distance to perfect
equivalent to: distance to perfect
Relations
| Other | Relation from | Relation to |
|---|---|---|
| acyclic chromatic number | exclusion | exclusion |
| arboricity | exclusion | exclusion |
| average degree | exclusion | exclusion |
| average distance | exclusion | exclusion |
| bandwidth | exclusion | exclusion |
| bipartite | constant | exclusion |
| bipartite number | exclusion | unknown to HOPS |
| bisection bandwidth | exclusion | exclusion |
| block | constant | exclusion |
| book thickness | exclusion | exclusion |
| boolean width | exclusion | exclusion |
| bounded components | exclusion | exclusion |
| boxicity | exclusion | exclusion |
| branch width | exclusion | exclusion |
| c-closure | exclusion | exclusion |
| carving-width | exclusion | exclusion |
| chordal | constant | exclusion |
| chordality | exclusion | exclusion |
| chromatic number | exclusion | exclusion |
| clique cover number | exclusion | exclusion |
| clique-tree-width | exclusion | exclusion |
| clique-width | exclusion | exclusion |
| cluster | constant | exclusion |
| co-cluster | constant | exclusion |
| cograph | constant | exclusion |
| complete | constant | exclusion |
| connected | unknown to HOPS | unknown to HOPS |
| cutwidth | exclusion | exclusion |
| cycle | constant | exclusion |
| cycles | unbounded | exclusion |
| d-path-free | exclusion | exclusion |
| degeneracy | exclusion | exclusion |
| degree treewidth | exclusion | exclusion |
| diameter | exclusion | exclusion |
| diameter+max degree | exclusion | exclusion |
| disjoint cycles | unbounded | exclusion |
| distance to bipartite | upper bound | exclusion |
| distance to block | upper bound | exclusion |
| distance to bounded components | exclusion | exclusion |
| distance to chordal | upper bound | exclusion |
| distance to cluster | upper bound | exclusion |
| distance to co-cluster | upper bound | exclusion |
| distance to cograph | upper bound | exclusion |
| distance to complete | upper bound | exclusion |
| distance to edgeless | upper bound | exclusion |
| distance to forest | upper bound | exclusion |
| distance to interval | upper bound | exclusion |
| distance to linear forest | upper bound | exclusion |
| distance to maximum degree | exclusion | exclusion |
| distance to outerplanar | exclusion | exclusion |
| distance to planar | exclusion | exclusion |
| distance to stars | upper bound | exclusion |
| domatic number | exclusion | exclusion |
| domination number | exclusion | exclusion |
| edge clique cover number | exclusion | exclusion |
| edge connectivity | exclusion | exclusion |
| edgeless | constant | exclusion |
| feedback edge set | upper bound | exclusion |
| feedback vertex set | upper bound | exclusion |
| forest | constant | exclusion |
| genus | exclusion | exclusion |
| girth | exclusion | exclusion |
| grid | constant | exclusion |
| h-index | exclusion | exclusion |
| inf-flip-width | exclusion | exclusion |
| interval | constant | exclusion |
| iterated type partitions | exclusion | exclusion |
| linear clique-width | exclusion | exclusion |
| linear forest | constant | exclusion |
| linear NLC-width | exclusion | exclusion |
| linear rank-width | exclusion | exclusion |
| maximum clique | exclusion | exclusion |
| maximum degree | exclusion | exclusion |
| maximum independent set | exclusion | exclusion |
| maximum induced matching | exclusion | exclusion |
| maximum leaf number | upper bound | exclusion |
| maximum matching | unknown to HOPS | exclusion |
| maximum matching on bipartite graphs | upper bound | exclusion |
| mim-width | exclusion | unknown to HOPS |
| minimum degree | exclusion | exclusion |
| mm-width | exclusion | exclusion |
| modular-width | exclusion | exclusion |
| module-width | exclusion | exclusion |
| neighborhood diversity | exclusion | exclusion |
| NLC-width | exclusion | exclusion |
| NLCT-width | exclusion | exclusion |
| odd cycle transversal | upper bound | exclusion |
| outerplanar | unbounded | exclusion |
| path | constant | exclusion |
| pathwidth | exclusion | exclusion |
| pathwidth+maxdegree | exclusion | exclusion |
| perfect | constant | unknown to HOPS |
| planar | unbounded | exclusion |
| radius-r flip-width | exclusion | unknown to HOPS |
| rank-width | exclusion | exclusion |
| shrub-depth | exclusion | exclusion |
| sim-width | exclusion | unknown to HOPS |
| star | constant | exclusion |
| stars | constant | exclusion |
| topological bandwidth | exclusion | exclusion |
| tree | constant | exclusion |
| tree-independence number | exclusion | unknown to HOPS |
| treedepth | exclusion | exclusion |
| treelength | exclusion | unknown to HOPS |
| treewidth | exclusion | exclusion |
| twin-cover number | upper bound | exclusion |
| twin-width | exclusion | exclusion |
| vertex connectivity | unknown to HOPS | exclusion |
| vertex cover | upper bound | exclusion |
| vertex integrity | exclusion | exclusion |
Results
- 2022 Expanding the Graph Parameter Hierarchy by Tran
- page 29 : bounded edge clique cover number does not imply bounded distance to perfect – Proposition 4.15. Edge Clique Cover Number is incomparable to Distance to Perfect.
- page 29 : bounded distance to perfect does not imply bounded edge clique cover number – Proposition 4.15. Edge Clique Cover Number is incomparable to Distance to Perfect.
- Comparing Graph Parameters by Schröder
- page 15 : bounded clique cover number does not imply bounded distance to perfect – Proposition 3.6
- page 19 : bounded distance to perfect does not imply bounded chordality – Proposition 3.11
- page 26 : bounded distance to outerplanar does not imply bounded distance to perfect – Proposition 3.23
- page 26 : bounded bandwidth does not imply bounded distance to perfect – Proposition 3.24
- page 26 : bounded genus does not imply bounded distance to perfect – Proposition 3.24
- page 26 : bounded treedepth does not imply bounded distance to perfect – Proposition 3.24
- unknown source
- graph class planar has unbounded distance to perfect
- graph class cycles has unbounded distance to perfect
- bounded bounded components does not imply bounded distance to perfect – By a disjoint union of small components with distance to perfect at least 1.
- assumed
- graph class perfect has constant distance to perfect – by definition