assumed

Is axiomatic knowledge from the viewpoint of HOPS website.

• connected – A graph is connected if there is a path between any pair of its vertices.
• bipartite – A graph is bipartite if it can be partitioned into two independent sets.
• block – In a block graph every block (maximal 2-connected subgraph) is a clique.
• cluster – Disjoint union of complete graphs.
• cluster – Every connected component is a complete graph.
• cluster – Does not include path on three vertices as an induced subgraph.
• co-cluster – Complete multipartite graph.
• graph class complete is included in graph class connected – by definition
• graph class complete is included in graph class cluster – by definition
• graph class tree is included in graph class connected – by definition
• graph class tree is included in graph class forest – by definition
• graph class path is included in graph class connected – by definition
• graph class path is included in graph class linear forest – by definition
• stars – Disjoint union of stars.
• graph class star is included in graph class connected – by definition
• graph class star is included in graph class stars – by definition
• cycles – Every component is a cycle.
• graph class cycle is included in graph class connected – by definition
• graph class cycle is included in graph class cycles – by definition
• disjoint cycles – All cycles in the graph are disjoint. Can contain arbitrary trees attached to and between the cycles.
• grid – Cartesian product of two paths.
• graph class chordal is included in graph class perfect
• graph class perfect is not included in graph class chordal
• graph class cograph is included in graph class perfect
• graph class perfect is not included in graph class cograph
• graph class bipartite is included in graph class perfect
• graph class perfect is not included in graph class bipartite
• graph class cluster is included in graph class interval
• graph class interval is not included in graph class cluster
• graph class linear forest is included in graph class interval
• graph class interval is not included in graph class linear forest
• graph class stars is included in graph class interval
• graph class interval is not included in graph class stars
• graph class interval is included in graph class chordal
• graph class chordal is not included in graph class interval
• graph class co-cluster is included in graph class cograph
• graph class cograph is not included in graph class co-cluster
• graph class forest is included in graph class bipartite
• graph class bipartite is not included in graph class forest
• graph class outerplanar is included in graph class planar
• graph class planar is not included in graph class outerplanar
• graph class complete is included in graph class co-cluster
• graph class co-cluster is not included in graph class complete
• graph class block is included in graph class chordal
• graph class chordal is not included in graph class block
• graph class cluster is included in graph class block
• graph class block is not included in graph class cluster
• graph class linear forest is included in graph class forest
• graph class forest is not included in graph class linear forest
• graph class disjoint cycles is included in graph class outerplanar
• graph class outerplanar is not included in graph class disjoint cycles
• graph class forest is included in graph class disjoint cycles
• graph class disjoint cycles is not included in graph class forest
• graph class forest is included in graph class block
• graph class block is not included in graph class forest
• graph class edgeless is included in graph class linear forest
• graph class linear forest is not included in graph class edgeless
• graph class stars is included in graph class forest
• graph class forest is not included in graph class stars
• graph class edgeless is included in graph class stars
• graph class stars is not included in graph class edgeless
• graph class edgeless is included in graph class co-cluster
• graph class co-cluster is not included in graph class edgeless
• graph class grid is included in graph class planar
• graph class planar is not included in graph class grid
• graph class grid is included in graph class bipartite
• graph class bipartite is not included in graph class grid
• graph class cycles is included in graph class disjoint cycles
• graph class disjoint cycles is not included in graph class cycles
• graph class grid is included in graph class connected
• graph class connected is not included in graph class grid
• graph class edgeless is included in graph class cluster
• graph class cluster is not included in graph class edgeless
• graph class star is included in graph class tree
• graph class tree is not included in graph class star
• graph class path is included in graph class tree
• graph class tree is not included in graph class path
• graph class path is included in graph class grid
• graph class grid is not included in graph class 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
• graph class complete has constant distance to complete – by definition
• graph class co-cluster has constant distance to co-cluster – by definition
• graph class cograph has constant distance to cograph – by definition
• graph class cluster has constant distance to cluster – by definition
• graph class linear forest has constant distance to linear forest – by definition
• graph class outerplanar has constant distance to outerplanar – by definition
• graph class block has constant distance to block – by definition
• graph class edgeless has constant distance to edgeless – by definition
• graph class forest has constant distance to forest – by definition
• graph class bipartite has constant distance to bipartite – by definition
• graph class planar has constant distance to planar – by definition
• graph class chordal has constant distance to chordal – by definition
• graph class stars has constant distance to stars – by definition
• graph class perfect has constant distance to perfect – by definition
• graph class interval has constant distance to interval – 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
• bipartite number – Bipartite number of $G$ is the maximum order of an induced bipartite subgraph.
• treelength – Treelength of a tree decomposition is the maxmimum distance of two vertices that appear in the same bag. Treelength of a graph is the minimum treelength over tree decompositions.
• 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
• graph class edgeless has constant bounded components – By definition
• graph class grid has constant maximum degree – By definition
• bounded components upper bounds maximum degree by a linear function – By definition
• graph class linear forest has constant maximum degree – By definition
• graph class cycles has constant maximum degree – By definition
• diameter upper bounds treelength 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
• diameter+max degree upper bounds bounded components by an exponential function – folklore
• maximum independent set upper bounds bipartite number by a linear function – folklore
• graphs with bounded maximum matching on bipartite graphs are included in graph class bipartite – by definition
• maximum matching on bipartite graphs upper bounds maximum matching by a linear function – by definition