Identifier
Values
0 => ([(0,1)],2) => ([],2) => 1
1 => ([(0,1)],2) => ([],2) => 1
00 => ([(0,2),(2,1)],3) => ([],3) => 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(2,3)],4) => 2
10 => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(2,3)],4) => 2
11 => ([(0,2),(2,1)],3) => ([],3) => 1
000 => ([(0,3),(2,1),(3,2)],4) => ([],4) => 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => 3
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => ([(2,5),(3,4)],6) => 2
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => 3
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => 3
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => ([(2,5),(3,4)],6) => 2
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => 3
111 => ([(0,3),(2,1),(3,2)],4) => ([],4) => 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => ([],7) => 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => ([],7) => 1
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Description
The tree-depth of a graph.
The tree-depth $\operatorname{td}(G)$ of a graph $G$ whose connected components are $G_1,\ldots,G_p$ is recursively defined as
$$\operatorname{td}(G)=\begin{cases} 1, & \text{if }|G|=1\\ 1 + \min_{v\in V} \operatorname{td}(G-v), & \text{if } p=1 \text{ and } |G| > 1\\ \max_{i=1}^p \operatorname{td}(G_i), & \text{otherwise} \end{cases}$$
Nešetřil and Ossona de Mendez [2] proved that the tree-depth of a connected graph is equal to its minimum elimination tree height and its centered chromatic number (fewest colors needed for a vertex coloring where every connected induced subgraph has a color that appears exactly once).
Tree-depth is strictly greater than pathwidth. A longest path in $G$ has at least $\operatorname{td}(G)$ vertices [3].
The tree-depth $\operatorname{td}(G)$ of a graph $G$ whose connected components are $G_1,\ldots,G_p$ is recursively defined as
$$\operatorname{td}(G)=\begin{cases} 1, & \text{if }|G|=1\\ 1 + \min_{v\in V} \operatorname{td}(G-v), & \text{if } p=1 \text{ and } |G| > 1\\ \max_{i=1}^p \operatorname{td}(G_i), & \text{otherwise} \end{cases}$$
Nešetřil and Ossona de Mendez [2] proved that the tree-depth of a connected graph is equal to its minimum elimination tree height and its centered chromatic number (fewest colors needed for a vertex coloring where every connected induced subgraph has a color that appears exactly once).
Tree-depth is strictly greater than pathwidth. A longest path in $G$ has at least $\operatorname{td}(G)$ vertices [3].
Map
poset of factors
Description
The poset of factors of a binary word.
This is the partial order on the set of distinct factors of a binary word, such that $u < v$ if and only if $u$ is a factor of $v$.
This is the partial order on the set of distinct factors of a binary word, such that $u < v$ if and only if $u$ is a factor of $v$.
Map
incomparability graph
Description
The incomparability graph of a poset.
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