vertex cover
abbr: vc
tags: vertex removal
equivalent to: vertex cover, distance to edgeless
Relations
| Other | Relation from | Relation to |
|---|---|---|
| acyclic chromatic number | exclusion | upper bound |
| arboricity | exclusion | upper bound |
| average degree | exclusion | upper bound |
| average distance | exclusion | upper bound |
| bandwidth | exclusion | exclusion |
| bipartite | unbounded | unknown to HOPS |
| bipartite number | exclusion | upper bound |
| bisection bandwidth | exclusion | exclusion |
| block | unbounded | unknown to HOPS |
| 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 | exclusion |
| carving-width | exclusion | exclusion |
| chordal | unbounded | unknown to HOPS |
| 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 | unknown to HOPS |
| co-cluster | unbounded | unknown to HOPS |
| cograph | unbounded | unknown to HOPS |
| complete | unbounded | exclusion |
| connected | unbounded | unknown to HOPS |
| cutwidth | exclusion | exclusion |
| cycle | unbounded | exclusion |
| cycles | unbounded | exclusion |
| d-path-free | exclusion | upper bound |
| degeneracy | exclusion | upper bound |
| degree treewidth | exclusion | exclusion |
| diameter | exclusion | upper bound |
| diameter+max degree | exclusion | exclusion |
| disjoint cycles | unbounded | exclusion |
| distance to bipartite | exclusion | upper bound |
| distance to block | exclusion | upper bound |
| distance to bounded components | exclusion | upper bound |
| distance to chordal | exclusion | upper bound |
| distance to cluster | exclusion | upper bound |
| distance to co-cluster | exclusion | upper bound |
| distance to cograph | exclusion | upper bound |
| distance to complete | exclusion | exclusion |
| distance to edgeless | equal | equal |
| distance to forest | exclusion | upper bound |
| distance to interval | exclusion | upper bound |
| distance to linear forest | exclusion | upper bound |
| distance to maximum degree | exclusion | upper bound |
| distance to outerplanar | exclusion | upper bound |
| distance to perfect | exclusion | upper bound |
| distance to planar | exclusion | upper bound |
| distance to stars | exclusion | upper bound |
| domatic number | exclusion | upper bound |
| domination number | exclusion | exclusion |
| edge clique cover number | exclusion | exclusion |
| edge connectivity | exclusion | upper bound |
| edgeless | constant | exclusion |
| feedback edge set | exclusion | exclusion |
| feedback vertex set | exclusion | upper bound |
| forest | unbounded | exclusion |
| genus | exclusion | exclusion |
| girth | exclusion | upper bound |
| grid | unbounded | exclusion |
| h-index | exclusion | upper bound |
| inf-flip-width | exclusion | upper bound |
| interval | unbounded | unknown to HOPS |
| iterated type partitions | exclusion | upper bound |
| linear clique-width | exclusion | upper bound |
| linear forest | unbounded | exclusion |
| linear NLC-width | exclusion | upper bound |
| linear rank-width | exclusion | upper bound |
| maximum clique | exclusion | upper bound |
| maximum degree | exclusion | exclusion |
| maximum independent set | exclusion | exclusion |
| maximum induced matching | exclusion | upper bound |
| maximum leaf number | exclusion | exclusion |
| maximum matching | unknown to HOPS | tight bounds |
| maximum matching on bipartite graphs | tight bounds | unknown to HOPS |
| mim-width | exclusion | upper bound |
| minimum degree | exclusion | upper bound |
| mm-width | exclusion | upper bound |
| modular-width | exclusion | upper bound |
| module-width | exclusion | upper bound |
| neighborhood diversity | exclusion | upper bound |
| NLC-width | exclusion | upper bound |
| NLCT-width | exclusion | upper bound |
| odd cycle transversal | exclusion | upper bound |
| outerplanar | unbounded | exclusion |
| path | unbounded | exclusion |
| pathwidth | exclusion | upper bound |
| pathwidth+maxdegree | exclusion | exclusion |
| perfect | unbounded | unknown to HOPS |
| planar | unbounded | exclusion |
| radius-r flip-width | exclusion | upper bound |
| rank-width | exclusion | upper bound |
| shrub-depth | exclusion | upper bound |
| sim-width | exclusion | upper bound |
| star | constant | exclusion |
| stars | unbounded | exclusion |
| topological bandwidth | exclusion | exclusion |
| tree | unbounded | exclusion |
| tree-independence number | exclusion | upper bound |
| treedepth | exclusion | upper bound |
| treelength | exclusion | upper bound |
| treewidth | exclusion | upper bound |
| twin-cover number | exclusion | upper bound |
| twin-width | exclusion | upper bound |
| vertex connectivity | unknown to HOPS | unknown to HOPS |
| vertex integrity | exclusion | upper bound |
Results
- 2022 Expanding the Graph Parameter Hierarchy by Tran
- page 18 : vertex cover upper bounds twin-cover number by a linear function – By definition
- page 18 : graph class complete has unbounded vertex cover – Note that a clique of size $n$ has … a vertex cover number of $n-1$
- page 23 : vertex cover upper bounds neighborhood diversity by an exponential function – Proposition 4.3. Vertex Cover Number strictly upper bounds Neighborhood Diversity.
- page 23 : bounded neighborhood diversity does not imply bounded vertex cover – Proposition 4.3. Vertex Cover Number strictly upper bounds Neighborhood Diversity.
- page 29 : bounded edge clique cover number does not imply bounded vertex cover – Proposition 4.13. Edge Clique Cover Number is incomparable to Vertex Cover Number.
- page 29 : bounded vertex cover does not imply bounded edge clique cover number – Proposition 4.13. Edge Clique Cover Number is incomparable to Vertex Cover Number.
- page 34 : bounded c-closure does not imply bounded vertex cover – Proposition 5.3. $c$-Closure is incomparable to Vertex Cover Number.
- page 34 : bounded vertex cover does not imply bounded c-closure – Proposition 5.3. $c$-Closure is incomparable to Vertex Cover Number.
- 2012 Twin-Cover: Beyond Vertex Cover in Parameterized Algorithmics by Ganian
- page 263 : bounded twin-cover number does not imply bounded vertex cover – The vertex cover of graphs of bounded twin-cover may be arbitrarily large.
- assumed
- vertex cover upper bounds twin-cover number by a linear function – By definition
- https://en.wikipedia.org/wiki/Vertex_cover
- vertex cover – … set of vertices that includes at least one endpoint of every edge of the graph.
- Comparing Graph Parameters by Schröder
- page 15 : bounded vertex cover does not imply bounded domination number – Proposition 3.5
- page 24 : bounded vertex cover does not imply bounded genus – Proposition 3.18
- page 24 : bounded vertex cover does not imply bounded maximum degree – Proposition 3.19
- page 24 : bounded vertex cover does not imply bounded bisection bandwidth – Proposition 3.20
- unknown source
- maximum matching on bipartite graphs upper and lower bounds vertex cover by a linear function – Kőnig’s theorem
- vertex cover upper and lower bounds maximum matching by a linear function – Every edge of the matching needs to be covered by at least one vertex. Path shows lower bound.
- vertex cover upper bounds neighborhood diversity by an exponential function
- vertex cover is equivalent to distance to edgeless
- distance to edgeless is equivalent to vertex cover
- graph class edgeless has constant vertex cover
- graph class star has constant vertex cover – trivially