A Polynomial Excluded-Minor Approximation of Treedepth

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Introduction
Treedepth is a well-studied graph invariant with several equivalent definitions.It appears in the literature under various names including vertex ranking number [30], ordered chromatic number [17], and minimum elimination tree height [23], before being systematically studied under the name treedepth by Ossona de Mendes and Nešetřil [20].Bounded treedepth graphs play an important role in areas such as the theory of sparse graph classes [21,22], parameterized complexity theory [13,15,24], and model theory [28,29].
Formally, the treedepth of an undirected graph G is the minimum height of a rooted forest F on the same set of vertices such that, for every edge {u, v} in G, vertices u and v have an ancestordescendant relationship in F (i.e., lie on a common branch).The forest F may be regarded as a decomposition of G into subgraphs of cliques that lie along the branches of F .This type of decomposition is related to the more familiar tree-decompositions in the definition of treewidth (see Section 3).Indeed, treedepth is a close relative of "width measures" like treewidth and pathwidth.Intuitively, treedepth measures how "star-like" a graph is (note that graphs of treedepth 1 are disjoint unions of stars), whereas treewidth and pathwidth measure the extent to which a graph is "tree-like" and "path-like".These three invariants, denoted td(G), tw(G) and pw(G), are related by inequalities (1) tw(G) + 1 ≤ pw(G) + 1 ≤ td(G) ≤ (tw(G) + 1) Treedepth is also related to the order of the longest path in G, denoted lp(G), by (2) log(lp(G) + 1) ≤ td(G) ≤ lp(G).
(Throughout this paper log(•) denotes the base-2 logarithm.See Ch. 6 of [21] for proofs of ( 1) and (2).)These four graph invariants -td, tw, pw and lp -share the property of being monotone under the graph-minor relation (a.k.a.minor-monotone).Recall that a graph H is a minor of G, denoted H G, if H can be obtained from G by a sequence of vertex deletions, edge deletions and edge contractions.A graph invariant f : {graphs} → N is minor-monotone if f (H) ≤ f (G) for all H G. This is equivalent to the class {G : f (G) ≤ k} being minor-closed for every k ∈ N, where a class C is minor-closed if G ∈ C ⇒ H ∈ C for all H G. By the Robertson-Seymour Graph Minor Theorem [25], every minor-closed class C is characterized by a finite set F of obstructions (a.k.a.excluded minors) with the property that G ∈ C ⇐⇒ (∀F ∈ F)(F G) for all graphs G; moreover, F is unique (up to isomorphism of its elements) subject to minimality (i.e., F F for all distinct F, F ∈ F).It follows that every minor-monotone graph invariant f is characterized by the sequence (F 1 , F 2 , . . ., F k , . . . ) of finite minimal obstruction sets F k for the class {G : f (G) ≤ k}.
Understanding the exact minimal obstruction sets F k (computing, classifying, counting, etc.) for specific minor-monotone graph invariants is an active topic of research in graph theory (see [1,11]).When it comes to treedepth, minimal obstructions have been studied by two sets of authors [4,5,14,16].However, a complete classification of minimal obstructions for treedepth ≤ k remains elusive even for small values of k (less than 10).Moreover, Dvořák et al [14] showed that the number of minimal obstructions grows enormously fast (at least doubly exponentially) as a function of k [14].The situation is similar for other width measures like treewidth.This severely limits the usefulness of minimal obstructions in applications, such as parameterized algorithms on bounded tree-depth graphs.
On the other hand, there are applications where having a reasonable approximation of a parameter like treedepth or treewidth serves a good enough purpose.(We describe one such application in Section 7, which was the original motivation for the results of this paper.)The question arises whether one (or a bounded number of) uniform families of non-minimal obstructions suffice for a polynomial approximation of a given minor-monotone graph invariant.A recent breakthrough of Chekuri and Chuzhoy [12] gave precisely such a result for treewidth (resolving a longstanding conjecture in graph minor theory).
Theorem 1.1 (Polynomial Grid-Minor Theorem for Treewidth [12]).There is an absolute constant c such that every graph with treewidth ≥ k c has a k × k grid minor.
Since the k × k grid has treewidth k, Theorem 1.1 establishes the treewidth of a graph is polynomially related to the size of its largest grid minor.(Prior to Theorem 1.1, treewidth only known to be exponential in the size of the largest grid minor.)In this paper, we establish an analogous "polynomial excluded-minor approximation" of treedepth in terms of three basic obstructions: grids, complete binary trees, and paths.Theorem 1.2 (Polynomial Grid/Tree/Path-Minor Theorem for Treedepth).There is an absolute constant c such that every graph with treedepth ≥ k c has one or more of the following minors: • the complete binary tree of height k, • the path of order 2 k .
Since each of the above graphs has treedepth ≥ k, the largest such obstruction gives a polynomial approximation of td(G).Moreover, all three obstructions in Theorem 1.2 are necessary for a polynomial approximation.(In light of (2), the length of the longest path in G gives a weaker exponential approximation of td(G).) Theorem 1.2 is obtained by a combination of Theorem 1.1 and the following result, which is the technical main theorem of this paper.

Theorem 1.3 (Main Theorem).
There is an absolute constant C such that every graph G with treedepth ≥ Ck 5 log 2 k satisfies one or more of the following conditions: • G has the complete binary tree of height k as a minor, • G contains a path of order 2 k .
Our proof of Theorem 1.3 is entirely self-contained (in particular, we do not rely on Theorem 1.1).Due to the constructive nature of the proofs, we get an algorithmic version of Theorem 1.3.Combined with the algorithmic version of Theorem 1.1 from [12], we get a randomized polynomialtime algorithm which, given a graph G, either determines that G is of small treedepth or outputs a certificate of one of the three cases in Theorem 1.2 (i.e., a minor-embedding of a grid, tree, or path.For details see Section 6).
The rest of this paper is organized as follows.In Section 2 we state some basic definitions.In Section 3 we prove some lemmas on tree decompositions.In Section 4 we prove some additional lemmas on rooted trees (essentially proving Theorem 1.3 in the case where G is a tree).In Section 5 we present the proof of Theorem 1.3.In Section 6 we describe polynomial-time algorithms which give effective versions of our main theorems.In Section 7 we describe a surprising application of Theorem 1.3 in circuit complexity and logic, which was the motivation for this paper.Finally, we conclude with some observations and open problems in Section 8.

Preliminaries
All graphs in this paper are finite simple graphs.Formally, a graph is a pair . A tree is a connected acyclic graph.A tree is subcubic if it has maximum degree at most 3. Examples of subcubic trees include paths and binary trees.• A tree decomposition of a graph G is a pair (T, W) where T is a tree and W = {W t } t∈V (T ) is a family of sets W t ⊆ V (G) such that , and every edge of G has both ends in some W t , if t, t , t ∈ V (T ) and t lies on the path in T between t and t , then • The width of a tree decomposition (T, W) is defined as max t∈V (T ) |W t | − 1.
• The treewidth of G, denoted tw(G), is the minimum width of a tree decomposition for G.
• The pathwidth of G, denoted pw(G), is the minimum width of a tree decomposition (T, W) for G such that T is a path.

Definition 2.2 (Rooted Trees).
A rooted tree is a tree T with a designated root vertex.The height of T is the maximum number of vertices on a root-to-leaf path.We use the following notation: • E(T ) is the set of ordered pairs xy such that x is a child of y in T .(We write xy instead of (x, y) and think of this pair as a directed edge.) • < T is the partial order on V (T ) defined by x < T y iff x is a proper descendent of y; we write x ≤ T y iff x < T y or x = y; for W ⊆ V (T ), we write W ≤ T x iff w ≤ T x for all w ∈ W .
• The closure of T , denoted Clos(T ), is the graph with vertex set V (T ) and edge set {{x, y} : x < T y or y < T x}.(In other words, two vertices are joined by an edge in Clos(T ) iff they lie on a common branch in T .)Definition 2.3 (Treedepth).The treedepth of a connected graph G, denoted td(G), is the minimum height of a rooted tree T such that G ⊆ Clos(T ).The treedepth of a disconnected graph is the maximum treedepth of its connected components. 1efinition 2.4 (Graph Minors and Minor-Monotonicity).
• A graph F is a minor of G, denoted F G, if F is isomorphic to a graph that can be obtained from G by a sequence of edge deletions and edge contractions.
Width measures tw(•), pw(•) and td(•) are easily seen to be minor-monotone.The parameter lp(•), the order of the longest path, is minor-monotone as well.

Lemmas on Tree Decompositions
Our first lemma bounds the treedepth of a graph G in terms of the width of one of its tree decomposition (T, W) and the treedepth of T .Although this lemma is essentially folklore (it is implicit in proofs of the inequality td(G) ≤ (tw(G) + 1) log |V (G)| [9,21]), we could not find a proof in the literature, so include one for completeness.
Proof.Suppose (T, W) be a width-w tree decomposition of a graph G.We will construct a rooted R of height at most (w + 1) • td(T ) such that G ⊆ Clos(R).(The construction is illustrated in Figure 1.The tree decomposition (T, W) in that example happens to be a path.) By definition of treedepth, there exists a rooted tree S such that T ⊆ Clos(S) and td(T ) = height(S).Without loss of generality, we may assume that V (S) = V (T ) (by deleting any vertices of V (S) \ V (T )).
Recall that W is a family and note that U forms a partition of V (G) (where some of sets U t may be empty).
For each t ∈ V (T ), fix an arbitrary linear order < t on U t .Define partial order < on V (G) by x, y ∈ U t and x < t y or t,u∈V (T ) : t< S u x ∈ U t and y ∈ U u .
That is, we have x < y iff either x, y belong to the same set U t and x < t y, or x, y belong to distinct U t , U u respectively where t < S u.
It is easy to see that < is equivalent to < R for a unique rooted R with V (R) = V (G).(This follows from the observation that < is a partial order on V (G); it has a unique maximal element (namely, the < t -maximal element of U t (= W t ) where t = root(S)); and for every x ∈ V (G), the set {y : x < y} is totally ordered by < .)Note that To complete the proof, it remains to establish that G ⊆ Clos(R).Consider an edge {x, y} ∈ E(G).By definition of (T, W) being a tree decomposition of G, the set {t ∈ V (T ) : {x, y} ⊆ W t } is non-empty; let p be any < S -maximal element in this set.Consider the set {u ∈ V (T ) : p ≤ S u and {x, y} ∩ W t = ∅}; let q be the unique < S -maximal element in this set.There are now two cases to consider: • Assume p = q.Then x, y ∈ U p .W.l.o.g., x < p y. Then we have x < R y and hence {x, y} ∈ E(Clos(R)).
• Assume p = q.Then |{x, y} ∩ W q | = 1.W.l.o.g., {x, y} ∩ W q = {y}.Then we have x ∈ U p and y ∈ U q and p < S q.It follows that x < R y and hence {x, y} ∈ E(Clos(R)).
Since {x, y} ∈ E(Clos(R)) in both cases, we conclude that G ⊆ Clos(R).We next introduce a normal form for tree decompositions of connected graphs, which witnesses tight upper bounds for both treewidth and treedepth (as shown in Lemmas 3.5 and 3.6).

Definition 3.2 (Greedy Rooted Tree Decomposition).
• A greedy rooted tree decomposition of a connected graph G is a rooted tree T with the following properties: 2. G ⊆ Clos(T ), 3. for every child-parent pair xy ∈ E(T ), there exists w ≤ T x such that {w, y} ∈ E(G).
(Given (1) and ( 2), note that condition (3) is equivalent to the following: for every x ∈ V (T ), the induced subgraph of G on {w : w ≤ T x} is connected.) Bag T,G (x) := {x} ∪ {y : there exists w such that w ≤ T x < T y and {w, y} ∈ E(G)}.
• The width of T with respect to G is defined by max Remark 3.3.Our notion of greedy rooted tree decompositions is defined only for connected graphs for simplicity.However, Definition 3.2 extends naturally to general graphs by considering rooted forests instead of rooted trees.
The same notion appears at least twice in the literature: in [13] under the name good treedepth decomposition and in [10] under the name reduced separation forest.An even "greedier" class of tree decompositions appears in [15] under the name minimal rooted trees.Every minimal rooted tree for a connected graph G (in the sense of [15]) is a greedy rooted tree decomposition of G (in our sense), but not conversely.(The notion of minimal rooted trees would not work for our purposes, as Lemma 3.6 is false with respect to this more restrictive class of tree decompositions.) The following three lemmas establish the key properties of greedy rooted tree decompositions.(These properties are also noted in [10,13].)The first lemma establishes that greedy rooted tree decompositions are, in fact, tree decompositions in the sense of Definition 2.1.Lemma 3.4.If T is a greedy rooted tree decomposition of a connected graph G, then T together with {Bag T,G (x)} x∈V (G) is a tree decomposition of G.
Proof.Straightforward from definitions.
The next two lemmas show that height-optimal (resp.width-optimal) greedy rooted tree decomposition witness the treedepth (resp.treewidth) of connected graphs.Lemma 3.5.Every connected graph G has a greedy rooted tree decomposition of height td(G).
Proof.By definition of treedepth, there exists a rooted tree T of height td(G) such that G ⊆ Clos(T ).W.l.o.g., we may assume that V (T ) = V (G) (by deleting any vertices in V (T ) \ V (G)).Thus, T satisfies conditions (i) and (ii) of Definition 3.2.If T satisfies condition (iii), then we are done.So we assume that T violates condition (iii).
Consider any child-parent pair xy ∈ E(T ) witnessing the violation of condition (iii), that is, y is the parent of x in T and there is no edge in G between y and any element of {w : w ≤ T x}.Note that y cannot be the root of T (since it would then follow from G ⊆ Clos(T ) that G is disconnected).Let z be the parent of y in T .Let T be the rooted tree obtained from T by removing the edge {x, y} and adding the edge {x, z}.Note the following: 1. T satisfies conditions (i) and (ii) (that is, V (T ) = V (G) and G ⊆ Clos(T )).

width(T , G) ≤ width(T, G).
4. We have φ(T ) < φ(T ) where φ : {rooted trees} → N is the potential function φ(S) := v∈V (S) depth S (v) where depth S (v) is the distance between v and the root of S. This is clear, since V (T ) = V (T ) and It follows from observations (1)-( 4) that finitely many operations T → T transform T into a greedy rooted tree decomposition of G of at most the same height and width.In particular, the height is at most td(T ), which proves the lemma.
Lemma 3.6.Every connected graph G has a greedy rooted tree decomposition of width tw(G).
Proof.By definition of treewidth, there exists a tree decomposition (T, W) of G of width tw(G).W.l.o.g., we may assume that W t is nonempty for all t ∈ V (T ).We now make T into a rooted tree by arbitrary fixing a choice of root(T ) ∈ V (T ).Without increasing width, we can massage2 the tree decomposition (T, W) in order that • |W s \ W t | = 1 for all every child-parent pair st ∈ E(T ).
We may now identify V (T ) with V (G) by identifying root(T ) with the unique element of W root(T ) and identifying each non-root t with the unique element of W t \ W u where u is the parent of t.
Thus identified, the rooted tree T now satisfies conditions (i) and (ii), that is, V (T ) = V (G) and G ⊆ Clos(T ).Moreover, we have width(T, G) ≤ width(T, W).Finally, we repeat the same operation T → T as in the proof of Lemma 3.5 until T satisfies condition (iii) with respect to G. Since this operation does not increase width, we obtain a greedy rooted tree decomposition of G of width at most tw(G), which proves the lemma.

Lemmas on Rooted Trees
In this section we some prove results about rooted trees.In particular, we prove the special case of our main theorem for trees.Namely, we show that every tree with treedepth k contains a path of length 2 Ω( √ k) or a complete binary tree of height Ω( √ k) as a minor.We begin with a few definitions.
Definition 4.1 (The Rooted Minor Relation rooted ).For rooted trees S and T , we say that S is a rooted minor of T , denoted S rooted T , if S is isomorphic to a rooted tree obtained from T by deleting non-root leaves and contracting edges.• For k ≥ 1, let P k denote the path of order k rooted at one of its endpoints.
• For h ≥ 1, let B h denote the rooted complete binary tree of height h (with 2 h − 1 vertices).
Note that P 1 and B 1 are both the rooted tree of size 1 (i.e., an isolated root).
The next definition gives some useful notation for describing the structure of rooted trees.• For rooted trees S and T , let S * T denote the rooted tree formed by taking the disjoint union of S and T and identifying the two roots.(For example, P 2 * • • • * P 2 is a star rooted at its central vertex.)This operation is associative and commutative with identity element P 1 .For a sequence of rooted trees T 1 , . . ., T m (m ∈ N), we adopt the convention that • For a rooted tree T , let T denote the rooted tree obtained from T by creating a new root ρ and drawing an edge between ρ and the old root of T .
• For a sequence of rooted trees T 1 , . . ., T m (m ≥ 1), let That is, T 1 , . . ., T m is the rooted tree obtained by identifying the root of T i with the ith vertex from the root on the rooted path P m+1 .
These operations on rooted trees are illustrated in Figure 2, below.
As a matter of notation, let and B 0 both denote the rooted tree P 1 (i.e., a single isolated root).Note that for k, h ≥ 1, (Note: The reader may regard B 0 as the empty tree with 0 vertices; this is not a rooted tree.On the other hand, B 0 is a rooted tree with 1 vertex.)Lemma 4.4.Every rooted tree T has a unique decomposition the form T 1 * • • • * T l for some l ∈ N and rooted trees T 1 , . . ., T l (unique up to ordering).
Proof.Straightforward from definitions.Here l is the degree of root(T ) and T 1 , . . ., T l are the subtrees rooted at the children of root(T ) (see Figure 1).(Note that l = 0 in this decomposition if, and only if, T is an isolated root.) The next two lemmas characterize the rooted minor relation in terms of the decomposition given by Lemma 4.4.Proof.Assume S rooted T and consider the edge in T between root( T ) and root(T ).If this edge is not contracted in the minor isomorphic to S, then S rooted T .If this edge is contracted, then S rooted T .The other direction is clear.If S rooted T , then clearly S rooted T (by the same sequence of deletions and contractions).If S rooted T , then S rooted T (by T rooted T and transitivity of rooted ).Proof.Straightforward from definitions.

Rooted trees that exclude B h minors
The next lemmas characterize the structure of rooted trees T that omit binary trees B h as rooted minors.(We use these results soon in §4.3.)Lemma 4.7.If T is a rooted tree such that B h rooted T and B h rooted T , then there exist m ≥ 1 and rooted trees S 1 , . . ., S m such that T = S 1 * S 2 , . . ., S m and B h−1 rooted S i for all i ∈ [m].
Proof.Assume B h rooted T and B h rooted T and note that this implies h ≥ 2 (since B 1 rooted T ).We argue by induction on |V (T )|.In the base case T = P 1 , we set m := 1 and S 1 := T .
For the induction step, assume |V (T )| ≥ 1 and let T = T 1 * • • • * T l be the decomposition given by Lemma 4.4.Observe that there exists at most one i ∈ [l] such that B h−1 rooted T i (since otherwise we would have Consider the case that B h−1 rooted T i for all i ∈ [l].In this case, we have B h−1 rooted T by Lemma 4.6.Therefore, the condition in the lemma is satisfied with m := 1 and S 1 := T .
Finally, consider the case that there exists a unique i Lemma 4.8.If T is a rooted tree such that B h rooted T , then there exist m ≥ 0 and l 1 , . . ., l m ≥ 1 and rooted trees S i,j (i the decomposition given by Lemma 4.4.For all i ∈ [m], we have B h rooted T i and B h rooted T i by Lemmas 4.5 and 4.6.By Lemma 4.7, there exist l i ∈ N and rooted trees S i,1 , . . ., S i,l i such that T i = S i,1 * S i,2 , . . ., S i,l i and B h−1 rooted S i,j for all j ∈ [l i ].We have T i = S i,1 , . . ., S i,l i , and hence T = S 1,1 , . . ., S 1,l 1 * • • • * S m,1 , . . ., S m,lm .

Treedepth bounds
The next lemmas give bounds on the treedepth of the underlying graph of a tree T (that is, ignoring the root).These lemmas play a role in the proof of Theorem 1.3 in §5.Lemma 4.9 ([20,21]).For all k, h ≥ 1, we have td(P k ) = log(k + 1) and td(B h ) = h.
Note that the embedding P 15 ⊆ Clos(B 4 ), which witnesses the bound td(P 15 ) ≤ 4, is depicted in Figure 1.Proof.The lemma is proved by induction on h.In the base case h = 1, the condition B 1 rooted T implies that T is an isolated root (since For the induction step, suppose h ≥ 2. By Lemma 4.8, there exist m ∈ N and l 1 , . . ., l m ∈ N and rooted trees S i,j such that T = S 1,1 , . . ., S 1,l 1 * • • • * S m,1 , . . ., S m,lm and B h−1 rooted S i,j for all i ∈ [m] and j ∈ [l m ].Note that m ≤ c and l i ≤ k − 1 and P k−1 rooted S i,j for all i ∈ [m] and j ∈ [l i ].By the induction hypothesis, we have |V (S i,j )| ≤ (c(k − 1)) h−2 for all i and j.Therefore, Proof.Let T be any spanning tree of G rooted at any of its leaves.Since G has maximum degree c + 1, every vertex has at most c children in T .The assumption that P c h G and B h G implies that P c h rooted T and B h rooted T .Therefore, by Lemma 4.14,

Proof of Theorem 1.3
In this section, we prove Theorem 1.3 by showing the following stronger result.
Theorem 5.1.Every graph G contains a path of order 2 h or has a B h -minor where (Obs: Note that the ratio r is at most 1 by inequality (1).) Proof of Theorem 1.3 assuming Theorem 5.1.Let k ≥ 1 and suppose G is a graph of treewidth < k which does not contains a path of order 2 k nor a B k -minor.Theorem 5.1 implies The rest of this section is devoted the proof of Theorem 5.1.
Proof of Theorem 5.1.It clearly suffices to prove the theorem for connected graphs G. Let G be any connected graph and let r = td(G)/(tw(G) + 1).We must show that G contains a path of length 2 h or a B h -minor where h = Ω(r 1/4 / log 1/2 (tw(G) + 1)).By Lemma 3.6, we may fix a greedy rooted tree decomposition T of width tw(G) for G.By Lemma 3.1, we have td(T ) ≥ r.
In the rest of the proof, we will construct a sequence of three trees: first a spanning tree F ⊆ G, then a subcubic rooted subtree S ⊆ T of order |V (S)| = 2 Ω( √ r) , and finally a subtree Q ⊆ F with maximum degree ≤ tw(G) + 2 and V (S) ⊆ V (Q).By Lemma 4.15, we conclude that Q contains a path of length 2 h or a B h -minor where h = Ω(r 1/4 / log 1/2 (tw(G)+1)).Since Q ⊆ G, this completes the proof.
The spanning tree F ⊆ G: x be the parent of x in T (i.e., the unique vertex in V (G) (= V (T )) such that x x ∈ E(T )).
• By condition (iii) of Definition 3.2, there exists a function x → x : V → V such that x ≤ T x and { x, x} ∈ E(G) for all x ∈ V .Fix any choice of such a function x → x.
Claim 1. F is a spanning tree for G (that is, F is a tree and The fact that F is a tree follows from the observation that x < T x for all x ∈ V (since x ≤ T x and x < T x).To see that V (F ) = V (G), note that V (G) = V (T ) and consider any vertex x ∈ V (G).If x is a non-leaf in T , then let w be any of its children (i.e., x = w); we have { w, x} ∈ E(F ) and hence x ∈ V (F ).If x is a leaf in T , then (assuming w.l.o.g. that |V (G)| ≥ 2 so that x ∈ V ) we have x = x (this is forced by the requirement x ≤ T x) and therefore { x, x} ∈ E(F ) and hence Let {u, v} ∈ E(F ).There exists x ∈ V such that {u, v} = { x, x}.Either u = x and v ∈ x (in which case u ≤ T x < T v), or u = x and v ∈ x (in which case v ≤ T x < T u).Let (p 0 , . . ., p t ) be the sequence of vertices on the unique path from x to x in F (with p 0 = x and p t = x).Let p i be the unique < T -maximum element in {p 0 , . . ., p t }.Toward a contradiction assume i = 0 (that is, p i = x).Then i ∈ {1, . . ., t − 1} and, moreover, p i−1 < T p i and p i+1 < T p i (by Claim 2 since {p i−1 , p i } ∈ E(F ) and {p i , p i+1 } ∈ E(G)).Since p i−1 = p i+1 , it must be the case that p i−1 = u and p i+1 = w for distinct u, w ∈ V such that u = w = p i .It may now be seen that p 0 , . . ., p i−1 ≤ T u and p i+1 , . . ., p t ≤ T w. 3 Since x = p 0 and x = p t , this means that x ≤ T u and x ≤ T w.But then x and x would be incomparable under < T (since u and w are siblings in T ).This yields the desired contradiction, since x is the parent of x in T .

Claim 3
The rooted subtree S ⊆ T : • By Lemma 4.13, T has a subcubic rooted subtree S of order 2 Ω( √ r) (with root(S) = root(T ) and E(S) ⊆ E(T )).Fix any choice of S.
The reason we need this subcubic tree S will become clear later on (in Claim 4).In short, the fact that S has degree ≤ 3 guarantees that the tree Q (which we are about to construct) will have maximum degree ≤ tw(G) + 2.
Trees {Q x ⊆ F } x∈W and paths {P x ⊆ F } x∈W : By simultaneous induction (upward from the leaves of S), we define families of subgraphs {Q x ⊆ F } x∈W and {P x ⊆ F } x∈W where each Q x is a tree satisfying x ∈ V (Q x ) and V (Q x ) ≤ T x and each P x is a path satisfying x ∈ V (P x ) and • Suppose x ∈ W where Q x is already defined.Let (p 0 , . . ., p t ) be the sequence of vertices on the unique path from x to x in F (with p 0 = x and p t = x).Let s ∈ {1, . . ., t} be the minimum index satisfying p s ∈ V (Q x ).(This is well-defined since p t = x ∈ V (Q x ).)Then P x is the subpath of F with V (P x ) = {p 0 , . . ., p s } and E(P x ) = {{p 0 , p 1 }, . . ., {p s−1 , p s }}.
(Note that P x satisfies x = p 0 ∈ V (P x ) and V (P x ) \ { x} = {p 1 , . . ., p s } ≤ T x (by Claim 3) and The tree Q ⊆ F and vertices {x ∈ . Note that Q = x∈W P x by the above definition.
• For each x ∈ W , let x be the unique element in V (P x ) ∩ V (Q x ).(That is, x is the vertex p s in the above definition of P x .)Thus, P x is the unique path in F between x (the parent of x in S) and x (the first vertex in V (Q x ) encountered on the unique path in F from x to x).
Consider any q ∈ V (Q).Recall that Bag T,G (q) = {q} ∪ {y : there exists w such that w ≤ T q < T y and {w, y} ∈ E(G)}.
For each x ∈ W such that q = x , define the set Three simple observations: • Let (p 0 , . . ., p s ) be the unique path in F from x to q (with p 0 = x and p s = q).Let i ∈ {1, . . ., s} be the minimum index such that p i ≤ T q.Then p i = p i−1 and p i ≤ T q < T p i−1 .Therefore, U x is nonempty.
• We have V (P x ) ∩ V (P y ) = {q} for all distinct x, y ∈ W such that q = x = y .Therefore, U x and U y are disjoint.
It follows from these three observations that |{x ∈ W : q = x }| ≤ |Bag T,G (q)| − 1.Finally, recall that T was chosen such that width(T, Claim 5 Claims 4 and 5 imply that Q has maximum degree ≤ tw(G) + 2. Since V (S) ⊆ V (Q), we have It now follows from Lemma 4.15 that Q (and hence also G since Q ⊆ G) contains either a path of order 2 h or a B h -minor where h = Ω(r 1/4 / log 1/2 (tw(G)+1)).This completes the proof of Theorem 5.1.

Algorithmic Results
In this section, we describe the algorithmic versions of our main results.From the constructive nature of the proof of Theorem 5.1, we have the following Corollary 6.1 (Algorithmic Version of Theorem 5.1).There is a polynomial-time algorithm which, given a graph G and a width-w tree decomposition of G, outputs a minor embedding of either P 2 h or B h where Results of Bodlaender et al [9,8] give a polynomial-time algorithm which, given a graph G, outputs a tree decomposition of G of width O(tw(G) 2 ).(This is actually a combination of two polynomial-time approximation algorithms for treewidth: an O(log n)-approximation for arbitrary n-vertex graphs G [9] and a 5-approximation algorithm in the case where tw(G) ≤ log n [9].)Combining this algorithm with Corollary 6.2, we get Definition 7.2.The colored G-subgraph isomorphism problem is the following problem: Given a graph X ⊆ G ↑n , does there exist α ∈ [n] V (G) such that G (α) ⊆ X?
The following lemma from Li et al [18] shows that the complexity of SUB(G) is a minormonotone graph invariant.Lemma 7.3.If H is a minor of G, then there is a monotone-projection reduction from SUB(H, n) to SUB(G, n) for every n ∈ N. 4As a consequence of Lemma 7.3, the function G → χ(SUB(G, n)) is a minor-monotone graph invariant for all standard complexity measures χ(•), including AC 0 circuit/formula size. 5t is known that SUB(G) is computable by AC 0 circuits of size O(n tw(G)+1 ), as well as by AC 0 formulas of size O(n td(G) ) (moreover, depth |V (G)| is sufficient in both cases). 6The next theorem summarizes known lower bounds on the AC 0 complexity of SUB(G).
Combining Theorem 1.3 with the three lower bounds in Theorem 7.4, and using the fact that the AC 0 formula size of SUB(G) is minor-monotone by Lemma 7.3, we get the following: Theorem 7.5.There is an absolute constant ε > 0 such that SUB(G) has AC 0 formula size n Ω(td(G) ε ) for all graphs G. Theorem 1.3(1) and Theorem 7.5 lend support to the conjectures that the unbounded-depth circuit (resp.formula) size of SUB(G) is n Ω(tw(G)) (resp.n Ω(td(G)) ).Since these conjectures imply P = NP and NC 1 = NL, it is an interesting and worthwhile first step to prove these lower bounds in the restricted bounded-depth setting.

An Improved Homomorphism Preservation Theorem on Finite Structures
Theorem 7.5 turns out to have a surprising corollary in finite model theory.The following result was proved in [28].
Theorem 7.6.Let ϕ be a first-order sentence of quantifier-rank r.If ϕ is preserved under homomorphisms on finite structures, then there is an existential-positive sentence ψ of quantifier-rank r O (1) such that ϕ and ψ are logically equivalent on finite structures.
The proof of Theorem 7.6 is based on a reduction to the AC 0 formula of SUB(G) and relies on Theorem 7.5 (and hence on Theorem 1.3) for the polynomial bound on the quantifier-rank of ψ.Theorem 7.6 dramatically improves an earlier result in [26], in which the bound on quantifierrank of ψ is a non-elementary function of r (i.e., growing faster than any constant-height tower of exponentials).

Open Questions
In light of Theorems 1.1 and 1.2, we conjecture the following "Polynomial Grid/Tree-Minor Theorem for Pathwidth": Conjecture 8.1.There is an absolute constant c such that every graph with pathwidth ≥ k c has one of the following minors: • the k × k grid, • the complete binary tree of height k.

Definition 4 . 2 (
Rooted Trees P k and B h ).

Lemma 4 . 5 .
For rooted trees S and T , we have S rooted T if, and only if, S rooted T or S rooted T .

Lemma 4 . 6 .
Suppose S = S 1 * • • • * S l and T = T 1 * • • • * T m .Then S rooted T if, and only if, there exists a one-to-one function j :[l] → [m] such that S i rooted T j(i) for all i ∈ [l].
1 and T := T l .Observe that B h−1 rooted S 1 and |V (T )| ≥ 1+|V (T )| (since T is a subtree of T ).By the induction hypothesis applied to T , there exists m ≥ 2 and rooted trees S 2 , . . ., S m such that T = S 2 * S 3 , . . ., S m and B h−1 rooted S i for all i ∈ [m].We are now done, as T = S 1 * T = S 1 * S 2 , . . ., S m .

Claim 2 Claim 3 .
If x ∈ V and P is the unique path in F between x and x, then V (P ) \ { x} ≤ T x.
The techniques introduced in this paper might be helpful in proving this conjecture.Another open problem is to improve the O(k 5 log 2 k) bound in Theorem 1.3.The optimal bound is likely smaller; however, examples show one cannot do better than O(k 2 ).
is a non-leaf with children w 1 , . . ., w k in S (i.e., {w 1 , . . ., w k } = {w ∈ W : wx ∈ E(S)} where x may have additional children in T ) such that Q w 1 , . . ., Q w k and P w 1 , . . ., P w k are already defined.(Obs: k ≤ 2 since S is subcubic.)We define Q x