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Identifier

St001963: Graphs ⟶ ℤ

Values

([],1) => 1

([],2) => 1

([(0,1)],2) => 2

([],3) => 1

([(1,2)],3) => 2

([(0,2),(1,2)],3) => 2

([(0,1),(0,2),(1,2)],3) => 3

([],4) => 1

([(2,3)],4) => 2

([(1,3),(2,3)],4) => 2

([(0,3),(1,3),(2,3)],4) => 2

([(0,3),(1,2)],4) => 2

([(0,3),(1,2),(2,3)],4) => 3

([(1,2),(1,3),(2,3)],4) => 3

([(0,3),(1,2),(1,3),(2,3)],4) => 3

([(0,2),(0,3),(1,2),(1,3)],4) => 3

([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3

([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4

([],5) => 1

([(3,4)],5) => 2

([(2,4),(3,4)],5) => 2

([(1,4),(2,4),(3,4)],5) => 2

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

([(1,4),(2,3)],5) => 2

([(1,4),(2,3),(3,4)],5) => 3

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

([(2,3),(2,4),(3,4)],5) => 3

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

([(1,4),(2,3),(2,4),(3,4)],5) => 3

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

([(1,3),(1,4),(2,3),(2,4)],5) => 3

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

([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3

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

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

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

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

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

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

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

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

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

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

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

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

([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4

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

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

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

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

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

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

([],6) => 1

([(4,5)],6) => 2

([(3,5),(4,5)],6) => 2

([(2,5),(3,5),(4,5)],6) => 2

([(1,5),(2,5),(3,5),(4,5)],6) => 2

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2

([(2,5),(3,4)],6) => 2

([(2,5),(3,4),(4,5)],6) => 3

([(1,2),(3,5),(4,5)],6) => 2

([(3,4),(3,5),(4,5)],6) => 3

([(1,5),(2,5),(3,4),(4,5)],6) => 3

([(0,1),(2,5),(3,5),(4,5)],6) => 2

([(2,5),(3,4),(3,5),(4,5)],6) => 3

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 3

([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 3

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 3

([(2,4),(2,5),(3,4),(3,5)],6) => 3

([(0,5),(1,5),(2,4),(3,4)],6) => 2

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 3

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3

([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 3

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 3

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 3

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 3

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 3

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 3

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 3

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 3

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

([(0,5),(1,4),(2,3)],6) => 2

([(1,5),(2,4),(3,4),(3,5)],6) => 3

([(0,1),(2,5),(3,4),(4,5)],6) => 3

([(1,2),(3,4),(3,5),(4,5)],6) => 3

([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3

([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 3

([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 3

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 3

([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 3

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 3

([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 4

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 4

([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 4

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 4

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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].

References

[1] wikipedia:Tree-depth
[2] Ne\u set\u ril, J., Ossona de Mendez, P. Tree-depth, subgraph coloring and homomorphism bounds MathSciNet:2226435
[3] Ne\u set\u ril, J., Ossona de Mendez, P. Sparsity MathSciNet:2920058
[4] Kowalik, Łu., Mucha, M., Nadara, W., Pilipczuk, M., Sorge, M., Wygocki, P. The PACE 2020 parameterized algorithms and computational experiments challenge: treedepth MathSciNet:4189443

Code

@cached_function
def statistic(G):
    if G.order() == 1:
        return 1
    if G.is_connected(): 
        return 1 + min(statistic(G.subgraph(vertices=(u for u in G if u != v))) for v in G)
    return max(statistic(Gi) for Gi in G.connected_components_subgraphs())

Created

Jan 24, 2025 at 22:58 by Lennard Hofmann

Updated

Jan 24, 2025 at 22:58 by Lennard Hofmann