distance to bipartite
functionally equivalent to: odd cycle transversal
Definition: Minimum number of vertices removed to make the graph into bipartite
Relations
Other | Relation from | Relation to | |
---|---|---|---|
acyclic chromatic number | ■ | exclusion | exclusion |
admissibility | ■ | exclusion | exclusion |
arboricity | ■ | exclusion | exclusion |
average degree | ■ | exclusion | exclusion |
average distance | ■ | exclusion | exclusion |
bandwidth | ■ | exclusion | exclusion |
bipartite | ■ | upper bound | exclusion |
bipartite number | ■ | exclusion | exclusion |
bisection bandwidth | ■ | exclusion | exclusion |
block | ■ | unbounded | exclusion |
book thickness | ■ | exclusion | exclusion |
boolean width | ■ | exclusion | exclusion |
bounded components | ■ | exclusion | exclusion |
bounded expansion | ■ | exclusion | avoids |
boxicity | ■ | exclusion | exclusion |
branch width | ■ | exclusion | exclusion |
c-closure | ■ | exclusion | exclusion |
carving-width | ■ | exclusion | exclusion |
chi-bounded | ■ | exclusion | unknown to HOPS |
chordal | ■ | unbounded | exclusion |
chordality | ■ | exclusion | upper bound |
chromatic number | ■ | exclusion | upper bound |
clique cover number | ■ | exclusion | exclusion |
clique-tree-width | ■ | exclusion | exclusion |
clique-width | ■ | exclusion | exclusion |
cluster | ■ | unbounded | exclusion |
co-cluster | ■ | unbounded | exclusion |
cograph | ■ | unbounded | exclusion |
complete | ■ | unbounded | exclusion |
connected | ■ | exclusion | avoids |
contraction complexity | ■ | exclusion | exclusion |
cutwidth | ■ | exclusion | exclusion |
cycle | ■ | upper bound | exclusion |
cycles | ■ | unbounded | exclusion |
d-admissibility | ■ | exclusion | unknown to HOPS |
d-path-free | ■ | exclusion | exclusion |
degeneracy | ■ | exclusion | exclusion |
degree treewidth | ■ | exclusion | exclusion |
diameter | ■ | exclusion | exclusion |
diameter+max degree | ■ | exclusion | exclusion |
distance to bipartite | ■ | equal | equal |
distance to block | ■ | exclusion | exclusion |
distance to bounded components | ■ | exclusion | exclusion |
distance to chordal | ■ | exclusion | exclusion |
distance to cluster | ■ | exclusion | exclusion |
distance to co-cluster | ■ | exclusion | exclusion |
distance to cograph | ■ | exclusion | exclusion |
distance to complete | ■ | exclusion | exclusion |
distance to edgeless | ■ | upper bound | exclusion |
distance to forest | ■ | upper bound | exclusion |
distance to interval | ■ | exclusion | exclusion |
distance to linear forest | ■ | upper bound | exclusion |
distance to maximum degree | ■ | exclusion | exclusion |
distance to outerplanar | ■ | exclusion | exclusion |
distance to perfect | ■ | exclusion | upper bound |
distance to planar | ■ | exclusion | exclusion |
distance to stars | ■ | upper bound | exclusion |
domatic number | ■ | exclusion | exclusion |
domination number | ■ | exclusion | exclusion |
domino treewidth | ■ | exclusion | exclusion |
edge clique cover number | ■ | exclusion | exclusion |
edge connectivity | ■ | exclusion | exclusion |
edge-cut width | ■ | unknown to HOPS | exclusion |
edge-treewidth | ■ | exclusion | exclusion |
edgeless | ■ | upper bound | avoids |
excluded minor | ■ | exclusion | avoids |
excluded planar minor | ■ | unknown to HOPS | avoids |
excluded top-minor | ■ | exclusion | avoids |
feedback edge set | ■ | upper bound | exclusion |
feedback vertex set | ■ | upper bound | exclusion |
flip-width | ■ | exclusion | unknown to HOPS |
forest | ■ | upper bound | exclusion |
genus | ■ | exclusion | exclusion |
grid | ■ | upper bound | exclusion |
h-index | ■ | exclusion | exclusion |
interval | ■ | unbounded | exclusion |
iterated type partitions | ■ | exclusion | exclusion |
linear clique-width | ■ | exclusion | exclusion |
linear forest | ■ | upper bound | exclusion |
linear NLC-width | ■ | exclusion | exclusion |
linear rank-width | ■ | exclusion | exclusion |
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 | ■ | upper bound | exclusion |
maximum matching on bipartite graphs | ■ | upper bound | exclusion |
merge-width | ■ | exclusion | unknown to HOPS |
mim-width | ■ | exclusion | unknown to HOPS |
minimum degree | ■ | exclusion | exclusion |
mm-width | ■ | exclusion | exclusion |
modular-width | ■ | exclusion | exclusion |
module-width | ■ | exclusion | exclusion |
monadically dependent | ■ | exclusion | unknown to HOPS |
monadically stable | ■ | exclusion | unknown to HOPS |
neighborhood diversity | ■ | exclusion | exclusion |
NLC-width | ■ | exclusion | exclusion |
NLCT-width | ■ | exclusion | exclusion |
nowhere dense | ■ | exclusion | unknown to HOPS |
odd cycle transversal | ■ | equal | equal |
outerplanar | ■ | unknown to HOPS | exclusion |
overlap treewidth | ■ | exclusion | exclusion |
path | ■ | upper bound | exclusion |
pathwidth | ■ | exclusion | exclusion |
pathwidth+maxdegree | ■ | exclusion | exclusion |
perfect | ■ | unbounded | exclusion |
planar | ■ | unbounded | exclusion |
radius-inf flip-width | ■ | exclusion | exclusion |
radius-r flip-width | ■ | exclusion | unknown to HOPS |
rank-width | ■ | exclusion | exclusion |
series-parallel | ■ | unknown to HOPS | unknown to HOPS |
shrub-depth | ■ | exclusion | exclusion |
sim-width | ■ | exclusion | unknown to HOPS |
size | ■ | upper bound | exclusion |
slim tree-cut width | ■ | exclusion | exclusion |
sparse twin-width | ■ | exclusion | exclusion |
star | ■ | upper bound | exclusion |
stars | ■ | upper bound | exclusion |
strong coloring number | ■ | exclusion | exclusion |
strong d-coloring number | ■ | exclusion | unknown to HOPS |
strong inf-coloring number | ■ | exclusion | exclusion |
topological bandwidth | ■ | exclusion | exclusion |
tree | ■ | upper bound | exclusion |
tree-cut width | ■ | exclusion | exclusion |
tree-independence number | ■ | exclusion | unknown to HOPS |
tree-partition-width | ■ | exclusion | exclusion |
treebandwidth | ■ | exclusion | exclusion |
treedepth | ■ | exclusion | exclusion |
treelength | ■ | exclusion | unknown to HOPS |
treespan | ■ | exclusion | exclusion |
treewidth | ■ | exclusion | exclusion |
twin-cover number | ■ | exclusion | exclusion |
twin-width | ■ | exclusion | exclusion |
vertex connectivity | ■ | unknown to HOPS | unknown to HOPS |
vertex cover | ■ | upper bound | exclusion |
vertex integrity | ■ | exclusion | exclusion |
weak coloring number | ■ | exclusion | exclusion |
weak d-coloring number | ■ | exclusion | unknown to HOPS |
weak inf-coloring number | ■ | exclusion | exclusion |
weakly sparse | ■ | exclusion | unknown to HOPS |
weakly sparse and merge width | ■ | exclusion | exclusion |
Results
- 2022 Expanding the Graph Parameter Hierarchy by Tran
- page 38 : graph classes with bounded twin-width are not bounded distance to bipartite – Proposition 6.6. Twin-width is incomparable to Distance to Bipartite.
- page 38 : graph classes with bounded distance to bipartite are not bounded twin-width – Proposition 6.6. Twin-width is incomparable to Distance to Bipartite.
- 2011 Chordal Bipartite Graphs with High Boxicity by Chandran, Francis, Mathew
- page 9 : graph classes with bounded distance to bipartite are not bounded boxicity – Theorem 2 For any $b \in \mathbb N^+$, there exists a chordal bipartite graph $G$ (ed: i.e. bipartite graph with no induced cycle on more than 4 vertices) with $\mathrm{box}(G) > b$.
- assumed
- bipartite upper bounds distance to bipartite by a constant – by definition
- distance to bipartite is equivalent to distance to bipartite – assumed
- Comparing Graph Parameters by Schröder
- page 20 : graph classes with bounded distance to bipartite are not bounded distance to chordal – Proposition 3.12
- page 20 : graph classes with bounded distance to bipartite are not bounded domatic number – Proposition 3.12
- page 28 : graph classes with bounded distance to bipartite are not bounded clique-width – Proposition 3.26
- page 28 : graph classes with bounded distance to bipartite are not bounded bisection bandwidth – Proposition 3.26
- unknown source
- odd cycle transversal is equivalent to distance to bipartite – Bipartite graphs is the graph class without any odd cycles.
- distance to bipartite is equivalent to odd cycle transversal – Bipartite graphs is the graph class without any odd cycles.
- distance to bipartite upper bounds chromatic number by a linear function – Removed vertices get one color each and we need only $2$ colors for the rest.