feedback vertex set

abbr: fvs

functionally equivalent to: distance to forest

providers: PACE

Definition: The minimum set of vertices $S$ such that every cycle in the graph contains at least one vertex of $S$.

Feedback vertex set, also called cycle transversal, is very simiar to vertex cover but focuses on cycles instead of edges. With bounded feedback vertex set, we can partition the graph into a modulator of $k$ vertices and a forest. It generalizes feedback edge set for the price of having an arbitrary connection from its modulator to the forest.

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	blue■	exclusion	exclusion
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	green■	upper bound	exclusion
distance to forest	yellow■	equal	equal
distance to interval	blue■	exclusion	exclusion
distance to linear forest	green■	upper bound	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	green■	upper bound	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	gray■	unknown to HOPS	unknown to HOPS
edge-treewidth	magenta■	exclusion	unknown to HOPS
edgeless	green■	upper bound	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	green■	upper bound	exclusion
feedback vertex set	yellow■	equal	equal
flip-width	red■	exclusion	upper bound
forest	green■	upper bound	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	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	green■	upper bound	exclusion
maximum matching on bipartite graphs	green■	upper bound	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	magenta■	exclusion	unknown to HOPS
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	magenta■	exclusion	unknown to HOPS
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	magenta■	exclusion	unknown to HOPS
tree-independence number	red■	exclusion	upper bound
tree-partition-width	magenta■	exclusion	unknown to HOPS
treebandwidth	magenta■	exclusion	unknown to HOPS
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	gray■	unknown to HOPS	unknown to HOPS
vertex cover	green■	upper bound	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

2019 The Graph Parameter Hierarchy by Sorge
- page 10 : feedback vertex set upper bounds distance to chordal by a linear function – Lemma 4.20. The feedback edge set number $f$ upper bounds the distance to a chordal graph $c$. We have $c \le f$.
assumed
- feedback vertex set is equivalent to feedback vertex 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 vertex set is equivalent to distance to forest
- distance to forest is equivalent to feedback vertex set
- feedback vertex set upper bounds distance to outerplanar by a computable function