Tree independence number I. (Even hole, diamond, pyramid)‐free graphs

The tree‐independence number tree‐ α , first defined and studied by Dallard, Milanič, and Štorgel, is a variant of treewidth tailored to solving the maximum independent set problem. Over a series of papers, Abrishami et al. developed the so‐called central bag method to study induced obstructions to bounded treewidth. Among others, they showed that, in a certain superclass C of (even hole, diamond, pyramid)‐free graphs, treewidth is bounded by a function of the clique number. In this paper, we relax the bounded clique number assumption, and show that C has bounded tree‐ α . Via existing results, this yields a polynomial‐time algorithm for the Maximum Weight Independent Set problem in this class. Our result also corroborates, for this class of graphs, a conjecture of Dallard, Milanič, and Štorgel that in a hereditary graph class, tree‐ α is bounded if and only if the treewidth is bounded by a function of the clique number.


Introduction
All graphs in this paper are finite and simple.Let G = (V (G), E(G)) be a graph.An induced subgraph of G is a subgraph of G obtained by deleting vertices.More explicitly, if X ⊆ V (G), then G[X] denotes the subgraph of G induced by X, with vertex set X and edge set E(G) ∩ X 2 .In this paper, we use induced subgraphs and their vertex sets interchangeably.For a graph H, we say that G contains H if H is isomorphic to an induced subgraph of G, and G is H-free if it does not contain H.If H is a set of graphs, then we say that G is H-free if G is H-free for every H ∈ H.
A tree decomposition (T, ϕ) of G consists of a tree T and a map ϕ : V (T ) → 2 V (G) satisfying the following: (i) For all v ∈ V (G), there exists t ∈ V (T ) such that v ∈ ϕ(t); (ii) For all v 1 v 2 ∈ E(G), there exists t ∈ V (T ) such that v 1 , v 2 ∈ ϕ(t); (iii) For all v ∈ V (G), the subgraph of T induced by {t ∈ V (T ) s.t.v ∈ ϕ(t)} is connected.
The width of a tree decomposition (T, ϕ) is max t∈V (T ) |ϕ(t)| − 1.The treewidth of G, denoted by tw(G), is the minimum width of a tree decomposition of G.
A stable set in a graph G is a set of pairwise non-adjacent vertices of G.The independence number α(G) of G is the size of a maximum stable set in G.The independence number of a tree decomposition (T, ϕ) of G is max t∈V (T ) α(G[ϕ(t)]).The tree independence number of G, denoted tree-α(G), is the minimum independence number of a tree decomposition of G.
The tree independence number was defined and studied by Dallard, Milanič and Štorgel in [21], in the context of studying the complexity of the Maximum Weight Independent Set (MWIS) problem on graph classes whose treewidth is large only due to the presence of a large clique.It is shown in [21] that if a graph is given together with a tree decomposition with bounded independence number, then the MWIS problem can be solved in polynomial time.In [19], it is then shown how to compute such tree decompositions efficiently in graphs of bounded tree-α, yielding an efficient algorithm for the MWIS problem for graphs of bounded tree-α.
In [20], a graph class G is called (tw, ω)-bounded if there exists a function f : N → N such that for every graph G ∈ G, and for every induced subgraph H of G, tw(H) ≤ f (ω(H)) (where ω(G) is the size of a largest clique in G).Such a function f is called a (tw, ω)-bounding function for the class G.By Ramsey's theorem, graph classes of bounded tree independence number are (tw, ω)-bounded.Furthermore, all of the following graph classes have bounded tree independence number: graph classes of bounded treewidth, intersection graphs of connected subgraphs of graphs with treewidth at most t, for any fixed positive integer t (these contain for example chordal graphs and circular arc graphs), and graph classes in which the size of a minimal separator is bounded.More examples and further discussion of these parameters is given in [20,21,22].The question remains whether the (tw, ω)-bounded property has algorithmic implications for the MWIS problem.This question would be settled if the following conjecture from [21] holds true: Conjecture 1.1 ([21]).A class C of graphs is (tw, ω)-bounded if and only if it has bounded tree-α.
Before we state our main result, we define several types of graphs.A clique in a graph is a set of pairwise adjacent vertices, and for an integer t ≥ 1, K t denotes the complete graph on t vertices.A path is a tree of maximum degree at most 2. A path in G is an induced subgraph of G that is a path.A diamond is the graph obtained by deleting an edge from K 4 .A hole of a graph G is an induced cycle of length at least four.By C 4 we denote the hole of length four.A wheel (H, w) is a hole H and a vertex w ∈ V (G) such that w has at least three neighbors in H.An even wheel is a wheel (H, w) such that w has an even number of neighbors in H.A theta is a graph consisting of two vertices a and b and three distinct paths P 1 , P 2 , P 3 from a to b such that any two of them induce a hole (so in particular a and b are nonadjacent, and hence all three paths are of length at least 2).A pyramid is a graph consisting of a vertex a, a triangle b 1 , b 2 , b 3 (disjoint from a), and three paths P 1 , P 2 , P 3 from a to b 1 , b 2 , b 3 , respectively, such that any two of them induce a hole (so in particular, at least two of the three paths are of length at least 2).A prism is a graph consisting of two disjoint triangles a 1 a 2 a 3 and b 1 b 2 b 3 and three paths P 1 , P 2 , P 3 , with P i from a i to b i , such that any two of them induce a hole.Thetas, pyramids, and prisms are called three-path configurations (3PCs).
We introduce the following notation, which we will use throughout the paper: • C is the class of (C 4 , diamond, theta, pyramid, prism, even wheel)-free graphs.
• C * is the class of (C 4 , diamond, theta, prism, even wheel)-free graphs (and in particular, C ⊆ C * ).Our main result is the following.

Theorem 1.2.
There exists an integer Γ such that tree-α(G) ≤ Γ for every graph G ∈ C. Theorem 1.2 strengthens the main result of [7], while building on some of the ideas of its proof: Theorem 1.3.( [7]) For each t > 0, there exists an integer c t such that for every G ∈ C with ω(G) ≤ t, we have tw(G) ≤ c t .
Since the class of (even hole, diamond, pyramid)-free graphs is a subclass of C (as even-holefree graphs are (C 4 , theta, prism, even wheel)-free), the above two theorems imply, respectively, the following two statements about (even hole, diamond, pyramid)-free graphs.

Theorem 1.4.
There exists an integer c such that, for every (even hole, diamond, pyramid)-free graph G, we have tree-α(G) ≤ c.Theorem 1.5.( [7]) For each t > 0, there exists an integer c t , such that for every (even hole, diamond, pyramid, K t )-free graph G, tw(G) ≤ c t .
Thus Theorem 1.4 yields a polynomial-time algorithm for the MWIS problem in this class.It is in fact known that for a certain superclass of this class, the MWIS problem can be solved in polynomial time [5], which we discuss further below.But the algorithmic method shown here is of independent interest, since it might be possible to extend it to (even-hole, diamond)-free graphs (for which we do not yet have a polynomial time algorithm to solve the MWIS problem).
Conjecture 1.6.The class of (even hole, diamond)-free graph has bounded tree-α.Further context.First, observe that the class C was already known to be (tw, ω)-bounded [7], so Theorem 1.2 corroborates Conjecture 1.1 for this class.Although many of the ingredients of our proof are class-specific, it is worth investigating whether any of our methods can be used for tackling the conjecture in general.
The second consideration is an algorithmic one.Even-hole-free graphs have been studied extensively since the 1990s, and yet despite their structural similarity to perfect graphs, the complexity of some fundamental optimization problems that are solvable in polynomial time for perfect graphs is not known for even-hole-free graphs.Indeed, while we can find maximum cliques in even-hole-free graphs efficiently by a result shown by Farber [23] and later Alekseev [9] stating that C 4 -free graphs have quadratically many maximal cliques, we still do not have efficient algorithms for the MWIS, minimum vertex coloring, and minimum clique cover problems.
In this direction, a number of subclasses of even-hole-free graphs have been studied: for instance, in [26] it was shown that (even hole, diamond)-free graphs can be colored in polynomial time.A sketch of that proof is as follows.First, a decomposition theorem is obtained for (even hole, diamond)-free graphs: these graphs are decomposed by star cutsets that partition into two cliques, and by 2-joins, into line graphs of a tree plus at most two vertices.This decomposition theorem is used to show that (even hole, diamond)-free graphs always have a vertex that is either simplicial or of degree 2. This then implies that the class is "β-perfect," which in turn yields a polynomial time coloring algorithm which proceeds by coloring greedily on a particular, easily constructible ordering of the vertices.
Our current paper continues this study by investigating the MWIS problem, in the hope of extending the methods presented here to (even hole, diamond)-free graphs (see Conjecture 1.6).We now survey some subclasses of even-hole-free graphs in relation to the MWIS problem.Recall that even-hole-free graphs are (C 4 , theta, prism, even wheel)-free.

A superclass of (even hole, pyramid)-free graphs
In [5], a superclass of (even hole, pyramid)-free graphs is studied, namely the class of (C 4 , theta, prism, pyramid, turtle)-free graphs (where a turtle is a graph derived from a wheel that cannot be present in even-hole-free graphs, or in fact, in even-wheel-free graphs).This yields, in particular, a polynomial-time algorithm for MWIS in our class C.However, their method (which we outline below) does not generalize beyond the pyramid-free setting.In contrast, we believe the method we present in this paper extends to the larger class of (even hole, diamond)-free graphs.
To understand why the results from [5] do not generalize, we outline their procedure for solving MWIS here.It is shown in [5] that the class under study has polynomially many minimal separators, and that the set of all minimal separators can be constructed in polynomial time.
The number of minimal separators of a graph is related to the number of potential maximal cliques (PMCs) introduced by Bouchitté and Todinca in [12,13].For a graph G, denote by n the number of vertices, m the number of edges, s the number of minimal separators, and p the number of PMCs in G.In [13], it is proved that p is in O(ns 2 + ns + 1), and that given the list of minimal separators, the PMCs of G can be listed in time O(n 2 ms 2 ).In [29], based on [24], it is proved that given the list of PMCs, the MWIS problem can be solved in O(n 5 mp).So, summing everything up, the ability to list all minimal separators in polynomial time implies a polynomial-time algorithm for the MWIS problem.
On the other hand, if pyramids are allowed in even-hole-free graphs, the polynomial minimal separator property is lost, as witnessed for example by a k-pyramid (a graph consisting of a clique B = {b 1 , . . ., b k }, a vertex a and paths P i = a . . .b i , for i = 1, . . ., k, such that any 3 of them induce a pyramid).

A subclass of (even hole, pyramid)-free graphs
The class of (even hole, K 3 )-free graphs is a proper subclass of (even hole, pyramid)-free graphs.
In [14] it is shown that (even hole, K 3 )-free graphs (and in fact (theta, even wheel, K 3 )-free graphs) have treewidth at most 5. On the other hand, in [32], a certain construction called a "layered wheel" is given, which shows that (even hole, K 4 )-free graphs do not have bounded treewidth.The construction is full of diamonds, so it was conjectured in [32] and proved in [3] that (even hole, diamond, K 4 )-free graphs have bounded treewidth (in fact, the result from [3] pertains to more general classes of even-hole-free graphs obtained by forbidding any clique and a certain type of graph which contains diamonds).
The question remains whether the following is true (it can be shown that the outcome holds when forbidding, in addition, another family of graphs, called "generalized k-pyramids" [2]).
Conjecture 1.7.The class of (even hole, K t )-free graph has logarithmic treewidth.

Terminology and notation.
If G is a path or a cycle, the length of G is |E(G)|.By P k we denote the path on k vertices.If P = p 1 -p 2 -• • • -p k , then P * = P \ {p 1 , p k } denotes the interior of the path P , and p 1 , p k are its ends.
Let G be a graph and is the union of X and N G (X).When the graph G is clear from context, we omit the subscript G from the open and closed neighborhoods.If X, Y ⊆ V (G), we say X is anticomplete to Y if there are no edges with one endpoint in X and one endpoint in Y .We say that X has a neighbor in Y if X is not anticomplete to Y .We say that v is anticomplete to X if {v} is anticomplete to X.
A clique is a set of pairwise adjacent vertices.The clique number of G, denoted by ω(G), is the size of a maximum clique in G.The chromatic number of G, denoted by χ(G), is the minimum number of sets of a partition of V (G) into stable sets.The clique cover number of G, denoted by χ(G), is the minimum number of sets of a partition of V (G) into cliques (note that some sources use "clique cover number" to refer the minimum number of cliques required to cover the edges of G, and use "node clique cover number" or "clique partition number" for what we call "clique cover number" here; for the avoidance of doubt, all clique covers in this paper will be vertex clique covers, that is, partitions of a vertex set into cliques).
Let (H, v) be a wheel.We call v the center, or hub of the wheel.A sector of (H, v) is a path P ⊆ H of length at least one such that the ends of P are adjacent to v and P * is anticomplete to v. A sector of (H, v) is long if it has length greater than one, and short otherwise.
If paths . .-b 3 induce a pyramid, then we say that the vertex a is the apex of this pyramid. A

Tools
In this section, we describe the tools and preliminary results needed to prove that graphs in C have bounded tree-α.
2.1.χ-boundedness of the complements and the tree-clique cover number.Many of the results from [7] that we use provide bounds on the size of certain sets in terms of ω, but in actuality the proofs work by exhibiting clique covers of those sets.Correspondingly, our arguments revolve around clique covers rather than stable sets.As such, it is natural to define the tree-clique cover number tree-χ(G) of a graph G analogously to tree-α.More explicitly, we define the clique cover number of a tree decomposition (T, ϕ) as the maximum value of χ(G[ϕ(t)]) over nodes t ∈ V (T ), and we let tree-χ(G) be the minimum value of the clique cover number over all tree decompositions of G.
We note that, for C 4 -free graphs, statements about tree-χ have tree-α analogues, and viceversa.Indeed, in one direction, α(G) ≤ χ(G) for any graph G.In the other direction, by a result of Wagon [34], the class of 2K 2 -free graphs is χ-bounded, meaning that there exists a non-decreasing function f : N + → N + such that, for any 2K 2 -free graph G, χ(G) ≤ f (ω(G)).In fact, from [34], one may take f (ω(G)) to be ω(G)+1

2
. We record this in a lemma, noting that the complement of a C 4 -free graph G is 2K 2 -free: The above discussion immediately yields the following: We remark that C 4 -freeness (or at least, χ-boundedness in the complement) is necessary to obtain a relationship between tree-χ and tree-α: Remark 2.3.For any C > 0, there exists a graph with tree-α ≤ 2 and tree-χ ≥ C.
Proof.It is pointed out in [21] that, in any tree decomposition (T, ϕ) of a graph G, there exists a vertex v ∈ G and a node t ∈ T with N [v] ⊆ ϕ(t); in particular, if we start with a graph G and construct G + from two copies of G by adding all possible edges between them, for any tree decomposition of G + , some bag will contain a copy of G.We can use this to construct an example as claimed.Start with any triangle-free graph G with chromatic number greater than C (such graphs are well-known to exist); take its complement G, and construct the graph G + as above.The resulting graph will have α(G + ) ≤ 2 (since its complement is still triangle-free), but by the above discussion, some bag in any tree decomposition of G + will have clique cover number more than C.
Finally, we also have the following simple observation relating tree-χ to treewidth: Throughout the paper, we will work with tree-χ rather than tree-α, with the understanding that our results can be stated in terms of tree-α via Lemma 2.2.

2.2.
Treewidth, tree-χ, clique cutsets and (3PC, wheel)-free graphs.We next state the observations that clique cutsets do not affect treewidth, and tree-χ.We note that [21] includes an analogous result for tree-α.In view of the treewidth and tree-α results, the proof of the tree-χ result is routine and left to the reader.Lemma 2.5.( [11]) Let G be a graph.Then tw(G) is equal to the maximum treewidth over all induced subgraphs of G with no clique cutset.Lemma 2.6.Let G be a graph.Then tree-χ(G) is equal to the maximum tree-χ over all induced subgraphs of G with no clique cutset.
In view of this, given a class X , we will write X nc for the class of graphs in X with no clique cutset.
We continue with a few observations about (3PC, wheel)-free graphs.
Proof.The first statement follows from Lemma 2.5, Lemma 2.6 and Lemma 2.7, and the second from the fact that (theta, wheel, K 3 )-free graphs are a subclass of (3PC, wheel)-free graphs.A different proof of the second statement is given in [7].
The next few lemmas state how (w, c)-balanced separators relate to tree-χ.They are modelled on analogous results concerning treewidth, which we also include.Lemma 2.9 ([4, 25], see also [30]).Let G be a graph, let c ∈ [ 1 2 , 1), and let k be a positive integer.If, for every weight function w : 1), and let k be a positive integer.Suppose that for every weight function w : V (G) → [0, 1] with w(G) = 1, G has a (w, c)-balanced separator with clique cover number at most k.Then there exists a number g(k, c) such that tree-χ(G) ≤ g(k, c).
Proof.We adapt the proof from [30] and g(k, c We claim that g(k, c) satisfies the statement of the lemma.In fact, we will recursively show the following slightly stronger statement: (*) Let (G, W ) be a pair where G is as in the lemma, and Then there exists a tree decomposition (T, ϕ) of G of clique cover number at most g(k, c), and such that W ⊆ ϕ(t) for some t ∈ V (T ).In particular, tree-χ(G) ≤ g(k, c).The statement (*) is clear if α(G) ≤ h(k, c).Indeed, in that case, Lemma 2.1 yields that a tree decomposition consisting of a single bag does the job.
If α(G) > h(k, c), we may assume without loss of generality that α(W ) ≥ h(k, c) (otherwise, we simply add vertices to it arbitrarily until α(W ) = h(k, c), and by Lemma 2.1, we do not lose the property that χ(W ) ≤ h(k,c)+1 2 in doing so).We then select a stable subset W ′ of W with ), and define a weight function w : V (G) → [0, 1] by putting w(x) := 1 h(k,c) for each x ∈ W ′ , and w(x) := 0 for all other x.By assumption, G has a (w, c)-balanced separator X with χ(X) ≤ k.Now for any component D of G\X, we have w(D) ≤ c, and so there are at least (1−c)h(k, c) ≥ 2k vertices of W ′ outside of D. In particular, since α(X) ≤ χ(X) ≤ k, there are k vertices of W ′ , say x 1 , . . ., x k , which do not belong to D ∪ X.
Let us now pick a clique cover of W with at most h(k,c)+1 2 cliques, and consider the set Y := W ∩ D. By the above, the (distinct) cliques containing x 1 , . . ., x k do not intersect Y , so , and we may recursively find a tree decomposition of D ∪ X by applying (*) to the pair (D ∪ X, Y ∪ X) (noting that D ∪ X is a smaller graph).
We do the above for each component D i of G \ X, defining Y i analogously to Y ; we obtain, for each i, a tree decomposition We then assemble those tree decompositions into a decomposition (T, ϕ) of G with the desired property by starting with the union of the T i 's and adding single vertex t adjacent to all t i 's, and with ϕ(t) = W ∪ X (and with ϕ restricting to each ϕ i on the corresponding k, c).Inductively, this is also true of every other bag.This yields a tree decomposition of G (see Fact 2.8 from page 111 of [30]).
In the converse direction, we have the following: Lemma 2.11 ([17]).Let G be a graph and let k be a positive integer.If tw(G) ≤ k, then for every weight function w : This can be generalized at no extra cost to obtain tree-α and tree-χ versions of the lemma (since the separator in the proof from [17] is contained in a single bag of the given tree decomposition).Lemma 2.12 (see [17]).Let G be a graph and let k be a positive integer.If tree-α(G) ≤ k, then for every weight function w : V (G) → [0, 1] with w(G) = 1 and every c ∈ [ 1 2 , 1), G has a (w, c)-balanced separator of independence number at most k.The analogous statement for tree-χ holds as well.
We note the following result from [7]: Let G be a graph and let w : , and unbalanced otherwise.Let U denote the set of unbalanced vertices of G. Let v ∈ U .The canonical star separation for v, denoted Let ≤ A be the relation on U where for x, y ∈ U , x ≤ A y if and only if x = y or y ∈ A x .We also need: Lemma 2.14 ([7]).Let G be a (C 4 , diamond)-free graph with no clique cutset, let w : V (G) → [0, 1] be a weight function on G with w(G) = 1, let U be the set of unbalanced vertices of G, and let ≤ A be the relation on U defined above.Then, ≤ A is a partial order.

Central bags.
Let G be a graph and let w : V (G) → [0, 1] be a weight function on G with w(G) = 1.We call a collection S of separations of G smooth if the following hold: (i) S 1 and S 2 are nearly non-crossing for all distinct S 1 , S 2 ∈ S; (ii) There is a set of unbalanced vertices v(S) ⊆ V (G) such that there is a bijection Let S be a smooth collection of separations of G.Then, the central bag for S, denoted β S , is defined as follows: By property (iii) of smooth collections of separations, it holds that v(S) ⊆ β S .Also note that G \ β S = S∈S A(S).Moreover, by property (i), every component for some i (to see this, we consider a maximal subset of D belonging to a single A(f (v i )), and use (i) to extend the subset if it does not equal the whole of D, contradicting maximality).
We next define an inherited weight function w S on β S by recording the weight of each component of G \ β S into a unique vertex v i ∈ β S .To this end, we start by fixing an ordering {v 1 , . . ., v k } of v(S).For every f (v i ) ∈ S, let A * (f (v i )) be the union of all connected components D of 1≤j≤k A(f (v j )) such that D ⊆ A(f (v i )), and D ⊆ A(f (v j )) for every j < i.In particular, (A * (f (v 1 )), . . ., A * (f (v k ))) is a partition of S∈S A(S).For a component D of S∈S A(S), we call the unique v i with D ⊆ A * (f (v i )) the anchor of D. In other words, the anchor of D is the first v i in our ordering with D ⊆ A(f (v i )).We define w S : ) for all v i ∈ v(S), and w S (v) = w(v) for all v ∈ v(S).

Lemma 2.15 ([7]
).Let G be a (theta, even wheel)-free graph, let H be a hole of G, and let

H) be adjacent vertices each with at least two non-adjacent neighbors in H. Then, v 1 and v 2 have a common neighbor in H.
Recall that a wheel (H, w) is a hole H and a vertex w ∈ V (G) such that w has at least three neighbors in H.A wheel (H, w) is a twin wheel if N (w) ∩ H is a path of length two.A wheel (H, w) is a bug if |N (w) ∩ H| = 3 and w has exactly two adjacent neighbors in H. Bugs are also known as short pyramids.A wheel (H, w) is a universal wheel if w is complete to H. We will also use the following result about wheels and star cutsets.

Lemma 2.16 ([7]
).Let G be a (C 4 , even wheel, theta, prism)-free graph and let (H, v) be a wheel of G that is not a bug, a twin wheel nor a universal wheel.Let (A, C, B) be a separation of

Trisimplicial elimination orderings.
A central ingredient in our argument is the fact that the graphs we consider admit elimination orderings v 1 , . . ., v |V (G)| of their vertex sets such that for each i, N (v i ) ∩ {v i , v i+1 , . . ., v |V (G)| } is the union of a bounded number of cliques (in particular, bisimplicial or trisimplicial elimination orderings qualify).
It is worth pointing out that every even-hole-free graph has a bisimplicial vertex [15], and as such admits a bisimplicial elimination ordering.This gives us hope that it might be possible to generalize the results of this paper in order to obtain a polynomial algorithm for MWIS in the whole class of even-hole-free graphs.However, for the purposes of this paper, we would like to derive and use an analogous result for the class C. It is not clear how to adapt the proof from [15] to this setting; nonetheless, there is another result from [26] that one may use as a basis for what we need here: Theorem 2.17.( [26]) Every (even hole, diamond)-free graph has a vertex which is either simplicial or of degree 2.
We note that this result is in a sense orthogonal to the one from [15]: it requires the additional condition of diamond-freeness, and yields a stronger outcome than just a bisimplicial vertex.While the proof of this result does generalize to the class C (and in fact to the even more general class of (C 4 , diamond, theta, prism, even wheel)-free graphs) in a relatively straightforward way, it is a bit long.Instead, we include here a shorter proof of a weaker result which is still strong enough for our purposes.

Lemma 2.18. Every graph in C has a trisimplicial vertex.
Proof.We first prove that graphs in C with no wheel satisfy a stronger statement: (1) If a graph G ∈ C contains no wheel, then it has a bisimplicial vertex.
Note that G is (3PC, wheel)-free, so by Lemma 2.7, it either has a clique cutset, or is a complete graph or a hole.In the latter two cases, G satisfies the statement, so assume G has a clique cutset.In that case, it is well-known (see e.g.Lemma 3.3 of [33]) that G has an extreme clique cutset C, that is, a clique cutset C such that, for some component D of G \ C, D ∪ C has no clique cutset.As above, D ∪ C is a complete graph or a hole; in particular, any vertex in D is bisimplicial in G, proving (1).
We are ready to show that general graphs in C have trisimplicial vertices.We do so using an approach conceptually similar to the proof of (1), by finding an "extreme" star cutset in G.
If G is wheel-free, then we are done by (1), so assume G contains a wheel (H, w).Since G is pyramid-free, (H, w) is not a bug, and since it is diamond-free, it is not a twin or universal wheel, and hence Lemma 2.16 implies that G has a star cutset.Let us pick a star cutset C with center v such that some component D of G \ C is minimal, in the sense that it does not properly contain a component of G \ C ′ for any other choice of star cutset C ′ .
Case 1: v is complete to D. In this case, D is a singleton (since otherwise we may add all but one vertex of D to the cutset, and removing the new cutset leaves a singleton component properly contained in D, contradicting the minimality of D).But in this case, the unique vertex in D is simplicial in G by diamond-freeness, since its neighborhood is contained in N [v], and we are done.
Case 2: v is not complete to D. In this case, once more by the minimality of D, v must be anticomplete to D (since otherwise we may add all neighbors of v in D to the cutset and obtain once more a smaller component).We now claim: (2) G[D] is wheel-free.Indeed, suppose not, and let (H, u) be a wheel in G [D].Put C ′ := N [u], and note that, by Lemma 2.16, C ′ is a star cutset in G with center u, and H Q∪C ′ for any component Q of G \ C ′ (in actuality, the statement of the lemma yields Now by (1), G[D] has a bisimplicial vertex x.But x has at most one neighbor in G\D, since N G\D (x) ⊆ N (v), and x and v are non-adjacent (and so by (C 4 , diamond)-freeness, they have at most one common neighbor).Thus x is trisimplicial in G, completing the proof.

Balanced separators and central bags
In this section, we construct a useful central bag for graphs in C and prove that the central bag has a balanced separator of bounded χ.We note that, because of Lemma 2.6, we often assume that the graphs we work with do not have clique cutsets.We also have the following simple characterisation of the neighborhood of vertices in diamond-free graphs.

Lemma 3.1 ([7]
).Let G be diamond-free and let v ∈ V (G).Then, N (v) is the union of disjoint and pairwise anticomplete cliques.
For X ⊆ V (G), let Hub(X) denote the set of all vertices x ∈ X for which there exists a wheel (H, x) which is not a bug and with H ⊆ X.We will need several results from [7] that we will now describe, introducing the relevant notation and terminology as necessary.Note that we will not explicitly use Lemmas 3.2 and 3.4 in this paper, but for completeness (and to give the reader some intuition of how the results that we do use explicitly were obtained), we include their statements.We start with a result showing that, under suitable conditions, the components of a central bag by a star cutset attach to the cutset in a controlled way: Lemma 3.2 ([7]).Let G be a (C 4 , theta, prism, even wheel, diamond)-free graph with no clique cutset and let w : V (G) → [0, 1] be a weight function on G with w(G) = 1.Let S be a smooth collection of separations of G, let β S be the central bag for S, and let w S be the inherited weight function on β S .Let v ∈ β S and (by Lemma 3.1) We next have a result about balanced separators in central bags with balanced vertices.This result follows directly from the proof of Lemma 20 of [7].The statement of that lemma gives a separator of size 6ω(β s ) + k, under the stronger assumption that |N (v) ∩ Hub(β S )| < k, but its proof works by actually providing a clique cover of N (v) \ Hub(β S ) by 6 cliques, which is strong enough to derive the following: Lemma 3.3 ([7]).Let G be a (C 4 , theta, pyramid, prism, diamond, even wheel)-free graph with no clique cutset, let w : 1), and let k be a positive integer.Let S be a smooth collection of separations of G, let β S be the central bag for S, and let w S be the inherited weight function on β S .Suppose that there exists v ∈ β S such that v is balanced in G, and assume that χ(N (v) ∩ Hub(β S )) ≤ k.Then β S has a (w S , c)-balanced separator of clique cover number at most 6 + k.
Recall that C * nc denotes the class of (C 4 , diamond, theta, prism, even wheel)-free graphs with no clique cutset.Let G ∈ C * nc , let w : V (G) → [0, 1] be a weight function on G with w(G) = 1, and let U be the set of unbalanced vertices of G. Let X ⊆ U .The X-revised collection of separations, denoted SX , is defined as follows.Let u ∈ X, and let Su = ( Ãu , Cu , Bu ) be such that Bu is the largest-weight connected component of , and Ãu = V (G) \ ( Cu ∪ Bu ).Then, SX = { Su : u ∈ X}.Note that the separations in SX are closely related to canonical star separations.Specifically, for all u ∈ X, the following hold: One way of thinking about the revised separations is that we begin with the canonical separation, and extend C u by extending every edge ux with x ∈ C u ∩ X to a maximal clique (each such clique has size at least 2, so it can be extended to a maximal clique in a unique way, by diamond-freeness).
The following lemma shows that separations from a revised collection are nearly non-crossing.It follows directly from Lemma 21 of [7], which additionally assumes K t -freeness in the statement, but never uses that assumption in the proof.

Lemma 3.4 ([7]
).Let G ∈ C * nc , let w : V (G) → [0, 1] be a weight function on G with w(G) = 1, let U be the set of unbalanced vertices of G, and let X ⊆ U be such that every vertex of X is minimal under the relation ≤ A .Let S = SX be the X-revised collection of separations.Then, Su and Sv are nearly non-crossing for all Su , Sv ∈ S.
We now show how to construct a useful collection of separations of G.By Theorem 2.18, any graph in C admits a trisimplicial elimination ordering.Let G ∈ C, and let v 1 , . . ., v ℓ be a trisimplicial elimination ordering of Hub(G).Let U be the set of unbalanced vertices of G. Let m be defined as follows.If Hub(G) ⊆ U , then m = ℓ+1.Otherwise, let m be minimum such that v m is an element of Hub(G) \ U .Now, {v 1 , . . ., v m−1 } ⊆ U .Let M be the set of minimal vertices of {v 1 , . . ., v m−1 } under the relation ≤ A , and let SM be the M -revised collection of separations.We call ({v 1 , . . ., v ℓ }, m, M, SM ) the hub division of G.The next two lemmas describe properties of the hub division.Once more, their statements in [7] assume K t -freeness, but their proofs do not.Moreover, the hub division was defined in [7] with respect to a different ordering of the vertices, but the proofs of these two lemmas apply to any ordering (and in particular, to our trisimplicial elimination ordering).
We continue with a short result compiling two earlier ones.

Extending balanced separators
In this section, we prove that we can construct a balanced separator of G of bounded χ given a balanced separator of β M of bounded χ.This is where our proof differs significantly from [7].Together with the main result of the previous section, this is sufficient to prove Theorem 1.2.First, we need the following lemma, which is once more an adaptation of a corresponding result from [7], and is directly implied by the proof from there.
We also need the next lemma, which examines how three vertices can have neighbors in a connected subgraph.

Lemma 4.2 ([4]
).Let x 1 , x 2 , x 3 be three distinct vertices of a graph G. Assume that H is a connected induced subgraph of G \ {x 1 , x 2 , x 3 } such that H contains at least one neighbor of each of x 1 , x 2 , x 3 , and that subject to these conditions V (H) is minimal subject to inclusion.Then one of the following holds: (i) For distinct i, j, k ∈ {1, 2, 3}, there exists P that is either a path from x i to x j or a hole containing the edge x i x j such that • H = P \ {x i , x j }, and • either x k has at least two non-adjacent neighbors in H or x k has exactly two neighbors in H and its neighbors in H are adjacent.(ii) There exists a vertex a ∈ H and three paths P 1 , P 2 , P 3 , where P i is from a to x i , such that • H = (P 1 ∪ P 2 ∪ P 3 ) \ {x 1 , x 2 , x 3 }, and • the sets P 1 \ {a}, P 2 \ {a} and P 3 \ {a} are pairwise disjoint, and • for distinct i, j ∈ {1, 2, 3}, there are no edges between P i \ {a} and P j \ {a}, except possibly x i x j .(iii) There exists a triangle a 1 a 2 a 3 in H and three paths P 1 , P 2 , P 3 , where P i is from a i to x i , such that Ki , and note that X ∩ β M is also a (w M , 1  2 )-balanced separator of β M , with χ( X ∩ β M ) ≤ t.We claim that we have: We first show that this is true if x ∈ M .Indeed, by Lemma 3.6, Hub(β M ) ⊆ {v m , v m+1 , . . ., v ℓ }.In particular, for each x ∈ M , N β M (x) ∩ Hub(β M ) is the union of three cliques.Moreover, by Lemma 4.1, χ(N β M (x) \ Hub(β M )) ≤ 2, and consequently N β M (x) can be covered by 5 cliques as claimed.Now assume that x ∈ {v 1 , . . ., v m−1 } \ M .In this case, we may find y ∈ M with y < A x.But then x ∈ A y , and N β M (x) ⊆ β M ∩ C y ⊆ {y} ∪ N β M (y), which can be covered by 5 cliques by the previous argument.This yields (1).
Consider now the bipartite graph H = (A, B) where A = {a 1 , . . ., a r }, B = {b 1 , . . ., b s }, each a i corresponds to a component D i of G\(β M ∪ X), and each b i corresponds to a component Q i of β M \ X, with an edge between a i and b j if there is an edge between D i and Q j .Recall that each Define γ := 8t + 3. We claim that: (2) H Core contains no hole of length at least γ.
Suppose for a contradiction that H Core contains such a hole L. Without loss of generality, the vertices of L are a We may find a hole with the following properties: • For each i with 1 ≤ i ≤ 2γ ′ , x i , respectively y i , belongs to the component corresponding to the tail, respectively head of f i .• For each i with 1 ≤ i ≤ γ ′ , P i is a (possibly one-vertex) path in Q i between y 2i−1 and x 2i .
• For each i with 1 ≤ i ≤ γ ′ , R i is a (possibly one-vertex) path in D i between y 2i and x 2i+1 (indices modulo 2γ ′ ).Since γ ′ > 4t, the pigeonhole principle yields five distinct indices i 1 , . . ., i 5 such that the anchors v(i 1 ), . . ., v(i 5 ) belong to the same clique, say K1 .It might be that some of those anchors coincide; however, we claim that at least two of them are distinct.Indeed, note that, by (1), a fixed v ∈ X ∩ M can have neighbors in at most five of the Q i , since χ(N β M (v)) ≤ 5.If all five anchors coincide, then we consider N L ({a i 1 , . . ., a i 5 }) (recall that L is the hole in H Core that we started with).Since L has size more than 10, this neighborhood consists of at least 6 nodes corresponding to components of )) has neighbors in all 6 of those components, which is not possible.Thus we may assume that two of the anchors, say v(i 1 ) and v(i 2 ) are distinct.
We note that each of v(i 1 ) and v(i 2 ) has two non-adjacent neighbors on W . Indeed, this is the case by construction, since D i 1 and D i 2 each have edges to a pair of distinct Q j s, and ) and v(i 2 ) are adjacent (since they are distinct vertices of K1 ), and they have no common neighbors in W (since all common neighbors of the two vertices belong to X).This is a contradiction to Lemma 2.15, which proves (2).
We next claim that: ( Indeed (3) follows directly from (1), since we have To see ( 4), note that, by the definition of H Core , any vertex a i ∈ A \ H Core has its anchor v(i) contained in some component Q of β M \ X.In particular, the anchor (and thus the component D i of G \ (β M ∪ X) corresponding to a i ) cannot have neighbors in any other component of β M \ X, meaning deg H (a i ) ≤ 1 in this case, as claimed.
In [35], it is shown that the treewidth of graphs without a complete bipartite subgraph and without a large induced hole is bounded (where the bound only depends on the sizes of the forbidden bipartite subgraph, and of the largest induced hole).Hence (2) and (3) imply that k := tw(H Core ) is bounded above by some constant depending only on t.By (4), tw(H) = k (assuming without loss of generality that it is at least 1).Let w ′ H be the weight function defined on H by letting w ′ H (a i ), respectively w ′ H (b j ), be the w-weight of D i , respectively Q j .Let w H be the normalisation of w ′ H , so that w H (H) = 1.Explicitly, w H (x) := w ′ H (x)/w ′ H (H) for each x ∈ H (and we note that we may assume w ′ H (H) = 0, since otherwise w( X) = 1, and X is a (w, 1  2 )-balanced separator whose clique cover number is bounded by t).From Lemma 2.11, it follows H has a (w H , 1  2 )-balanced separator of size at most k + 1, which we call Z.Put Z ′ := (Z ∩ B) ∪ N H (Z ∩ A), and note that (3) and (4) immediately imply: (5) |Z ′ | ≤ 5(k + 1).
We now claim the following: To see this, assume for a contradiction that this is not the case.Then we may find distinct i 1 , i 2 , i 3 such that v i 1 , v i 2 and v i 3 all belong to the same clique in the clique partition of X and, say, N (v iα ) ∩ Q 1 = ∅ for each α ∈ [3].We now apply Lemma 4.2 to those three vertices and a minimal induced subgraph of Q 1 containing neighbors of all three.We note that Q 1 contains no common neighbors of any two of the three vertices (since any such common neighbors belong to X, by diamond-freeness).In view of this, the second and third outcomes of the lemma imply that β M contains a pyramid, respectively a prism, which is impossible.In the first outcome, one of v i 1 , v i 2 , v i 3 is the center of a wheel in β M which is not a bug -once more impossible, by Lemma 3.6.This yields our desired contradiction and proves (6).
We are now ready to construct our (w, 1  2 )-balanced separator for G.To do so, we let J := {j ∈ [s] : b j ∈ Z ′ }, and We then put By (1), ( 5) and ( 6), we know that Y can be covered by at most t + 5 • (2t + 1) • 5(k + 1) cliques (t cliques for X, and 5 cliques for each of the neighborhoods, of which there are at most |I| ≤ (2t + 1)|Z ′ |).This yields our desired upper bound on χ(Y ) which depends only on t.It remains to check that Y is, indeed, a (w, 1  2 )-balanced separator of G. To do so, we first note that every component of H \ Z ′ is either contained inside a component of H \ Z, or consists of a single vertex in Z ∩ A. In particular, we have: (7) For every component F of H \ Z ′ , if F G is the union of all D i and Q j corresponding to vertices of F , then w(F G ) ≤ 1 2 .Indeed, this is true if F consists of a single vertex in Z ∩ A, since then F G belongs to A(f (v)) for some unbalanced v, and so It is also true if F belongs to some component of H \ Z, since then w H (F ) ≤ 1 2 , and we have Let S H := H[{a i : S ∩ D i = ∅} ∪ {b j : S ∩ Q j = ∅}], and note that S H is a connected induced subgraph of H. Finally, we look at two cases: • S H ∩ Z ′ = ∅.In this case, S H is contained in a connected component of H \ Z ′ , and S ⊆ (S H ) G (where as above, (S H ) G is the union of the connected components corresponding to vertices of S H ), so w(S) ≤ 1 2 by (7).
• S H ∩ Z ′ = ∅.In this case, we first note that |S H ∩ Z ′ | = 1.Indeed, let b j ∈ S H ∩ Z ′ , and note that, by construction of Y , the only D i with a neighbor in Q j \ Y are D i with v(i) ∈ Q j .These D i have no neighbors in any other Q j (since N (D i ) ⊆ N [v(i)]).In particular, S H is a star centered at b j , and v(i) ∈ Q j for all a i ∈ S H . Now as required.
This finishes the proof.

2. 4 .
Separations.A separation of a graph G is a triple (A, C, B), where A, B, C ⊆ V (G), A∪C∪B = V (G), A, B, and C are pairwise disjoint, and A is anticomplete to B. If S = (A, C, B) is a separation, we let A(S) = A, B(S) = B, and since if that were the case, any vertex in H \(Q∪N (Q)∪{u}) would be the center of aP 3 in H ∩N (u)).Let B ′ be the component of G \ C ′ containing v. By the above, there exists x ∈ D \ (B ′ ∪ C ′ ).Let D ′ be the component of G \ C ′ containing x.Since every path from x to G \ D intersects C, and since C ⊆ B ′ ∪ C ′ , it follows that D ′ ⊆ D, and this containment is proper (as u ∈ D \ D ′ ), contradicting the choice of C and D.

Lemma 3 . 5 (
[7]).Let G ∈ C, let w : V (G) → [0, 1] be a weight function on G with w(G) = 1, and let ({v 1 , . . ., v ℓ }, m, M, SM ) be the hub division of G.Then, SM is a smooth collection of separations of G.By Lemma 3.5, there is a central bag β M for SM and an inherited weight function w M on β M .Lemma 3.6 ([7]).Let G ∈ C, let w : V (G) → [0, 1] be a weight function on G with w(G) = 1, and let ({v 1 , . . ., v ℓ }, m, M, SM ) be the hub division of G. Let β M be the central bag for SM and let w M be the inherited weight function on β M .Then, for all 1

Theorem 3 . 8 .
Let G ∈ C nc , let w : V (G) → [0, 1] be a weight function on G, and let ({v 1 , . . ., v ℓ }, m, M, SM ) be the hub division of G. Let β M be the central bag for SM and let w M be the inherited weight function on β M .Then, β M has a (w M ,1  2 )-balanced separator of clique cover number at most 9.
x 2 , x 3 }, and • the sets P 1 , P 2 , and P 3 are pairwise disjoint, and • for distinct i, j ∈ {1, 2, 3}, there are no edges between P i and P j , except a i a j and possibly x i x j .For any t ∈ N + , there exists Γ ′ (t) with the following property.Let G ∈ C nc , let w : V (G) → [0, 1] be a weight function on G with w(G) = 1, and let ({v 1 , ..., v ℓ }, m, M, SM ) be the hub division of G. Let β M be the central bag for SM and let w M be the inherited weight function on β M .Assume that β M has a (w M ,12 )-balanced separator of clique cover number at most t.Then G has a (w,12 )-balanced separator whose clique cover number is at most Γ ′ (t).Proof.Let X be a (w M ,12 )-balanced separator of β M with χ(X) ≤ t.Let K 1 , . . ., K t be cliques covering X.Now for each i with 1 ≤ i ≤ t, let Ki equal K i if |K i | = 1, and let Ki be a maximal clique in G containing K i otherwise.We note that, since G is diamond-free, Ki is unique.Put X :=