feedback edge set
tags: edge removal
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 | ■ | exclusion | upper bound |
arboricity | ■ | exclusion | upper bound |
average degree | ■ | exclusion | upper bound |
average distance | ■ | exclusion | exclusion |
bandwidth | ■ | exclusion | exclusion |
bipartite | ■ | unbounded | exclusion |
bipartite number | ■ | exclusion | unknown to HOPS |
bisection bandwidth | ■ | exclusion | exclusion |
block | ■ | unbounded | exclusion |
book thickness | ■ | exclusion | upper bound |
boolean width | ■ | exclusion | upper bound |
bounded components | ■ | exclusion | exclusion |
boxicity | ■ | exclusion | upper bound |
branch width | ■ | exclusion | upper bound |
c-closure | ■ | exclusion | upper bound |
carving-width | ■ | exclusion | exclusion |
chordal | ■ | unbounded | exclusion |
chordality | ■ | exclusion | upper bound |
chromatic number | ■ | exclusion | upper bound |
clique cover number | ■ | exclusion | exclusion |
clique-tree-width | ■ | exclusion | upper bound |
clique-width | ■ | exclusion | upper bound |
cluster | ■ | unbounded | exclusion |
co-cluster | ■ | unbounded | exclusion |
cograph | ■ | unbounded | exclusion |
complete | ■ | unbounded | exclusion |
connected | ■ | unbounded | exclusion |
contraction complexity | ■ | exclusion | exclusion |
cutwidth | ■ | exclusion | exclusion |
cycle | ■ | upper bound | exclusion |
cycles | ■ | unbounded | exclusion |
d-path-free | ■ | exclusion | exclusion |
degeneracy | ■ | exclusion | upper bound |
degree treewidth | ■ | exclusion | exclusion |
diameter | ■ | exclusion | exclusion |
diameter+max degree | ■ | exclusion | exclusion |
disconnected | ■ | unknown to HOPS | unknown to HOPS |
disjoint cycles | ■ | unbounded | exclusion |
distance to bipartite | ■ | exclusion | upper bound |
distance to block | ■ | exclusion | upper bound |
distance to bounded components | ■ | exclusion | exclusion |
distance to chordal | ■ | exclusion | upper bound |
distance to cluster | ■ | exclusion | exclusion |
distance to co-cluster | ■ | exclusion | exclusion |
distance to cograph | ■ | exclusion | exclusion |
distance to complete | ■ | exclusion | exclusion |
distance to disconnected | ■ | exclusion | upper bound |
distance to edgeless | ■ | exclusion | exclusion |
distance to forest | ■ | exclusion | upper bound |
distance to interval | ■ | exclusion | exclusion |
distance to linear forest | ■ | exclusion | exclusion |
distance to maximum degree | ■ | exclusion | exclusion |
distance to outerplanar | ■ | exclusion | upper bound |
distance to perfect | ■ | exclusion | upper bound |
distance to planar | ■ | exclusion | upper bound |
distance to stars | ■ | exclusion | exclusion |
domatic number | ■ | exclusion | upper bound |
domination number | ■ | exclusion | exclusion |
edge clique cover number | ■ | exclusion | exclusion |
edge connectivity | ■ | exclusion | upper bound |
edgeless | ■ | upper bound | exclusion |
feedback edge set | ■ | equal | equal |
feedback vertex set | ■ | exclusion | upper bound |
forest | ■ | upper bound | exclusion |
genus | ■ | exclusion | upper bound |
girth | ■ | exclusion | exclusion |
grid | ■ | unbounded | exclusion |
h-index | ■ | exclusion | exclusion |
inf-flip-width | ■ | exclusion | upper bound |
interval | ■ | unbounded | exclusion |
iterated type partitions | ■ | exclusion | exclusion |
linear clique-width | ■ | exclusion | unknown to HOPS |
linear forest | ■ | upper bound | exclusion |
linear NLC-width | ■ | exclusion | unknown to HOPS |
linear rank-width | ■ | exclusion | unknown to HOPS |
maximum clique | ■ | exclusion | upper bound |
maximum degree | ■ | exclusion | exclusion |
maximum independent set | ■ | exclusion | exclusion |
maximum induced matching | ■ | exclusion | exclusion |
maximum leaf number | ■ | upper bound | exclusion |
maximum matching | ■ | exclusion | exclusion |
maximum matching on bipartite graphs | ■ | unknown to HOPS | exclusion |
mim-width | ■ | exclusion | upper bound |
minimum degree | ■ | exclusion | upper bound |
mm-width | ■ | exclusion | upper bound |
modular-width | ■ | exclusion | exclusion |
module-width | ■ | exclusion | upper bound |
neighborhood diversity | ■ | exclusion | exclusion |
NLC-width | ■ | exclusion | upper bound |
NLCT-width | ■ | exclusion | upper bound |
odd cycle transversal | ■ | exclusion | upper bound |
outerplanar | ■ | unbounded | exclusion |
path | ■ | upper bound | exclusion |
pathwidth | ■ | exclusion | exclusion |
pathwidth+maxdegree | ■ | exclusion | exclusion |
perfect | ■ | unbounded | exclusion |
planar | ■ | unbounded | exclusion |
radius-r flip-width | ■ | exclusion | upper bound |
rank-width | ■ | exclusion | upper bound |
shrub-depth | ■ | exclusion | unknown to HOPS |
sim-width | ■ | exclusion | upper bound |
size | ■ | upper bound | exclusion |
star | ■ | upper bound | exclusion |
stars | ■ | upper bound | exclusion |
topological bandwidth | ■ | exclusion | exclusion |
tree | ■ | upper bound | exclusion |
tree-independence number | ■ | exclusion | upper bound |
treedepth | ■ | exclusion | exclusion |
treelength | ■ | exclusion | unknown to HOPS |
treewidth | ■ | exclusion | upper bound |
twin-cover number | ■ | exclusion | exclusion |
twin-width | ■ | exclusion | upper bound |
vertex connectivity | ■ | exclusion | upper bound |
vertex cover | ■ | exclusion | exclusion |
vertex integrity | ■ | exclusion | exclusion |
Results
- 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 : bounded c-closure does not imply 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 $g \le f$.
- 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)
- assumed
- feedback edge set is equivalent to feedback edge set – assumed
- Comparing Graph Parameters by Schröder
- page 23 : bounded feedback edge set does not imply bounded pathwidth – Proposition 3.16
- page 25 : bounded feedback edge set does not imply bounded distance to interval – Proposition 3.21
- page 25 : bounded feedback edge set does not imply bounded h-index – Proposition 3.22
- page 30 : bounded feedback edge set does not imply bounded bisection bandwidth – Proposition 3.29