Shrub-Depth Calculator

labeled tree model depth

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About Shrub-Depth Calculator

A shrub-depth calculator computing sd(G): minimum depth d of a labeled tree T such that G = connection model on leaves of T with bounded labels. Ganian et al. Captures dense shallow structure. sd=1 for threshold. Related to treedepth. Client-side.

Shrub-Depth Calculator Features

  • sd(G)
  • Tree model
  • Threshold=1
  • vs td
  • Common graphs
Shrub-depth sd(G): depth of optimal tree model with bounded labels. Vertices are leaves; adjacency determined by labels and tree structure. Dense analogue of treedepth. sd=1: threshold graphs. Bounded sd implies bounded clique-width.

How to Use

Select graph:

  • sd: Shrub-depth
  • Model: Tree+labels
  • td: Compare

Tree Model

Build a rooted tree T. Vertices of G are leaves. Each leaf gets a label (bounded number). Adjacency: two leaves connected iff their labels and least common ancestor in T satisfy a connection rule.

Parameter Hierarchy

Bounded shrub-depth implies bounded clique-width. Shrub-depth is to clique-width what treedepth is to treewidth — a depth-bounded version. Captures graphs with shallow dense structure.

Step-by-Step Instructions

  1. 1Select graph.
  2. 2Build tree model.
  3. 3Assign labels.
  4. 4Set connections.
  5. 5Measure depth.

Shrub-Depth Calculator — Frequently Asked Questions

How does shrub-depth relate to treedepth?+

Treedepth: sparse graphs, tree decomposition depth. Shrub-depth: dense graphs, tree model depth. Analogous concepts for different graph density regimes. Both capture 'shallow structure'.

What's special about sd=1?+

Shrub-depth 1 = threshold graphs (nested neighborhood structure). Very simple: can be characterized by a single weight function. Many problems trivial on threshold graphs.

When is shrub-depth useful?+

For dense graphs where treewidth/treedepth are large but structure is simple (few neighborhood types). Social networks, biological networks. Many problems FPT by shrub-depth.

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