion	exclusion
weakly sparse	red■	exclusion	upper bound
weakly sparse and merge width	red■	exclusion	upper bound

Results

2024 Slim Tree-Cut Width by Ganian, Korchemna
- page 2716 : feedback edge set upper bounds edge-cut width by a computable function – Figure 1
2022 Expanding the Graph Parameter Hierarchy by Tran
- page 32 : feedback edge set upper bounds c-closure by a computable function – Theorem 5.2. Feedback Edge Number strictly upper bounds $c$-Closure.
- page 32 : graph classes with bounded c-closure are not bounded feedback edge set – Theorem 5.2. Feedback Edge Number strictly upper bounds $c$-Closure.
2019 The Graph Parameter Hierarchy by Sorge
- page 10 : feedback edge set upper bounds genus by a linear function – Lemma 4.19. The feedback edge set number $f$ upper bounds the genus $g$. We have $g \le f$.
assumed
- feedback edge set is equivalent to feedback edge set – assumed
unknown source
- feedback edge set upper bounds feedback vertex set by a linear function – Given solution to feedback edge set one can remove one vertex incident to the solution edges to obtain feedback vertex set.
- feedback edge set upper bounds genus by a linear function – Removing $k$ edges creates a forest that is embeddable into the plane. We now add one handle for each of the $k$ edges to get embedding into $k$-handle genus.
- maximum leaf number upper bounds feedback edge set by a polynomial function
- feedback edge set upper bounds c-closure by a computable function
- forest upper bounds feedback edge set by a constant
- maximum leaf number upper bounds feedback edge set by a polynomial function – M. Bentert (personal communication)
Comparing Graph Parameters by Schröder
- page 23 : graph classes with bounded feedback edge set are not bounded pathwidth – Proposition 3.16
- page 25 : graph classes with bounded feedback edge set are not bounded distance to interval – Proposition 3.21
- page 25 : graph classes with bounded feedback edge set are not bounded h-index – Proposition 3.22
- page 30 : graph classes with bounded feedback edge set are not bounded bisection bandwidth – Proposition 3.29