t -sails and sparse hereditary classes of unbounded tree-width

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Introduction
Tree-width is a graph parameter that became of great interest in the fields of structural and algorithmic graph theory following the series of papers published by Robertson and Seymour on graph minors (for example [21]).In particular, their Grid Minor Theorem [22] states that every graph of large enough tree-width must contain a minor isomorphic to a large grid (or equivalently a large wall).Consequently, tree-width has been primarily associated with minorclosed graph classes.However, recent interest in tree-width has focussed on hereditary graph Thus, a hereditary graph class with an excluded minor or of bounded vertex degree has unbounded tree-width if and only if it contains arbitrarily large subdivisions of a wall or the line graph of a subdivision of a wall.
It has long been known that the following t-basic obstructions are obstructions to bounded treewidth: for arbitrarily large t, (1) the complete graph K t , (2) the complete bipartite graph K t,t , (3) a subdivision of the (t × t)-wall and (4) the line graph of a subdivision of the (t × t)-wall.It is tempting to conclude that these four objects are the only obstructions to bounded treewidth in sparse hereditary classes.However, counterexamples have recently been found by Sintiari and Trotignon [23], Davies [11] and Bonamy, Bonnet, Déprés, Esperet, Geniet, Hilaire, Thomassé and Wesolek [5].Tree-width in hereditary graph classes is currently a very active area of research.In particular, Abrishami, Alecu, Chudnovsky, Dibek, Hajebi, Rza ¸żewski, Spirkl and Vušković have contributed to a series of papers on the topic of induced subgraphs and tree decompositions, e.g.[2,3].
A hereditary graph class is KKW-free if there exists t ∈ N such that the class does not contain the complete graph K t , the complete bipartite graph K t,t , a subdivision of the (t×t)-wall or the line graph of a subdivision of the (t × t)-wall.Our interest is in KKW-free classes that nevertheless have unbounded tree-width.
The least number of forests that can cover the edges of a graph is called its arboricity.A graph with arboricity bounded by k ∈ N is called k-uniformly sparse or just sparse.A graph class is k-uniformly sparse if it does not contain a graph of arboricity greater than k.Uniformly sparse graph classes are a larger family than either excluded minor or bounded degree classes.
Using Ramsey theory [13,Proposition 9.4.1]we know that for every r ∈ N there is an n ∈ N such that every connected graph of order at least n contains a clique, a star or a path of order r as an induced subgraph.Thus, large trees contain long induced paths or big induced stars, and this suggests that 'path-path' (two forests of paths), 'path-star'(a forest of paths and a forest of stars) and 'star-star' (two forests of stars) are the three natural structures to look for in graph classes of arboricity two.Given that walls are 'path-path' graphs we seek further obstructions to bounded tree-width in another family of classes of arboricity two, namely 'path-star' classes.A hereditary path-star class is the collection of all the finite induced subgraphs of a single infinite graph whose edges can be decomposed into a path (or forest of paths) with a forest of stars, where the leaves of the stars (but not the internal vertices of the stars) may embed in the paths.We denote a path-star class R α where α is an infinite word over alphabet N (see Definition 4.1).Our study of path-star graph classes has led to the discovery of a family of objects, t-sails, with tree-width at least t − 1 (Lemma 2.4).Definition 1.3.A t-sail is a graph G = (V, E) which decomposes into t disjoint stars with star-vertices {s i } and t disjoint path components {P j } 1 j t of arbitrary length, such that star vertex s i is adjacent to one or more path vertices in P j whenever 1 i j t.
Note that no two star leaves can be incident to the same path vertex so that only the star-vertices can have degree greater than three.An example is shown in Figure 1.
We show that a path-star class defined by an infinite path-star graph with an infinite number of stars each of which connects to a continuous path an unbounded number of times, contains a tsail for arbitrarily large t and therefore has unbounded tree-width (Theorem 4.2).To distinguish this work from that recently undertaken on circle graphs in [16], we show that no path-star class defined by a word where at least two letters alternate more than four times is a subclass of circle graphs (Theorem 4.3).
We identify a collection of nested words (see Section 3.1) with a recursive structure that exhibit interesting characteristics when used to define a hereditary path-star graph class: Theorem 5.1.If α is a nested word then the path-star class R α is KKW-free.Theorem 5.8.If α is a nested word then for every t 1 there is a positive integer valued function f α (t) such that every graph in R α of tree-width at least f α (t) contains an induced subgraph isomorphic to a subdivision of a t-sail.
This shows that t-sails perform the same role for these sparse classes as walls do for minor excluded or bounded degree classes.
It has previously been shown (reproduced here as Theorem 2.6) that in sparse graph classes tree-width and another parameter, clique-width, are either both bounded or unbounded, so the behaviour of clique-width may shed light on the behaviour of tree-width.
By contrast with sparse hereditary classes, all dense hereditary classes have unbounded treewidth (Theorem 2.2), but this is not true for clique-width (Section 2.4), where there are dense classes of both bounded and unbounded clique-width.
A hereditary class of graphs C is minimal of unbounded tree-width/clique-width if every proper hereditary subclass D has bounded tree-width/clique-width (if it is clear from the context whether we are referring to tree-width or clique-width we will just call the class minimal).In other words, C is minimal if for any proper subclass D formed by adding just one more forbidden graph then D has bounded tree-width/clique-width.
The discovery of the first minimal hereditary classes of unbounded clique-width was made by Lozin [18].However, more recently many more such classes have been identified, in Atminas, Brignall, Lozin and Stacho [4], Collins, Foniok, Korpelainen, Lozin and Zamaraev [8] and Dawar and Sankaran [12].Most recently Brignall and Cocks demonstrated an uncountably infinite family of minimal hereditary classes of unbounded clique-width in [6] and created a framework for minimal classes in [7].
It is therefore natural to ask whether there are any sparse minimal classes of unbounded treewidth (or clique-width).
We show the following: Theorem 6.2.If C is a hereditary class of graphs of bounded vertex degree or has an excluded minor then it does not contain a minimal class.Theorem 6.4.If R α is a path-star hereditary class of graphs defined by a nested word α then it does not contain a minimal class.This suggests the following conjecture: Conjecture 1.4.Sparse hereditary graph classes of unbounded tree-width do not contain a minimal class of unbounded tree-width.

Graphs -General
Unless otherwise stated, all graphs in this paper are simple, i.e. undirected, without loops or multiple If vertex u is adjacent to vertex v we write u ∼ v and if u is not adjacent to v we write u ∼ v.We denote N(v) as the neighbourhood of a vertex v, that is, the set of vertices adjacent to v. A set of vertices is independent if no two of its elements are adjacent and is a clique if all the vertices are pairwise adjacent.We denote a clique with r vertices as K r and an independent set of r vertices as K r .A graph is bipartite if its vertices can be partitioned into two independent sets, V 1 and V 2 , and is complete bipartite if, in addition, each vertex of V 1 is adjacent to each vertex of V 2 .We denote this by K r,s where A tree is a graph in which any two vertices are connected by exactly one path and a forest is a collection of disjoint trees.A star S k is the complete bipartite graph K 1,k : a tree with one internal vertex, which we will refer to as the star-vertex, and k leaves.
A k-cycle is a closed path with k vertices.A k-cycle has a chord if two of its k vertices are joined by an edge which is not itself part of the cycle.A hole is a chordless cycle of length at least 4.
An (m×n)-wall is a graph whose edges are visually equivalent to the mortar lines of a stretcherbonded clay brick wall with m rows of bricks each of which is n bricks long.More precisely, we can define the wall W m×n = (V, E) using a square grid of the usual (x, y) Cartesian coordinates.
See example of W 4×4 in Figure 2.
The line graph of a graph G = (V, E) is the graph L(G) on E vertices where two vertices in L(G) are adjacent if and only if they are adjacent as edges in G.A graph H is a subdivision of a graph G if H can be obtained from G by inserting new (degree 2) vertices on some of the edges of G.
We will use the notation H I G to denote graph H is an induced subgraph of graph G, meaning V(H) ⊆ V(G) and two vertices of V(H) are adjacent in H if and only if they are adjacent in G.We will denote the subgraph of G = (V, E) induced by the set of vertices U ⊆ V by G[U].If a graph G does not contain an induced subgraph isomorphic to H we say that G is H-free.
An embedding of a graph H in a graph G is an injective map φ : V(H) → V(G) such that the subgraph of G induced by the vertices φ(V(H)) is isomorphic to H.In other words, vw ∈ E(H) if and only if φ(v)φ(w) ∈ E(G).If H is an induced subgraph of G, then this can be witnessed by one or more embeddings.
A class of graphs C is hereditary if it is closed under taking induced subgraphs, that is G ∈ C implies H ∈ C for every induced subgraph H of G.It is well known that for any hereditary class C there exists a unique (but not necessarily finite) set of minimal forbidden graphs {H 1 , H 2 , . . .} such that C = Free(H 1 , H 2 , . . . ) (i.e.every graph G ∈ C is H i -free for i = 1, 2, . . .).We will use the notation C ⊆ R to denote that C is a hereditary subclass of a hereditary graph class R (C R for a proper subclass).

Sparsity and Density
Recall from Section 1 that a k-uniformly sparse graph is one whose edges can be covered by at most k forests.The following theorems are useful: Theorem 2.1 (Nash-Williams, 1964 [20]).The edges of a graph G = (V, E) can be covered by at most k forests if and only if G[U] k(|U| − 1) for every non-empty set U ⊆ V.
Theorem 2.2 (Kostochka, 1982 [13]).There exists a constant c ∈ R such that, for every t ∈ N, every graph G of average degree d(G) ct log t contains K t as a minor.
A dense hereditary graph class is one that is not k-uniformly sparse for some k ∈ N, or in other words, by Theorem 2.1, for any k there is a graph in the class that has average degree greater than k.By Theorem 2.2, for any t ∈ N we can set k ct log t so that the class contains a graph with a K t minor.

Tree-width
Let G = (V, E) be a graph , T a tree, and let V = (V t ) t∈T be a family of vertex sets V t ⊆ V(G) (called bags) indexed by the vertices t of T .The pair (T , V) is called a tree-decomposition of G if it satisfies the following three conditions: 1. Every vertex of G is in at least one of the bags V t , 2. If (u, v) ∈ E, then u and v are together in some bag, 3. for all v ∈ V, the graph induced by the bags containing v is connected in T .
The width of a tree-decomposition is the maximum bag size minus 1.The tree-width of G, denoted by tw(G), is the least width of any tree-decomposition of G.
It is easy to show that tw(K t ) = t − 1 for all positive integers t 2. Theorems 2.1 and 2.2 tell us that all dense hereditary graph classes contain a graph with a K t minor for any positive integer t 2 and so by Lemma 2.3 have unbounded tree-width.
Similarly, recalling the definition of t-sails (see Definition 1.3), we have: If G is a t-sail for positive integer t 2 then tw(G) t − 1.
Proof.Contracting star-vertex s i with the vertices of path P i for 1 i t gives a K t -minor, then using Lemma 2.3 and the fact that tw(K t ) = t − 1 we have tw(G) t − 1.
It follows that hereditary graph classes containing arbitrarily large t-sails have unbounded tree-width.

Clique-width
Clique-width is a graph width parameter introduced by Courcelle, Engelfriet and Rozenberg in the 1990s [9].The clique-width of a graph is denoted cw(G) and is defined as the minimum number of labels needed to construct G by means of four graph operations -for definition see [10].
Courcelle and Olariu [10] showed that bounded tree-width always implies bounded cliquewidth with the following result: However, bounded clique-width does not always imply bounded tree-width.For example, the class of complete graphs has bounded clique-width but unbounded tree-width.Combining results from [10] with results from Gurski and Wanke [15] gives us certain graph classes for which tree-width and clique-width are either both bounded or both unbounded: Theorem 2.6 ([10, 15]).If C is a collection of graphs such that every graph G ∈ C either (i) has bounded vertex degree of no more than a constant ∆, or (ii) excludes a fixed graph H as a minor, or (iii) has bounded arboricity, then C has unbounded clique-width if and only if it has unbounded tree-width.

Symbolic sequences (words)
In this paper we use subsets of the natural numbers as letters in our alphabet creating infinite sequences of symbols to define our graph classes.
We refer to a (finite or infinite) sequence of letters chosen from a (finite or infinite) alphabet as a word.We denote by α i the i-th letter of the word α.A factor of α is a contiguous subword α [i,j] being the sequence of letters from the i-th to the j-th letter of α.The length of a word (or factor) is the number of letters the word contains, and the distance between the i-th and j-th letter in a word is |i − j|.
Given a word α over an alphabet A, and a sub-alphabet S ⊂ A, the subword of α restricted to S is the word derived from α by deleting all letters not in S and concatenating the remaining factors in the same order as they appear in α.We denote this subword as α S .
Nested words have a recursive structure and are defined as follows: Definition 3.1.A word α over alphabet A = N is branched if A can be partitioned into a finite base set B and ordered (finite or infinite) branch sets {H 1 , H 2 , . . .}, so that in α an occurrence of letter n ∈ H i can only appear • immediately after a base letter (only the first letter in H i ) or the letter in H i that immediately preceeds it in the H i order, and • immediately before a base letter or the letter in H i that immediately succeeds it in the H i order.
A maximal factor of α containing no base letters is called a branch of α.A branch is preceded and succeeded by a base letter.The letters in the branch come from one branch set, say H i , and appear, starting with the first letter in H i , in the defined order.

Definition 3.2.
A branched word α over alphabet A = N is b-nested if there exists a fixed positive integer b such that any subset S ⊆ A can be partitioned into a base set B S and ordered branch sets • α S is a branched word with base set B S , and • the branch sets of α S are (possibly empty) subsets of the branch sets of α (i.e H S i ⊆ H i with the same ordering, for each i).
We refer generally to nested words when referring to a collection of b-nested words for possibly more than one b.

Examples : Arithmetic nested words
The following nested words have infinite branch sets.
• α(1): One branch set with increasing branch sizes (base letters shown in blue): Let S be any subset of N.

Examples: Power nested words
The following nested words have an infinite number of single letter branch sets.

q-ary representation
For natural numbers q and n, let n q be the representation of n in q-ary.Let k be the number of trailing zeros of n q and let j be the first non-zero digit from the right.Alternatively, there exist unique j, k ∈ N and m ∈ N 0 such that n = jq k + mq k+1 (1 j q − 1).
. .} be a subset of N, where x 1 < x 2 . . . .Let n be a position in κ(q) with the letter x 1 and let j, k, m be the unique integers such that n = jq k + mq k+1 and x 1 = k(q − 1) + j as described above.Further, let x i x 1 be the highest element of S such that x i < kq.We claim that κ(q) is nested over S with base B S = {x 1 , x 2 , . . .x i } of maximum size q − 1 with single letter branch sets H S1 = {x i+1 }, . . ., H St = {x i+t }, . . . .For any positive integer t the letter immediately preceding or succeeding an appearance of x i+t in κ(q) S is either x 1 or x i .Hence, κ(q) S is branched with base B S , |B S | q − 1 and branch sets H St , and κ(q) satisfies the conditions of Definition 3.2.

Fibonacci representation
The well-known Fibonacci sequence of numbers is defined recursively as The Fibonacci representation of a number is a sequence of 0s and 1s, rather like binary, except that a 1 in position k counting from the right represents F k+1 instead of 2 k−1 .Note that to avoid having a repeated 1 in the sequence we start with F 2 .So, for example, 101101 represents as a Fibonacci representation.In general, a Fibonacci representation is not unique.However, Zeckendorf [25] showed that every positive integer can be represented uniquely as the sum of non-consecutive Fibonacci numbers, so we will only use the Zeckendorf representation here.
We define the infinite word η such that η n = i if the first 1 in the Fibonacci representation of n (from the right) appears in position i (e.g.n = 45 which has Fibonacci representation 10010100 gives η 45 = 3).Thus: The first six iterates are as follows:

Generating new nested words
If α is a nested word and L a finite collection of letters (that may or may not be letters in A, the alphabet of α) then inserting an arbitrary number of letters from L into arbitrary positions in α creates a new word that is also nested, since we can add the finite number of letters in L to the base B. Hence, nested words are not rare.Proposition 3.6.There are uncountably many distinct nested words.
Proof.Let α be a nested word and β an infinite (non-nested) binary word.Interlace the letters of α and β to create a new word γ, so that the even letters of γ are α and the odd letters β. γ is nested since we can just add the two letters of β to the base of α to create a finite base for γ.There are uncountably many distinct binary words which gives the result.
However, the existence of a nested subword β in α is not sufficient to make α a nested word.The subword α \ β may have an unbounded base and contain elements that contradict our desired characteristics -see Section 5.1.4 Path-star hereditary graph classes and t-sails A path-star graph is a graph with arboricity two that can be decomposed into a path (or forest of paths) and a forest of stars, where the leaves of the stars may embed in the paths.If we are considering a single infinite path then a convenient way to define the sequence of connections between the forest of stars and the path is using an infinite word so that the i-th letter in the word indicates the star that connects to the i-th vertex in the path.We assume that all leaves of the stars embed in the path since non-embedding leaves have no effect on tree-width, i.e. if G is a finite path-star graph and H is isomorphic to G except for the removal of all non-embedding leaves then tw(H) = tw(G).
Let α be an infinite word over the alphabet A = N.We denote the path P = (V P , E P ) with vertices V P = {p j : j ∈ N} and edges E P = {(p j , p j+1 ) : j ∈ N}.The star-vertices are denoted V S = {s i : i ∈ N} and star edges E S = {(p j , s α j ) : j ∈ N}.
Definition 4.1.We define an infinite path-star graph R α = (V, E) where V = V P ∪ V S and E = E P ∪ E S (see example in Figure 3).We define the corresponding path-star class R α to be the finite induced subgraphs of R α .
Any graph G ∈ R α can be witnessed by an embedding φ(G) into the infinite graph R α .To simplify the presentation we will associate G with a particular embedding in R α depending on the context.
To avoid confusion when referring to different types of path, we will refer to the class-path when referring to the (infinite) path of the path-star class, or a path component when referring to a finite section of it.We use the shorthand m-path-vertex for a vertex in the class-path corresponding to the letter m in α.
In addition, if α is a nested word over alphabet A, and α S a nested subword restricted to the sub-alphabet S, then we refer to base star-vertices and base path-vertices for vertices that correspond to base letters in S and branch star-vertices and branch path-vertices for vertices that correspond to branch letters in S.These vertices will depend on the choice of S.

Path-star classes with unbounded tree-width and clique-width
Throughout this section let A be an alphabet and α an infinite word over A. Let A α ⊆ A be the set of letters in A that appear an infinite number of times in α.That is, these are the letters of A corresponding to the infinite stars in R α .
Theorem 4.2.If A α is infinite then the graph class R α has unbounded tree-width and clique-width.
Proof.We will show that R α contains a t-sail for all t and thus has unbounded tree-width by Lemma 2.4.As R α has arboricity two it follows from Theorem 2.6 that R α also has unbounded clique-width.
Let A α = {i 1 , i 2 , . . .}.For any t ∈ N we can create a set of t factors of α as follows.
Let I 1 = {j}, where j is the position of the first occurrence of letter i 1 in α.For 2 k t let I k = [x, y] be the next interval beyond I k−1 where α I k contains all of i 1 , . . ., i k .Such intervals can always be found because the letters in A α repeat infinitely in α.This gives us a set of t disjoint factors of α, {α I k : 1 k t}.

Defining the vertex set
is a t-sail and the result follows.

Path-star classes are not subclasses of circle graphs
A circle graph is an intersection graph of finitely many chords of a circle.Circle graphs are a much studied hereditary class, in particular, because the class is vertex-minor closed (for definition see [14]).Geelen, Kwon, McCarty and Wollan [14] showed that a vertex-minor-closed graph class has bounded clique-width if and only if it excludes a circle graph as a vertex-minor.
More recently, in [16], the authors describe the unavoidable induced subgraphs of circle graphs with large tree-width.To distinguish the results in this paper from those in [16] we show that path-star graph classes are not subclasses of circle graphs.
Let α be a word over an alphabet with at least two letters.We will call a factor of α that starts with one letter i and ends with another j, with no other occurrences of either letter in the factor, an (i, j)-alternance.If G is a graph in the path-star class R α {1,2} induced by the two stars s 1 and s 2 and a path component, we show that it is not possible to construct a chord representation of G when the sequence in α corresponding to the path component has more than four (1, 2)alternances, i.e.G is not a circle graph.For example, the word 11212221112 alternates 5 times between 1 and 2 and therefore does not represent a circle graph.
We may refer to G by name or by α letter sequence (e.g.G = 1221221).We will always label the path vertices of G starting with p 1 so that 1221221 has path vertices p 1 , . . ., p 7 .Proof.Every graph in R α {1,2} is a vertex-minor of a graph in R α .As circle graphs are vertexminor closed , if R α is a subclass of circle graphs then so is R α {1,2} .Thus, if we can find a graph in R α {1,2} that is not a circle graph then we are done.Also note that there is a 1−1 correspondence between the (1, 2)-alternances in α and α {1,2} .Suppose, for a contradiction, that there exists a factor β of α {1,2} in which the letters 1 and 2 alternate more than four times, with the property that the graph G induced by the stars s 1 and s 2 and the path component corresponding to the factor β is a circle graph.
We try to construct a chord representation for G -see examples in Figure 4. Without loss of generality, we assume that the first letter of β is 1 and the second 2.
Note that the chords representing s 1 and s 2 do not cross, shown as vertical lines in Figure 4. Designating the arcs between s 1 and s 2 A and B (shown in red), and the arcs bounded by s 1 and s 2 C and D respectively (shown in blue), note that every path vertex adjacent to s 1 must be represented by a chord with one end in C and the other in either A or B, and similarly for s 2 , a chord with one end in D and the other in either A or B. Therefore, if p 1 and p 2 are the two chords representing the first (1, 2)-alternance then they must cross and both have an end in either A or B. Without loss of generality, let them both have an end in A.
Notice that for any i, the chords representing p i+2 , p i+3 , . . .must all be on the same side of the chord representing p i as none of them can cross this chord.Also notice that if i < j < k and α p i = α p j = 1, α p k = 2 then the chord for p j must be situated on the s 2 side of the chord for p i to accommodate the next alternance (and likewise with 1 and 2 reversed).
Suppose that no path-vertex chord has an end in B. If p 3 is a 1 (i.e. a second (1, 2)-alternance) then its chord must cross only s 1 and p 2 .If this is on the 'non-s 2 ' side of p 1 then this prevents  any further alternance since the path is blocked from star s 2 by chord p 1 .So for there to be a third alternance, p 3 must be on the s 2 side as shown in the 1212 example in Figure 4.
The chord representing path-vertex p 4 cannot cross p 1 or p 2 .Furthermore, if it is on the 'nons 1 ' side of p 2 then this prevents any further alternance since the path is blocked from star s 1 by chord p 2 .Hence, without using arc B, we can have at most three (1, 2)-alternances.
Now suppose that a path-vertex chord may have an end in B. We may have at most two alternances through A before switching to B, as if we start with three alternances through A, as shown in the 1212 example in Figure 4, then p 4 is blocked from B.
If we switch to B after two alternances then p 4 is a 1 with chord ends in C and B. It is possible to have at most two alternances through B before we reach p 6 which is blocked by p 4 , as shown in the 121121 example in Figure 4, and thereafter no further alternance is possible either via A or B.
It follows that the maximum number of alternances possible is four.By assumption, the letters in β alternate more than four times and hence we cannot construct a chord representation of G, and we have a contradiction.

Nested path-star hereditary graph classes
We now focus on path-star graph classes defined by nested words.

Nested path-star classes are KKW-free
It is quite possible for path-star graph classes to contain a large wall -see example in Figure 5.However, we show that path-star graph classes created from nested words (see Definition 3.2) are KKW-free.
Theorem 5.1.If α is a nested word then R α is KKW-free.
Proof.R α has arboricity two so does not contain K 5 or K 4,4 .
We show that R α does not contain a subdivision of a t × t wall for t > 12b where α is b-nested.
Suppose, for a contradiction, that R α contains a graph G that is a subdivision of a t × t wall for some t > 12b.Fix some embedding of this wall into R α , and let S ⊆ A denote the letters whose star-vertices appear in this embedding.
Since α is b-nested, S has a base B where |B| b.There can be at most three x-path-vertices in G for x ∈ S since s x is a vertex of degree at most three.Hence, V(G) can contain at most 3b < t 4 base-path-vertices.
We claim that for every degree three vertex in G there must be a path to a base-path-vertex that passes through at most two other degree three vertices.
Case 1: Let our degree three vertex be star-vertex s x .If x ∈ B we are done.Otherwise, s x is adjacent to three x-path-vertices each of which is adjacent to at least one path-vertex.If these path-vertices correspond to distinct letters then from the construction of the nested word at least one of them is in B and we are done.
Otherwise, suppose two of them are branch-path-vertices corresponding to the same letter, say y / ∈ (See Figure 6).We now have a path vertex sequence y, x, s x , x, y.If there is no s y starvertex in the wall then the y-path-vertices must be only degree two and the next vertices at either end of the path must be the same -either a base-path-vertex or the next vertex in the branch order.
If it is the next vertex in branch order, then we repeat the process until we reach a base-pathvertex or a star-vertex.
If a star-vertex, say s z , is at both ends of the path then we have a hole formed.As G is a wall this must have 6 degree three vertices and the only candidates are s x , x, z, s z , z, x.Thus, the two x-path-vertices must each be adjacent to another non-y-path-vertex.If these are different then one is a base letter and we are done; if the same, say i / ∈ B, then we have the same branch sequence zyxi . . .as before and the path ends in a star, say s j .But now, considering the third x-path-vertex (See Figure 6), this can only be a path to s j or s z (or both) which creates a hole with only 4 degree three vertices which contradicts the structure of a wall.
Case 2: Let our degree three vertex be an x-path-vertex.Then one edge from the vertex must be incident to the star-vertex s x and the other two edges incident to path-vertices.The star-vertex s x must be incident to at least one other x-path-vertex followed by another path-vertex.If our three path-vertices adjacent to x-path-vertices correspond to distinct letters then at least one of them must be a base path-vertex.
Suppose two of them correspond to the same letter y / ∈ B (See Figure 6).Then we are in the same position as Case 1.
In each case our degree three vertex is in a path to a base-path-vertex that passes through at most two other degree three vertices.So every induced subdivision of a 2 × 2 wall in G must contain a base-path-vertex.
Hence, allowing at least one base-path-vertex in each induced subdivision of a 2 × 2-wall, V(G) must contain at least t 4 base-path-vertices.But we know V(G) can contain at most 3b < t 4 base- path-vertices, so we have a contradiction.Thus, R α cannot contain a subdivision of a t × t wall when t > 12b.
A similar argument can be applied to a line graph of a subdivision of a t × t wall noting that each triangle in the line graph contains a star-vertex.

A nested path-star graph with large tree-width contains a large t-sail
A k-block in a graph G is a maximal set of at least k vertices no two of which can be separated in G by deleting fewer than k vertices.A k-block can be thought of as a highly connected part of a graph and has been used in a number of ways.In particular, Weißauer showed in [24] that for k 1 every graph of tree-width at least 2k 2 has a minor containing a k-block.
In [2] a more restricted type of k-block was introduced.A strong k-block in G is a set B of at least k vertices such that for every 2-subset {x, y} of B, there exists a collection P x,y of at least k distinct and pairwise internally disjoint paths in G from x to y, where for every two distinct 2-subsets {x, y}, {x ′ , y ′ } ⊆ B and every choice of paths P ∈ P x,y and P ′ ∈ P x ′ ,y ′ we have We show that all large subdivisions of t-sails contain large strong k-blocks and that in nested path-star graph classes, large strong k-blocks only occur in graphs containing a large induced subgraph of the subdivision of a t-sail.We use this to conclude that a nested path-star graph has large tree-width if and only if it contains an induced subgraph of a large t-sail.
Lemma 5.2.For any t 1 a subdivision of a t 3 -sail contains a strong t-block.
Proof.Let B = {s 1 , . . .s t }, i.e. the first t star-vertices.We claim B is a strong t-block.
For every two distinct 2-subsets {x, y}, {x ′ , y ′ } ⊆ B and every choice of paths P ∈ P x,y and P ′ ∈ P x ′ ,y ′ we have To show that in nested path-star graph classes a large strong k-block contains a large induced subgraph of the subdivision of a t-sail, we explore the structure of nested words further.
Let α be a b-nested word over the alphabet S 1 , and define nested subwords α S i for i 2 by S i = S i−1 \ B i−1 with base B i .Observation 5.3.For any positive integer t, the letters of ∪ t i=1 B i appear in α only in factors containing at least one letter from each B i , which we will refer to as B t factors.
Observation 5.4.If x and y are in S t+1 in different branches of α S t then between them in α there is at least one B t factor.Observation 5.5.If x and y are in S t+1 in the same branch set H = {h 1 , h 2 , . . .} of α S t where h 1 < h 2 < . . ., x = h r and y = h s where t < r < r + t < s then between any incidence of x and y in α there is at least one factor of either h 1 . . .h t or h r+1 . . .h r+t .Lemma 5.6.Let G be a graph from a path-star class defined by a b-nested word α over the infinite alphabet A. If G contains a strong k-block, where k max{tb t + tb, t(b + 2) + 2} for some integer t 1, then it also contains as an induced subgraph a subdivision of a t-sail.
Proof.Fix some embedding of G into R α , and let S 1 ⊆ A denote the letters whose star-vertices appear in this embedding.For any two vertices in a strong k-block there must be k internally disjoint paths between them.The vertices in this strong k-block must be star-vertices since only star-vertices can have degree greater than three, so let L ⊆ S 1 be the letters corresponding to the vertices of the strong k-block in G where k tb t + tb.
Let subword α S 1 have base B 1 , and define subwords α S i for i 2 by S i = S i−1 \ B i−1 with base B i .
Observe that star-vertices that correspond to letters in S 1 \ L appear in at most one of the internally disjoint paths between two vertices of the strong k-block otherwise the paths would not be disjoint.
As k t(b + 2) + 2, and there are at most tb letters in ∪ t i=1 B i , then L contains at least 2t + 2 letters in S t+1 .Therefore, either there exists a pair x, y ∈ L that are in different branch sets of α S t (Case 1) or there are at least 2t + 2 letters in the same branch set of α S t , say H = {h 1 , h 2 , . . .} where h 1 < h 2 < . . .(Case 2) .In Case 2, since there are at least 2t + 2 letters, we can choose the t + 1-th and 2t + 2th letters as x = h r and y = h s so that t < r < r + t < s.From Observations 5.3, 5.4 and 5.5, one of the following cases holds: Case 1: (x and y in different branch sets) between every occurrence of x and occurrence of y in α S 1 there is a factor consisting of only letters from ∪ t i=1 B i with at least one letter from each B i .
At most one of the disjoint paths from s x to s y can pass through each star associated with a letter in ∪ t i=1 B i (i.e. at most tb paths).This leaves k − tb paths that do not pass through such a star-vertex.The remaining disjoint paths must all include a set of consecutive class-pathvertices corresponding to a B t factor in α (Observations 5.3 and 5.4).
Given a collection of B t factors of size k − tb, using the pigeonhole principle, as k tb t + tb, there are at least k−tb b t t of them that contain the same letter from each set B i , 1 i t.Call this set of at least t letters T ⊂ ∪ t i=1 B i .It follows that at least t of the disjoint paths from s x to s y contain a component of the class-path incorporating a path-vertex corresponding to each letter in T.
Case 2: (x and y in the same branch set H) between every occurrence of x and occurrence of y in α S 1 there is a set of t consecutive letters from H (either the first t letters in H or the (at least) t letters between x and y in the H order).
At most one of the disjoint paths from s x to s y can pass through each star associated with one of these letters in H.This leaves k − 2t disjoint paths that do not pass through such a starvertex.The remaining disjoint paths must all include a set of consecutive class-path-vertices corresponding to one of our two sets of consecutive letters from H (Observation 5.5).
From the pigeonhole principle at least k−2t 2 t of the disjoint paths must correspond to the same set of t consecutive letters from H.Call this set of letters T ⊆ H.It follows that at least t of the disjoint paths from s x to s y contain a component of the class-path incorporating a path-vertex corresponding to each letter in T.
In either Case 1 or Case 2, G contains at least t disjoint paths between s x and s y that contain class-path components containing path-vertices corresponding to each letter in T. The graph induced by the vertices of these class-path components plus the star-vertices corresponding to the letters in T form a subdivision of a t-sail.
Letting B k be the class of all graphs with no strong k-block and remembering that the k-basic obstructions are (1) the complete graph K k , (2) the complete bipartite graph K k,k , (3) a subdivision of the (k × k)-wall and (4) a line graph of a subdivision of the (k × k)-wall, we use the following result: Theorem 5.7 ([2]).For every integer k 1 there exists a positive integer w(k) such that every graph in B k with tree-width more than w(k) contains an induced subgraph isomorphic to one of the k-basic obstructions.
Theorem 5.8.If α is a nested word then for every t 1 there is a positive integer valued function f α (t) such that every graph in R α of tree-width at least f α (t) contains an induced subgraph isomorphic to a subdivision of a t-sail.
Proof.Let k = t(b+2) t and f α (t) = w(k) as defined by Theorem 5.7.Suppose for graph G ∈ R α , we have tw(G) f α (t).Then by Theorem 5.7, G cannot be in B k because by Theorem 5.1 R α is KKW-free, G does not contain a k-basic obstruction, and therefore, G contains a strong k-block.It follows by Lemma 5.6 that G contains an induced subgraph isomorphic to the subdivision of a t-sail.

Minimal sparse hereditary classes of unbounded tree-width 6.1 Hereditary graph classes of bounded vertex degree or with an excluded minor do not contain a minimal subclass
The structure of a wall allows us to delete vertices and leave the fundamental structure intact, ignoring subdivisions, and this quality is used in the following: Lemma 6.1.A subdivision (or line graph of a subdivision) of a W kt×kt wall for k, t 1 contains an induced subgraph isomorphic to a subdivision (or line graph of a subdivision) of a W t×t wall that does not contain a cycle smaller than C 8k−2 (other than C 3 in the case of the line graph).
Proof.Let G = (V, E) be a subdivision of a W kt×kt wall.Let V 3 ⊆ V be the set of degree three vertices in G. Every 'brick' (or hole) contains six vertices from V 3 .An induced subgraph G ′ isomorphic to a subdivision of a W t×t wall can be constructed by overlaying a lattice of k × k sub-walls, the new 'bricks', and deleting all vertices of G internal to every new brick, as shown in the example in Figure 7 where k = 2 and t = 4.Each new 'brick' contains 8k − 2 vertices from V 3 , and thus G ′ does not contain a cycle smaller than C 8k−2 .
An identical argument works if G is the line graph of a subdivision of a W kt×kt wall.Theorem 6.2.If C is a hereditary class of graphs of bounded vertex degree or has an excluded minor then it does not contain a minimal class.
Proof.If D is a minimal hereditary subclass of C then by Theorems 1.1 or 1.2, as it has unbounded tree-width, it contains (as a member of the class) a graph G which is isomorphic to a subdivision (or line graph of a subdivision) of a W kt×kt wall for arbitrarily large k and t.
Suppose C m (m > 3) is the shortest cycle in G. Set k > m+2 8 .Then from Lemma 6.1 G contains as an induced subgraph a subdivision (or line graph of a subdivision) of a W t×t wall that does not contain a cycle smaller than C 8k−2 which is longer than C m (other than C 3 ).
But now the proper hereditary subclass D ∩ Free(C m ) contains a subdivision of W t×t for arbitrarily large t, so D ∩ Free(C m ) also has unbounded tree-width, which contradicts D being minimal.

Nested path-star hereditary graph classes do not contain a minimal subclass
We will show that no path-star hereditary graph class defined by a nested word contains a minimal subclass.Proof.Let G be a T -sail in R α .Fix some embedding of G into R α , and let S 1 ⊆ A denote the letters whose star-vertices appear in this embedding.
Let subword α S 1 have base B 1 , and define nested subwords Let G ′ be the subgraph of G induced by the vertices of G excluding the star-vertices corresponding to the letters of ∪ m i=1 B i (at most mb) and excluding the star-vertices corresponding to alternate letters in the branch sets of S m (i.e.half the remaining star-vertices).
We claim G ′ contains an induced subgraph isomorphic to a subdivision of a t-sail which does not contain C m .
That G ′ contains an induced subgraph isomorphic to a subdivision of a t-sail follows from the fact that it contains at least T −mb 2 t of the star-vertices of G and all the path-vertices from the path components of G.
Suppose that G ′ contains an m-cycle with only one star-vertex, say s x .Then the path-vertices adjacent to s x in the cycle must both correspond to a branch letter x of S m .The rest of the cycle must consist of path-vertices corresponding to a factor x . . .x of α.Using Observations 5.3 and 5.4 there must be at least one base path-vertex from each of the m base sets B i in the cycle so it has more than m vertices, a contradiction.Suppose that G ′ contains an m-cycle with three (or more) star-vertices, s x , s y and s z .This must contain path components corresponding to α factors of the form x . . .y, x . . .z and y . . .z (or their reverse).These factors must be contained in branches of α S m , and hence x, y and z must be from the same branch set, otherwise the cycle would have more than m vertices for the same reason as for the one star-vertex case.But now, assuming without loss of generality that their branch order is x < y < z, then there must be a y in the x . . .z factor and a shorter cycle exists, which contradicts the fact that the shortest cycle in G is of length m.
Lastly, suppose that G ′ contains an m-cycle with precisely two star-vertices, s x and s z .This must contain path components corresponding to two α factors of the form x . . .z (or the reverse).Hence, x and z are from the same branch set.In fact they must be consecutive letters from the same branch set, since if there was another letter y between them in the branch order then there would be a shorter cycle than C m , either containing the two stars s x and s y or the two stars s y and s z .But the construction of G ′ requires the removal of star-vertices corresponding to alternate letters in the branch sets of S m , so x and z cannot be consecutive branch letters, a contradiction.Hence G ′ does not contain C m .Theorem 6.4.If R α is a path-star hereditary class of graphs defined by a nested word α then it does not contain a minimal class.
Proof.If D is a minimal subclass of R α then by Theorem 5.8 for every positive integer T , D contains a subdivision of a T -sail.
Suppose the shortest cycle in D is C m (m > 3).Then from Lemma 6.3 for any positive integer t there exists a positive integer T such that any T -sail in D contains an induced subgraph isomorphic to a subdivision of a t-sail which does not contain a C m cycle.Thus the subclass D ∩ Free(C m ) still contains a subdivision of a t-sail for arbitrarily large t and has unbounded tree-width, which contradicts D being minimal.

Concluding remarks
This paper is, as far as we know, the first time an attempt has been made to use combinatorics on words in the study of treewidth.We believe the results are sufficient enough to justify further use of this technique.Likewise, path-star hereditary graph classes seem to be significant in respect of the study of tree-width and clique-width in sparse graph classes and warrant a more thorough study.
We have shown that path-star graph classes defined by nested words block large walls and large line graphs of walls.However, we have not resolved whether there are other such words, or whether, if we forbid a large wall and a large line graph of a wall in a path-star graph class R α , then α contains a large nested subword.
Although we have added a new object, a t-sail, the identification of a full list of boundary objects that are obstructions to bounded tree-width in hereditary graph classes is still some way from being achieved.Our approach has been to consider graphs of bounded arboricity, in particular, those graphs of arboricity two constructed using forests of paths and stars.We believe this approach could be a fruitful way to identify further boundary objects.

Figure 3 :
Figure 3:The first section of path-star graph R κ(2)

Figure 5 :
Figure 5: Example of a subdivision of a 4 × 4-wall embedded in a path-star graph

Figure 6 :
Figure 6: Attempt to construct a wall in R α without base-path-vertices

Lemma 6 . 3 .
Let α be a b-nested word over an infinite alphabet A. Then for integers t 2, m > 3 and T 2t + mb, a T -sail in R α with smallest cycle C m contains an induced subgraph isomorphic to a subdivision of a t-sail which does not contain C m .