- FindStat

Identifier

Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001391: Graphs ⟶ ℤ

Values

['A',1] => ([],1) => ([],1) => ([(0,1)],2) => 1
['A',2] => ([(0,2),(1,2)],3) => ([(1,2)],3) => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
['B',2] => ([(0,3),(1,3),(3,2)],4) => ([(2,3)],4) => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 3

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$F_{1} = q$

Description

The disjunction number of a graph.
Let

$V_n$ be the power set of

$\{1,\dots,n\}$ and let

$E_n=\{(a,b)| a,b\in V_n, a\neq b, a\cap b=\emptyset\}$ . Then the disjunction number of a graph

$G$ is the smallest integer

$n$ such that

$(V_n, E_n)$ has an induced subgraph isomorphic to

$G$ .

Map

incomparability graph

Description

The incomparability graph of a poset.

Map

to root poset

Description

The root poset of a finite Cartan type.
This is the poset on the set of positive roots of its root system where

$\alpha \prec \beta$ if

$\beta - \alpha$ is a simple root.

Map

cone

Description

The cone of a graph.
The cone of a graph is obtained by joining a new vertex to all the vertices of the graph. The added vertex is called a universal vertex or a dominating vertex.