assumed

Is axiomatic knowledge from the viewpoint of HOPS website.

• complete upper bounds connected by a constant – by definition
• complete upper bounds cluster by a constant – by definition
• tree upper bounds connected by a constant – by definition
• tree upper bounds forest by a constant – by definition
• path upper bounds connected by a constant – by definition
• path upper bounds linear forest by a constant – by definition
• star upper bounds connected by a constant – by definition
• star upper bounds stars by a constant – by definition
• cycle upper bounds connected by a constant – by definition
• cycle upper bounds cycles by a constant – by definition
• graph class chordal has contained in perfect
• graph class perfect does not have contained in chordal
• graph class cograph has contained in perfect
• graph class perfect does not have contained in cograph
• graph class bipartite has contained in perfect
• graph class perfect does not have contained in bipartite
• graph class cluster has contained in interval
• graph class interval does not have contained in cluster
• graph class cluster has contained in cograph
• graph class cograph does not have contained in cluster
• graph class linear forest has contained in interval
• graph class interval does not have contained in linear forest
• graph class stars has contained in interval
• graph class interval does not have contained in stars
• graph class interval has contained in chordal
• graph class chordal does not have contained in interval
• graph class co-cluster has contained in cograph
• graph class cograph does not have contained in co-cluster
• graph class forest has contained in bipartite
• graph class bipartite does not have contained in forest
• graph class outerplanar has contained in planar
• graph class planar does not have contained in outerplanar
• graph class outerplanar has contained in series-parallel
• graph class series-parallel does not have contained in outerplanar
• graph class complete has contained in co-cluster
• graph class co-cluster does not have contained in complete
• graph class block has contained in chordal
• graph class chordal does not have contained in block
• graph class cluster has contained in block
• graph class block does not have contained in cluster
• graph class linear forest has contained in forest
• graph class forest does not have contained in linear forest
• graph class forest has contained in block
• graph class block does not have contained in forest
• graph classes that are edgeless have contained in linear forest
• graph class linear forest is not edgeless
• graph class stars has contained in forest
• graph class forest does not have contained in stars
• graph classes that are edgeless have contained in stars
• graph class stars is not edgeless
• graph classes that are edgeless have contained in co-cluster
• graph class co-cluster is not edgeless
• graph class grid has contained in planar
• graph class planar does not have contained in grid
• graph class grid has contained in bipartite
• graph class bipartite does not have contained in grid
• graph class grid is connected
• graph classes that are connected do not have contained in grid
• graph classes that are edgeless have contained in cluster
• graph class cluster is not edgeless
• graph class star has contained in tree
• graph class tree does not have contained in star
• graph class path has contained in tree
• graph class tree does not have contained in path
• graph class path has contained in grid
• graph class grid does not have contained in path
• pathwidth+maxdegree upper bounds pathwidth by a linear function – by definition
• pathwidth+maxdegree upper bounds maximum degree by a linear function – by definition
• degree treewidth upper bounds maximum degree by a linear function – by definition
• degree treewidth upper bounds treewidth by a linear function – by definition
• complete upper bounds distance to complete by a constant – by definition
• co-cluster upper bounds distance to co-cluster by a constant – by definition
• cograph upper bounds distance to cograph by a constant – by definition
• cluster upper bounds distance to cluster by a constant – by definition
• linear forest upper bounds distance to linear forest by a constant – by definition
• outerplanar upper bounds distance to outerplanar by a constant – by definition
• block upper bounds distance to block by a constant – by definition
• edgeless upper bounds distance to edgeless by a constant – by definition
• forest upper bounds distance to forest by a constant – by definition
• bipartite upper bounds distance to bipartite by a constant – by definition
• planar upper bounds distance to planar by a constant – by definition
• chordal upper bounds distance to chordal by a constant – by definition
• stars upper bounds distance to stars by a constant – by definition
• perfect upper bounds distance to perfect by a constant – by definition
• interval upper bounds distance to interval by a constant – by definition
• maximum degree upper bounds distance to maximum degree by a linear function – by definition
• bounded components upper bounds distance to bounded components by a linear function – by definition
• diameter+max degree upper bounds diameter by a linear function – by definition
• diameter+max degree upper bounds maximum degree by a linear function – by definition
• vertex integrity upper bounds distance to bounded components by a linear function – By definition
• distance to bounded components upper bounds vertex integrity by a linear function – By definition
• bandwidth upper bounds topological bandwidth by a linear function – By definition
• twin-cover number upper bounds distance to cluster by a linear function – By definition
• vertex cover upper bounds twin-cover number by a linear function – By definition
• average degree upper bounds minimum degree by a linear function – By definition
• diameter upper bounds average distance by a linear function – By definition
• maximum matching upper bounds maximum induced matching by a linear function – By definition
• distance to interval upper bounds boxicity by a linear function – By definition
• bisection bandwidth upper bounds edge connectivity by a linear function – By definition
• edgeless upper bounds bounded components by a constant – By definition
• grid upper bounds maximum degree by a constant – By definition
• bounded components upper bounds maximum degree by a linear function – By definition
• linear forest upper bounds maximum degree by a constant – By definition
• cycles upper bounds maximum degree by a constant – By definition
• diameter upper bounds treelength by a linear function – By definition
• graph classes that are edgeless are not connected – By definition
• graph classes that are edgeless do not have contained in cycles – By definition
• diameter+max degree upper bounds bounded components by an exponential function – folklore
• maximum independent set upper bounds bipartite number by a linear function – folklore: Each of the parts of the maximum induced bipartite subgraph is an independent set. Hence, the bipartite number is at most twice the size of the maximum independent set.
• bipartite number upper bounds maximum independent set by a linear function – As one can pick the maximum independent set as one side of an induced bipartite subgraph we know that the maximum one has size at least the size of the maximum independent set.
• graph classes with bounded minimum degree are not bounded average degree – folklore
• size upper bounds vertex cover by a linear function – By definition
• size upper bounds maximum leaf number by a linear function – By definition
• size upper bounds distance to complete by a linear function – By definition
• size upper bounds bounded components by a linear function – By definition
• graph classes with bounded size do not have contained in planar – By definition
• graph classes with bounded size do not have contained in perfect – By definition
• graph classes with bounded size are not connected – By definition
• sparse twin-width upper bounds twin-width by a linear function – by definition
• sparse twin-width upper bounds bounded expansion by a linear function – by definition
• maximum matching on bipartite graphs upper bounds bipartite by a linear function – by definition
• maximum matching on bipartite graphs upper bounds maximum matching by a linear function – by definition
• weakly sparse and merge width upper bounds weakly sparse by a constant – by definition
• weakly sparse and merge width upper bounds merge-width by a linear function – by definition
• connected is equivalent to connected – assumed
• bipartite is equivalent to bipartite – assumed
• block is equivalent to block – assumed
• chordal is equivalent to chordal – assumed
• cluster is equivalent to cluster – assumed
• co-cluster is equivalent to co-cluster – assumed
• cograph is equivalent to cograph – assumed
• complete is equivalent to complete – assumed
• forest is equivalent to forest – assumed
• tree is equivalent to tree – assumed
• interval is equivalent to interval – assumed
• edgeless is equivalent to edgeless – assumed
• linear forest is equivalent to linear forest – assumed
• path is equivalent to path – assumed
• outerplanar is equivalent to outerplanar – assumed
• perfect is equivalent to perfect – assumed
• planar is equivalent to planar – assumed
• stars is equivalent to stars – assumed
• star is equivalent to star – assumed
• cycles is equivalent to cycles – assumed
• cycle is equivalent to cycle – assumed
• grid is equivalent to grid – assumed
• series-parallel is equivalent to series-parallel – assumed
• size is equivalent to size – assumed
• vertex cover is equivalent to vertex cover – assumed
• maximum matching is equivalent to maximum matching – assumed
• vertex integrity is equivalent to vertex integrity – assumed
• treedepth is equivalent to treedepth – assumed
• clique cover number is equivalent to clique cover number – assumed
• maximum independent set is equivalent to maximum independent set – assumed
• domination number is equivalent to domination number – assumed
• twin-cover number is equivalent to twin-cover number – assumed
• edge clique cover number is equivalent to edge clique cover number – assumed
• neighborhood diversity is equivalent to neighborhood diversity – assumed
• modular-width is equivalent to modular-width – assumed
• iterated type partitions is equivalent to iterated type partitions – assumed
• maximum leaf number is equivalent to maximum leaf number – assumed
• feedback edge set is equivalent to feedback edge set – assumed
• genus is equivalent to genus – assumed
• cutwidth is equivalent to cutwidth – assumed
• carving-width is equivalent to carving-width – assumed
• bandwidth is equivalent to bandwidth – assumed
• topological bandwidth is equivalent to topological bandwidth – assumed
• bisection bandwidth is equivalent to bisection bandwidth – assumed
• maximum degree is equivalent to maximum degree – assumed
• c-closure is equivalent to c-closure – assumed
• feedback vertex set is equivalent to feedback vertex set – assumed
• shrub-depth is equivalent to shrub-depth – assumed
• linear clique-width is equivalent to linear clique-width – assumed
• pathwidth is equivalent to pathwidth – assumed
• pathwidth+maxdegree is equivalent to pathwidth+maxdegree – assumed
• d-path-free is equivalent to d-path-free – assumed
• treewidth is equivalent to treewidth – assumed
• mm-width is equivalent to mm-width – assumed
• tree-partition-width is equivalent to tree-partition-width – assumed
• edge-cut width is equivalent to edge-cut width – assumed
• tree-cut width is equivalent to tree-cut width – assumed
• slim tree-cut width is equivalent to slim tree-cut width – assumed
• edge-treewidth is equivalent to edge-treewidth – assumed
• overlap treewidth is equivalent to overlap treewidth – assumed
• degree treewidth is equivalent to degree treewidth – assumed
• domino treewidth is equivalent to domino treewidth – assumed
• treespan is equivalent to treespan – assumed
• treebandwidth is equivalent to treebandwidth – assumed
• contraction complexity is equivalent to contraction complexity – assumed
• branch width is equivalent to branch width – assumed
• clique-width is equivalent to clique-width – assumed
• clique-tree-width is equivalent to clique-tree-width – assumed
• rank-width is equivalent to rank-width – assumed
• linear rank-width is equivalent to linear rank-width – assumed
• boolean width is equivalent to boolean width – assumed
• twin-width is equivalent to twin-width – assumed
• radius-inf flip-width is equivalent to radius-inf flip-width – assumed
• radius-r flip-width is equivalent to radius-r flip-width – assumed
• flip-width is equivalent to flip-width – assumed
• weak d-coloring number is equivalent to weak d-coloring number – assumed
• weak inf-coloring number is equivalent to weak inf-coloring number – assumed
• weak coloring number is equivalent to weak coloring number – assumed
• strong d-coloring number is equivalent to strong d-coloring number – assumed
• strong inf-coloring number is equivalent to strong inf-coloring number – assumed
• strong coloring number is equivalent to strong coloring number – assumed
• d-admissibility is equivalent to d-admissibility – assumed
• admissibility is equivalent to admissibility – assumed
• merge-width is equivalent to merge-width – assumed
• book thickness is equivalent to book thickness – assumed
• h-index is equivalent to h-index – assumed
• acyclic chromatic number is equivalent to acyclic chromatic number – assumed
• odd cycle transversal is equivalent to odd cycle transversal – assumed
• degeneracy is equivalent to degeneracy – assumed
• chromatic number is equivalent to chromatic number – assumed
• average degree is equivalent to average degree – assumed
• minimum degree is equivalent to minimum degree – assumed
• maximum clique is equivalent to maximum clique – assumed
• edge connectivity is equivalent to edge connectivity – assumed
• vertex connectivity is equivalent to vertex connectivity – assumed
• boxicity is equivalent to boxicity – assumed
• chordality is equivalent to chordality – assumed
• maximum induced matching is equivalent to maximum induced matching – assumed
• diameter is equivalent to diameter – assumed
• average distance is equivalent to average distance – assumed
• domatic number is equivalent to domatic number – assumed
• arboricity is equivalent to arboricity – assumed
• mim-width is equivalent to mim-width – assumed
• sim-width is equivalent to sim-width – assumed
• module-width is equivalent to module-width – assumed
• tree-independence number is equivalent to tree-independence number – assumed
• NLC-width is equivalent to NLC-width – assumed
• NLCT-width is equivalent to NLCT-width – assumed
• linear NLC-width is equivalent to linear NLC-width – assumed
• bounded components is equivalent to bounded components – assumed
• distance to complete is equivalent to distance to complete – assumed
• distance to co-cluster is equivalent to distance to co-cluster – assumed
• distance to cograph is equivalent to distance to cograph – assumed
• distance to cluster is equivalent to distance to cluster – assumed
• distance to linear forest is equivalent to distance to linear forest – assumed
• distance to outerplanar is equivalent to distance to outerplanar – assumed
• distance to block is equivalent to distance to block – assumed
• distance to edgeless is equivalent to distance to edgeless – assumed
• distance to forest is equivalent to distance to forest – assumed
• distance to bipartite is equivalent to distance to bipartite – assumed
• distance to planar is equivalent to distance to planar – assumed
• distance to chordal is equivalent to distance to chordal – assumed
• distance to stars is equivalent to distance to stars – assumed
• distance to perfect is equivalent to distance to perfect – assumed
• distance to interval is equivalent to distance to interval – assumed
• distance to maximum degree is equivalent to distance to maximum degree – assumed
• distance to bounded components is equivalent to distance to bounded components – assumed
• bipartite number is equivalent to bipartite number – assumed
• treelength is equivalent to treelength – assumed
• diameter+max degree is equivalent to diameter+max degree – assumed
• nowhere dense is equivalent to nowhere dense – assumed
• bounded expansion is equivalent to bounded expansion – assumed
• sparse twin-width is equivalent to sparse twin-width – assumed
• monadically stable is equivalent to monadically stable – assumed
• monadically dependent is equivalent to monadically dependent – assumed
• excluded minor is equivalent to excluded minor – assumed
• excluded planar minor is equivalent to excluded planar minor – assumed
• excluded top-minor is equivalent to excluded top-minor – assumed
• chi-bounded is equivalent to chi-bounded – assumed
• weakly sparse is equivalent to weakly sparse – assumed
• maximum matching on bipartite graphs is equivalent to maximum matching on bipartite graphs – assumed
• weakly sparse and merge width is equivalent to weakly sparse and merge width – assumed