Shrub-Depth a Successful Depth Measure for Dense Graphs Graphs Petr Hlinˇen´Y Faculty of Informatics, Masaryk University Brno, Czech Republic

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Shrub-Depth a successful depth measure for dense graphs graphs Petr Hlinˇen´y Faculty of Informatics, Masaryk University Brno, Czech Republic

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 1 / 19 Shrub-depth measure for dense graphs page.19

Shrub-Depth a successful depth measure for dense graphs graphs Petr Hlinˇený Faculty of Informatics, Masaryk University Brno, Czech Republic Ingredients: joint results with J. Gajarský,R. Ganian, O. Kwon, J. Neˇsetˇril,J. Obdrˇzálek, S. Ordyniak, P. Ossona de Mendez

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 1 / 19 Shrub-depth measure for dense graphs page.19 Measuring Width or Depth?

• Being close to a TREE – “•-width”

sparse dense

tree-width / branch-width – showing a structure clique-width / rank-width – showing a construction

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 2 / 19 Shrub-depth measure for dense graphs page.19 Measuring Width or Depth?

• Being close to a TREE – “•-width”

sparse dense

tree-width / branch-width – showing a structure clique-width / rank-width – showing a construction

• Being close to a STAR – “•-depth”

sparse dense

tree-depth – containment in a structure ??? (will show)

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 2 / 19 Shrub-depth measure for dense graphs page.19 1 Recall: Width Measures

Tree-width tw(G) ≤ k if whole G can be covered by bags of size ≤ k + 1, arranged in a “tree-like fashion”.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 3 / 19 Shrub-depth measure for dense graphs page.19 1 Recall: Width Measures

Tree-width tw(G) ≤ k if whole G can be covered by bags of size ≤ k + 1, arranged in a “tree-like fashion”.

The underlying idea: G is recursively de- composed along small v. separators.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 3 / 19 Shrub-depth measure for dense graphs page.19 1 Recall: Width Measures

Tree-width tw(G) ≤ k if whole G can be covered by bags of size ≤ k + 1, arranged in a “tree-like fashion”.

The underlying idea: G is recursively de- composed along small v. separators.

Structural properties

• Monotone under subgraphs and minors,

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 3 / 19 Shrub-depth measure for dense graphs page.19 1 Recall: Width Measures

Tree-width tw(G) ≤ k if whole G can be covered by bags of size ≤ k + 1, arranged in a “tree-like fashion”.

The underlying idea: G is recursively de- composed along small v. separators.

Structural properties

• Monotone under subgraphs and minors, • degenerate and “very” sparse,

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 3 / 19 Shrub-depth measure for dense graphs page.19 1 Recall: Width Measures

Tree-width tw(G) ≤ k if whole G can be covered by bags of size ≤ k + 1, arranged in a “tree-like fashion”.

The underlying idea: G is recursively de- composed along small v. separators.

Structural properties

• Monotone under subgraphs and minors, • degenerate and “very” sparse, • asymptotically equivalent to NO large grid minor.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 3 / 19 Shrub-depth measure for dense graphs page.19

Clique-width cwd(G) ≤ k if G given by a k-expression (over k-labelled gr.), k-expression ∼ disjoint unions, rela- belling, edge-add. between labels.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 4 / 19 Shrub-depth measure for dense graphs page.19

Clique-width cwd(G) ≤ k if G given by a k-expression (over k-labelled gr.), k-expression ∼ disjoint unions, rela- belling, edge-add. between labels.

The underlying idea: G rec. constructed in a way that only k groups of vertices can be distiguished at any moment.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 4 / 19 Shrub-depth measure for dense graphs page.19

Clique-width cwd(G) ≤ k if G given by a k-expression (over k-labelled gr.), k-expression ∼ disjoint unions, rela- belling, edge-add. between labels.

The underlying idea: G rec. constructed in a way that only k groups of vertices can be distiguished at any moment.

Structural properties

• Preserved by ind. subgraphs and “vertex-minors” (asympt.),

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 4 / 19 Shrub-depth measure for dense graphs page.19

Clique-width cwd(G) ≤ k if G given by a k-expression (over k-labelled gr.), k-expression ∼ disjoint unions, rela- belling, edge-add. between labels.

The underlying idea: G rec. constructed in a way that only k groups of vertices can be distiguished at any moment.

Structural properties

• Preserved by ind. subgraphs and “vertex-minors” (asympt.), • no nice “excluded something” characterization known so far,

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 4 / 19 Shrub-depth measure for dense graphs page.19

Clique-width cwd(G) ≤ k if G given by a k-expression (over k-labelled gr.), k-expression ∼ disjoint unions, rela- belling, edge-add. between labels.

The underlying idea: G rec. constructed in a way that only k groups of vertices can be distiguished at any moment.

Structural properties

• Preserved by ind. subgraphs and “vertex-minors” (asympt.), • no nice “excluded something” characterization known so far,

• but can be characterized by MSO1 interpretations into trees.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 4 / 19 Shrub-depth measure for dense graphs page.19 and Depth Measures

Tree-depth td(G) ≤ k if whole G is contained in the closure of a rooted forest of height ≤ k + 1.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 5 / 19 Shrub-depth measure for dense graphs page.19 and Depth Measures

Tree-depth td(G) ≤ k if whole G is contained in the closure of a rooted forest of height ≤ k + 1.

Structural properties

• Monotone under subgraphs and minors,

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 5 / 19 Shrub-depth measure for dense graphs page.19 and Depth Measures

Tree-depth td(G) ≤ k if whole G is contained in the closure of a rooted forest of height ≤ k + 1.

Structural properties

• Monotone under subgraphs and minors, • again degenerate and “very” sparse,

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 5 / 19 Shrub-depth measure for dense graphs page.19 and Depth Measures

Tree-depth td(G) ≤ k if whole G is contained in the closure of a rooted forest of height ≤ k + 1.

Structural properties

• Monotone under subgraphs and minors, • again degenerate and “very” sparse, • equiv. also to bounding the height of a tree-decomposition,

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 5 / 19 Shrub-depth measure for dense graphs page.19 and Depth Measures

Tree-depth td(G) ≤ k if whole G is contained in the closure of a rooted forest of height ≤ k + 1.

Structural properties

• Monotone under subgraphs and minors, • again degenerate and “very” sparse, • equiv. also to bounding the height of a tree-decomposition, • asymptotically equivalent to a no long path subgraph,

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 5 / 19 Shrub-depth measure for dense graphs page.19 and Depth Measures

Tree-depth td(G) ≤ k if whole G is contained in the closure of a rooted forest of height ≤ k + 1.

Structural properties

• and well-behaved wrt. MSO2 interpretations.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 5 / 19 Shrub-depth measure for dense graphs page.19

“Clique-depth” ???

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 6 / 19 Shrub-depth measure for dense graphs page.19

“Clique-depth” ??? How to restrict the height of k-expression? Not working. . . (though, NLC-width provides some hint, and further simpliﬁed)

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 6 / 19 Shrub-depth measure for dense graphs page.19

“Clique-depth” ??? How to restrict the height of k-expression? Not working. . . (though, NLC-width provides some hint, and further simpliﬁed)

Desired structural properties

• Ones similar to clique-width, but of small depth? (Related to clique-width as tree-depth related to tree-width. . . )

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 6 / 19 Shrub-depth measure for dense graphs page.19

“Clique-depth” ??? How to restrict the height of k-expression? Not working. . . (though, NLC-width provides some hint, and further simpliﬁed)

Desired structural properties

• Ones similar to clique-width, but of small depth? (Related to clique-width as tree-depth related to tree-width. . . ) • Small also on some dense graphs, stable under complements. Recursive construction with limited inform. and of small depth.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 6 / 19 Shrub-depth measure for dense graphs page.19

“Clique-depth” ??? How to restrict the height of k-expression? Not working. . . (though, NLC-width provides some hint, and further simpliﬁed)

Desired structural properties

• Stable under FO / MSO1 interpretations!

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 6 / 19 Shrub-depth measure for dense graphs page.19 2 Tree-models and Shrub-depth T Tree-model of m colours and depth d: • a rooted tree T of height d,

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 7 / 19 Shrub-depth measure for dense graphs page.19 2 Tree-models and Shrub-depth T Tree-model of m colours and depth d: • a rooted tree T of height d,

• leaves are the vertices of G, G each leaf has one of m colours,

• an associated signature determining the edge set of G as follows: for i = 1, 2, . . . , d, let u and v be leaves with the least common ancestor at height i in T ,

• leaves are the vertices of G, G each leaf has one of m colours,

• an associated signature determining the edge set of G as follows: for i = 1, 2, . . . , d, let u and v be leaves with the least common ancestor at height i in T , i=2 then uv ∈ E(G) iﬀ the colour pair of u, v is in the signature at i=1 height i.

• leaves are the vertices of G, G each leaf has one of m colours,

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 7 / 19 Shrub-depth measure for dense graphs page.19

Shrub-depth T

Class TMm(d) = { graphs with a tree-model of m colours and depth d }

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 8 / 19 Shrub-depth measure for dense graphs page.19

Shrub-depth T

Class TMm(d) = { graphs with a tree-model of m colours and depth d }

– closed under complements and induced subgraphs,

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 8 / 19 Shrub-depth measure for dense graphs page.19

Shrub-depth T

Class TMm(d) = { graphs with a tree-model of m colours and depth d }

– closed under complements and induced subgraphs, – but neither under disjoint unions nor under subgraphs.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 8 / 19 Shrub-depth measure for dense graphs page.19

Shrub-depth T

Class TMm(d) = { graphs with a tree-model of m colours and depth d }

– closed under complements and induced subgraphs, – but neither under disjoint unions nor under subgraphs.

Deﬁnition A graph class G is of shrub-depth d iﬀ

there exists m such that G ⊆ TMm(d) (same m for all G!),

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 8 / 19 Shrub-depth measure for dense graphs page.19

Shrub-depth T

Class TMm(d) = { graphs with a tree-model of m colours and depth d }

– closed under complements and induced subgraphs, – but neither under disjoint unions nor under subgraphs.

Deﬁnition A graph class G is of shrub-depth d iﬀ

there exists m such that G ⊆ TMm(d) (same m for all G!), 0 while for all m we have G 6⊆ TMm0 (d − 1).

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 8 / 19 Shrub-depth measure for dense graphs page.19

Examples and simple properties

• Shrub-depth 1: e.g., cliques, stars, . . . gen. BND – bounded neighbourhood diversity.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 9 / 19 Shrub-depth measure for dense graphs page.19

Examples and simple properties

• Shrub-depth 1: e.g., cliques, stars, . . . gen. BND – bounded neighbourhood diversity.

• Shrub-depth 2: e.g., matchings, union of cliques,

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 9 / 19 Shrub-depth measure for dense graphs page.19

Examples and simple properties

• Shrub-depth 1: e.g., cliques, stars, . . . gen. BND – bounded neighbourhood diversity.

• Shrub-depth 2: e.g., matchings, union of cliques, generally – take a BND graph, and expand each vertex by any (other) BND graph.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 9 / 19 Shrub-depth measure for dense graphs page.19

Examples and simple properties

• Shrub-depth 1: e.g., cliques, stars, . . . gen. BND – bounded neighbourhood diversity.

• Shrub-depth 2: e.g., matchings, union of cliques, generally – take a BND graph, and expand each vertex by any (other) BND graph.

• The previous can be generalized to d levels. . .

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 9 / 19 Shrub-depth measure for dense graphs page.19

Examples and simple properties

• Shrub-depth 1: e.g., cliques, stars, . . . gen. BND – bounded neighbourhood diversity.

• Shrub-depth 2: e.g., matchings, union of cliques, generally – take a BND graph, and expand each vertex by any (other) BND graph.

• The previous can be generalized to d levels. . .

• Bounded tree-depth ⇒ bounded shrub-depth.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 9 / 19 Shrub-depth measure for dense graphs page.19

Examples and simple properties

• Shrub-depth 1: e.g., cliques, stars, . . . gen. BND – bounded neighbourhood diversity.

• Shrub-depth 2: e.g., matchings, union of cliques, generally – take a BND graph, and expand each vertex by any (other) BND graph.

• The previous can be generalized to d levels. . .

• Bounded tree-depth ⇒ bounded shrub-depth.

• Bounded shrub-depth ⇒ bounded linear clique-width.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 9 / 19 Shrub-depth measure for dense graphs page.19 Alternative: SC-depth Goal: to get a depth parameter valid for a single graph. . .

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 10 / 19 Shrub-depth measure for dense graphs page.19 Alternative: SC-depth Goal: to get a depth parameter valid for a single graph. . . SC-classes (“subset-complementation”):

– SC(0) = {K1},

– SC(0) = {K1}, – having G1,G2,...,Gp ∈SC(k),

take the disjoint union H := G1 ∪ G2 ∪ ... ∪ Gp and,

– SC(0) = {K1}, – having G1,G2,...,Gp ∈SC(k),

take the disjoint union H := G1 ∪ G2 ∪ ... ∪ Gp and, for every X ⊆ V (H), H with complemented edges on X,→ SC(k + 1).

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 10 / 19 Shrub-depth measure for dense graphs page.19

Deﬁnition A graph G is of SC-depth k iﬀ G ∈ SC(k) but G 6∈ SC(k − 1).

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 11 / 19 Shrub-depth measure for dense graphs page.19

Deﬁnition A graph G is of SC-depth k iﬀ G ∈ SC(k) but G 6∈ SC(k − 1).

Theorem For a graph class G; bounded shrub-depth ⇐⇒ bounded SC-depth.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 11 / 19 Shrub-depth measure for dense graphs page.19

Deﬁnition A graph G is of SC-depth k iﬀ G ∈ SC(k) but G 6∈ SC(k − 1).

Theorem For a graph class G; bounded shrub-depth ⇐⇒ bounded SC-depth. A proof sketch:

– SC(k) ⊆ TM2k (k), and

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 11 / 19 Shrub-depth measure for dense graphs page.19

Deﬁnition A graph G is of SC-depth k iﬀ G ∈ SC(k) but G 6∈ SC(k − 1).

Theorem For a graph class G; bounded shrub-depth ⇐⇒ bounded SC-depth. A proof sketch:

– SC(k) ⊆ TM2k (k), and

– TMm(d) ⊆ SC(dm(m + 1)).

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 11 / 19 Shrub-depth measure for dense graphs page.19 Alternative II: Rank-depth Warning: just a very brief sketch (assuming you know branch-width).

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 12 / 19 Shrub-depth measure for dense graphs page.19 Alternative II: Rank-depth Warning: just a very brief sketch (assuming you know branch-width).

• Branch-width of a submodular function ; rank-width of a graph: consider the cut-rank function on the vertex set of a graph.

• [Geelen] Branch-width ; branch-depth:

• Branch-width of a submodular function ; rank-width of a graph: consider the cut-rank function on the vertex set of a graph.

• [Geelen] Branch-width ; branch-depth: allowing high-degree nodes, and node width measured over all bipartitions of incident subtrees.

• Branch-width of a submodular function ; rank-width of a graph: consider the cut-rank function on the vertex set of a graph.

• [Geelen] Branch-width ; branch-depth: allowing high-degree nodes, and node width measured over all bipartitions of incident subtrees.

• Branch-depth of a submodular function ; rank-depth of a graph.

• Branch-width of a submodular function ; rank-width of a graph: consider the cut-rank function on the vertex set of a graph.

• [Geelen] Branch-width ; branch-depth: allowing high-degree nodes, and node width measured over all bipartitions of incident subtrees.

• Branch-depth of a submodular function ; rank-depth of a graph.

Theorem [DeVos, Kwon, Oum] For a graph class G; bounded shrub-depth ⇐⇒ bounded rank-depth.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 12 / 19 Shrub-depth measure for dense graphs page.19 3 Further Combinatorial Properties

Theorem Each of TMm(d) and SC(k) have ﬁnite sets of forbidden induced subgraphs (obstructions).

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 13 / 19 Shrub-depth measure for dense graphs page.19 3 Further Combinatorial Properties

Theorem Each of TMm(d) and SC(k) have ﬁnite sets of forbidden induced subgraphs (obstructions).

Proof sketch:

– e.g., min. obstructions for TMm(d) contained in TM2m+1(d);

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 13 / 19 Shrub-depth measure for dense graphs page.19 3 Further Combinatorial Properties

Theorem Each of TMm(d) and SC(k) have ﬁnite sets of forbidden induced subgraphs (obstructions).

Proof sketch:

– e.g., min. obstructions for TMm(d) contained in TM2m+1(d); – plus [Ding] WQO of coloured rooted trees of bounded height (simply iterate classical Higman’s lemma. . . ).

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 13 / 19 Shrub-depth measure for dense graphs page.19 3 Further Combinatorial Properties

Theorem Each of TMm(d) and SC(k) have ﬁnite sets of forbidden induced subgraphs (obstructions).

Proof sketch:

– e.g., min. obstructions for TMm(d) contained in TM2m+1(d); – plus [Ding] WQO of coloured rooted trees of bounded height (simply iterate classical Higman’s lemma. . . ).

Theorem (shrub-depth “from” tree-depth) A class G of bounded shrub-depth ⇒ exists d such that each graph of G is a vertex-minor of a graph of tree-depth d.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 13 / 19 Shrub-depth measure for dense graphs page.19 3 Further Combinatorial Properties

Theorem Each of TMm(d) and SC(k) have ﬁnite sets of forbidden induced subgraphs (obstructions).

Proof sketch:

– e.g., min. obstructions for TMm(d) contained in TM2m+1(d); – plus [Ding] WQO of coloured rooted trees of bounded height (simply iterate classical Higman’s lemma. . . ).

Theorem (shrub-depth “from” tree-depth) A class G of bounded shrub-depth ⇒ exists d such that each graph of G is a vertex-minor of a graph of tree-depth d. Proof sketch: – start from an SC-depth tree, and “simulate” subset complem. via extra vert. with local complem.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 13 / 19 Shrub-depth measure for dense graphs page.19

Theorem (“no long paths” in tree-models)

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 14 / 19 Shrub-depth measure for dense graphs page.19

Theorem (“no long paths” in tree-models) m Let P` := path of length `, where ` = 3 · 2 − 4. Then

P` ∈ TMm(2m + 1) but ∀d : P`+1 6∈ TMm(d).

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 14 / 19 Shrub-depth measure for dense graphs page.19

Theorem (“no long paths” in tree-models) m Let P` := path of length `, where ` = 3 · 2 − 4. Then

P` ∈ TMm(2m + 1) but ∀d : P`+1 6∈ TMm(d). Proof sketch: – the tight bound comes from a delicate induction (skipped),

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 14 / 19 Shrub-depth measure for dense graphs page.19

Theorem (“no long paths” in tree-models) m Let P` := path of length `, where ` = 3 · 2 − 4. Then

P` ∈ TMm(2m + 1) but ∀d : P`+1 6∈ TMm(d). Proof sketch: – the tight bound comes from a delicate induction (skipped), – easy weaker argument: every large tree-model of bounded d, m has triplicate subtrees → cannot represent a path.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 14 / 19 Shrub-depth measure for dense graphs page.19

Observation Forbidding induced paths is not enough! (to bound shrub-depth/SC-depth)

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 15 / 19 Shrub-depth measure for dense graphs page.19

Observation Forbidding induced paths is not enough! (to bound shrub-depth/SC-depth)

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 15 / 19 Shrub-depth measure for dense graphs page.19

Observation Forbidding induced paths is not enough! (to bound shrub-depth/SC-depth)

Conjecture A class G is of bounded shrub-depth ⇐⇒

there exists t such that no graph of G contains Pt as a vertex minor.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 15 / 19 Shrub-depth measure for dense graphs page.19 4 Shrub-depth and Interpretations

Observation (tree-model → interpretation)

Graphs have FO interpretations in their tree-models from TMm(d).

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 16 / 19 Shrub-depth measure for dense graphs page.19 4 Shrub-depth and Interpretations

Observation (tree-model → interpretation)

Graphs have FO interpretations in their tree-models from TMm(d). Theorem (interpretation → tree-model)

A graph class G has a simple CMSO1 interpretation in a class Td of coloured rooted trees of height ≤ d ⇒ G is of shrub-depth ≤ d.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 16 / 19 Shrub-depth measure for dense graphs page.19 4 Shrub-depth and Interpretations

Observation (tree-model → interpretation)

Graphs have FO interpretations in their tree-models from TMm(d). Theorem (interpretation → tree-model)

A graph class G has a simple CMSO1 interpretation in a class Td of coloured rooted trees of height ≤ d ⇒ G is of shrub-depth ≤ d. Some consequences • Shrub-depth preserved under vertex-minors (exactly); using [Courcelle–Oum].

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 16 / 19 Shrub-depth measure for dense graphs page.19 4 Shrub-depth and Interpretations

Observation (tree-model → interpretation)

Graphs have FO interpretations in their tree-models from TMm(d). Theorem (interpretation → tree-model)

• The ﬁnite levels of the MSO1 transduction hierarchy are given (almost) by shrub-depth 1, 2, 3 ...; cf. [Blumensath–Courcelle].

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 16 / 19 Shrub-depth measure for dense graphs page.19

Proof sketch

• CMSO1 interpretation I(Td) = G, given by I = (α, β): – vertex set V (G) := {u : T |= α(u)} – can be ignored,

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 17 / 19 Shrub-depth measure for dense graphs page.19

Proof sketch

• CMSO1 interpretation I(Td) = G, given by I = (α, β): – vertex set V (G) := {u : T |= α(u)} – can be ignored, – edge set E(G) := {uv : T |= β(u, v)} – crucial.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 17 / 19 Shrub-depth measure for dense graphs page.19

Proof sketch

• CMSO1 interpretation I(Td) = G, given by I = (α, β): – vertex set V (G) := {u : T |= α(u)} – can be ignored, – edge set E(G) := {uv : T |= β(u, v)} – crucial.

• Avoid free vars.: β0 ≡ ∃x, y[L(x) ∧ L(y) ∧ β(x, y)], and give the new label L to u, v, as needed,

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 17 / 19 Shrub-depth measure for dense graphs page.19

Proof sketch

• CMSO1 interpretation I(Td) = G, given by I = (α, β): – vertex set V (G) := {u : T |= α(u)} – can be ignored, – edge set E(G) := {uv : T |= β(u, v)} – crucial.

• Avoid free vars.: β0 ≡ ∃x, y[L(x) ∧ L(y) ∧ β(x, y)], and give the new label L to u, v, as needed,

T |= β(u, v) ⇐⇒ T [L(u),L(v)] |= β0.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 17 / 19 Shrub-depth measure for dense graphs page.19

Proof sketch

• CMSO1 interpretation I(Td) = G, given by I = (α, β): – vertex set V (G) := {u : T |= α(u)} – can be ignored, – edge set E(G) := {uv : T |= β(u, v)} – crucial.

• Avoid free vars.: β0 ≡ ∃x, y[L(x) ∧ L(y) ∧ β(x, y)], and give the new label L to u, v, as needed,

T |= β(u, v) ⇐⇒ T [L(u),L(v)] |= β0.

• Now, . . . * miracle happens *. . . , reducing every

T [L(u),L(v)] ; bounded T0[L(u),L(v)] such that

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 17 / 19 Shrub-depth measure for dense graphs page.19

Proof sketch

• CMSO1 interpretation I(Td) = G, given by I = (α, β): – vertex set V (G) := {u : T |= α(u)} – can be ignored, – edge set E(G) := {uv : T |= β(u, v)} – crucial.

• Avoid free vars.: β0 ≡ ∃x, y[L(x) ∧ L(y) ∧ β(x, y)], and give the new label L to u, v, as needed,

T |= β(u, v) ⇐⇒ T [L(u),L(v)] |= β0.

• Now, . . . * miracle happens *. . . , reducing every

T [L(u),L(v)] ; bounded T0[L(u),L(v)] such that 0 0 T [L(u),L(v)] |= β ⇐⇒ T0[L(u),L(v)] |= β .

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 17 / 19 Shrub-depth measure for dense graphs page.19

Proof sketch

• CMSO1 interpretation I(Td) = G, given by I = (α, β): – vertex set V (G) := {u : T |= α(u)} – can be ignored, – edge set E(G) := {uv : T |= β(u, v)} – crucial.

• Avoid free vars.: β0 ≡ ∃x, y[L(x) ∧ L(y) ∧ β(x, y)], and give the new label L to u, v, as needed,

T |= β(u, v) ⇐⇒ T [L(u),L(v)] |= β0.

• Now, . . . * miracle happens *. . . , reducing every

T [L(u),L(v)] ; bounded T0[L(u),L(v)] such that 0 0 T [L(u),L(v)] |= β ⇐⇒ T0[L(u),L(v)] |= β .

Moreover, doing this carefully, there is such universal T ; T0 to which L(u0),L(v0) can be added afterwards!

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 17 / 19 Shrub-depth measure for dense graphs page.19

• CMSO1 interpretation I(Td) = G, given by I = (α, β): – vertex set V (G) := {u : T |= α(u)} – can be ignored, – edge set E(G) := {uv : T |= β(u, v)} – crucial.

......

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 18 / 19 Shrub-depth measure for dense graphs page.19

• CMSO1 interpretation I(Td) = G, given by I = (α, β): – vertex set V (G) := {u : T |= α(u)} – can be ignored, – edge set E(G) := {uv : T |= β(u, v)} – crucial.

......

? • Recall univ. T ; T0, and choose L(u),L(v) to “query” uv ∈ E(G)

– this T0 is bounded → part of the signature, but

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 18 / 19 Shrub-depth measure for dense graphs page.19

• CMSO1 interpretation I(Td) = G, given by I = (α, β): – vertex set V (G) := {u : T |= α(u)} – can be ignored, – edge set E(G) := {uv : T |= β(u, v)} – crucial.

......

? • Recall univ. T ; T0, and choose L(u),L(v) to “query” uv ∈ E(G)

– this T0 is bounded → part of the signature, but

0 0 – how to get appropriate T0[L(u ),L(v )] then?

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 18 / 19 Shrub-depth measure for dense graphs page.19

• CMSO1 interpretation I(Td) = G, given by I = (α, β): – vertex set V (G) := {u : T |= α(u)} – can be ignored, – edge set E(G) := {uv : T |= β(u, v)} – crucial.

......

? • Recall univ. T ; T0, and choose L(u),L(v) to “query” uv ∈ E(G)

– this T0 is bounded → part of the signature, but

0 0 – how to get appropriate T0[L(u ),L(v )] then? • The ﬁnal step:

0 each u ∈ V (T ) 7→ orbit O(u ) of Aut(T0),

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 18 / 19 Shrub-depth measure for dense graphs page.19

• CMSO1 interpretation I(Td) = G, given by I = (α, β): – vertex set V (G) := {u : T |= α(u)} – can be ignored, – edge set E(G) := {uv : T |= β(u, v)} – crucial.

......

? • Recall univ. T ; T0, and choose L(u),L(v) to “query” uv ∈ E(G)

– this T0 is bounded → part of the signature, but

0 0 – how to get appropriate T0[L(u ),L(v )] then? • The ﬁnal step:

0 each u ∈ V (T ) 7→ orbit O(u ) of Aut(T0),

and we can assign O(u0) as the colour of u in our tree-model. 2

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 18 / 19 Shrub-depth measure for dense graphs page.19 Concluding Remarks

• Asymptotic shrub-depth *or* concrete SC-depth (rank-depth)?

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 19 / 19 Shrub-depth measure for dense graphs page.19 Concluding Remarks

• Asymptotic shrub-depth *or* concrete SC-depth (rank-depth)? – actually, choose any one you want (always good to have options).

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 19 / 19 Shrub-depth measure for dense graphs page.19 Concluding Remarks

• Asymptotic shrub-depth *or* concrete SC-depth (rank-depth)? – actually, choose any one you want (always good to have options).

• Generalize shrub-depth to relational structures? Straightforward. . .

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 19 / 19 Shrub-depth measure for dense graphs page.19 Concluding Remarks

• Asymptotic shrub-depth *or* concrete SC-depth (rank-depth)? – actually, choose any one you want (always good to have options).

• Generalize shrub-depth to relational structures? Straightforward. . .

• Characterise (combinatorially) classes of unbounded shrub-depth? (Recall long paths as vertex minors. . . )

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 19 / 19 Shrub-depth measure for dense graphs page.19 Concluding Remarks

• Asymptotic shrub-depth *or* concrete SC-depth (rank-depth)? – actually, choose any one you want (always good to have options).

• Generalize shrub-depth to relational structures? Straightforward. . .

• Characterise (combinatorially) classes of unbounded shrub-depth? (Recall long paths as vertex minors. . . )

• Get a better grip on shrub-depth (similarly to tree-depth)?

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 19 / 19 Shrub-depth measure for dense graphs page.19 Concluding Remarks

• Asymptotic shrub-depth *or* concrete SC-depth (rank-depth)? – actually, choose any one you want (always good to have options).

• Generalize shrub-depth to relational structures? Straightforward. . .

• Characterise (combinatorially) classes of unbounded shrub-depth? (Recall long paths as vertex minors. . . )

• Get a better grip on shrub-depth (similarly to tree-depth)?

Thank you for your attention.

Petr Hlinˇen´y, Sparsity, Logic . . . , Warwick, 2018 19 / 19 Shrub-depth measure for dense graphs