Bounds on the Twin-Width of Product Graphs

Twin-width is a graph width parameter recently introduced by Bonnet, Kim, Thomass\'{e}&Watrigant. Given two graphs $G$ and $H$ and a graph product $\star$, we address the question: is the twin-width of $G\star H$ bounded by a function of the twin-widths of $G$ and $H$ and their maximum degrees? It is known that a bound of this type holds for strong products (Bonnet, Geniet, Kim, Thomass\'{e}&Watrigant; SODA 2021). We show that bounds of the same form hold for Cartesian, tensor/direct, corona, rooted, replacement, and zig-zag products. For the lexicographical product it is known that the twin-width of the product of two graphs is exactly the maximum of the twin-widths of the individual graphs (Bonnet, Kim, Reinald, Thomass\'{e}&Watrigant; IPEC 2021). In contrast, for the modular product we show that no bound can hold. In addition, we provide examples showing many of our bounds are tight, and give improved bounds for certain classes of graphs.


Introduction
Twin-width is a recently introduced graph parameter which, roughly speaking, measures how much the neighbourhoods differ as pairs of (not necessarily adjacent) vertices are iteratively contracted until a single vertex remains Bonnet et al. (2022e).Twin-width has already proven very useful in parameterised complexity.In particular, given a contraction sequence as input, first-order model checking is FPT in the formula length on classes of bounded twin-width Bonnet et al. (2022e).Additionally, for graphs of bounded twin-width with the contraction sequence as input, there is a linear time algorithm for triangle counting Kratsch et al. (2022), and an FPT-algorithm for maximum independent set Bonnet et al. (2021b).Polynomial kernels for several problems Bonnet et al. (2022d) are also known which do not require the contraction sequence to be given as input.However, deciding if the twin-width of a graph is at most four is NP-complete Bergé et al. (2022).Bounds on twin-width are known for many graph classes Balabán and Hlinený (2021); Bonnet et al. (2021aBonnet et al. ( , 2022bBonnet et al. ( , 2021c)); Gajarský et al. (2022); Jacob and Pilipczuk (2022); Simon and Toruńczyk (2021).In particular, twin-width is bounded on some classes of dense graphs (e.g., the class of complete graphs has twin-width zero) and on some classes of sparse graphs (e.g., the class of grid graphs has bounded twin-width), and indeed both graph classes of bounded tree-width and

Our Results
As mentioned above the main question we address is, given a graph product ⋆, what can be said of tww(G ⋆ H) for any graphs G and H? Our main finding is that for all graphs products studied (with the exception of the modular product) the twin-width of a product of two graphs is functionally bounded by the twin-width and maximum degrees of the individual graphs.These upper bounds for the twinwidth of graph products are summarised in Table 1.Note that twin-width is not monotone with respect to non-induced subgraphs, thus none of these bounds are necessarily implied by any of the others.Tab.1: Bounds on the twin-width of product graphs, see relevant sections for definitions of the graph products.The lexicographic bound is an equality, i.e. tww(G • H) = max{tww(G), tww(H)}.

Product
Using our upper bounds we can determine the twin-width of several families of graphs, for example rook graphs, Hamming graphs, and weak powers of cliques.We show tightness of our bounds on the twin-width of Cartesian, tensor and corona products.We also show tightness of the bound for strong and lexicographical products proved by Bonnet et al. (2021a) and Bonnet et al. (2022d) respectively.We also show that our bound on rooted products is almost tight.We prove that twin-width of a modular product of two graphs cannot (in general) be functionally bounded by twin-width and maximum degrees of the individual graphs by showing that the twin-width of the modular product of two paths diverges.
For the replacement product we prove tightness up-to a constant factor, however we leave open whether our bound for the zig-zag product is tight -we strongly suspect this can be improved.Additionally, we prove a lower bound for replacement product graphs which is a constant times the square root of their degree.
It is known (Bonnet et al., 2021a, Theorem 2.9) that if C is a hereditary class of graphs that has bounded twin-width, then it is K t,t free if and only if the subgraph closure of C, denoted by Sub(C), has bounded twin-width.This fact can then be used in combination with a bound on the twin-width of strong products to establish a further bound on the twin-width of the subgraph closure of a strong products of two classes (Bonnet et al., 2021a, Theorem 2.8).Our bounds can be applied in the exact same way to extend this result, as follows.
Theorem 1.1 (extension of (Bonnet et al., 2021a, Theorem 2.8)).Let ⋆ be one of the Cartesian , tensor ×, strong ⊠, lexicographic •, corona , rooted ♯, replacement r , or zig-zag z products and let G and H two classes such that G ⋆ H is K t,t -free.Then there exists some function f such that

Organisation of the Paper
We begin in Section 2 with some notation and definitions.Sections 3 & 4 contain our findings for the twin-width of each graph product.Each subsection of Sections 3 & 4 begins with the definition of a graph product ⋆ before stating our results for it.We finish with some conclusions in Section 5.

Notation and Definitions
Elementary Definitions.Throughout the paper we assume all graphs are simple, that is they have no loops or multiple edges.Given a graph G, the neighbourhood of a vertex We drop subscripts when the graph is clear from the context.We let ∆(G) = max v∈V (G) d G (v) denote the maximum degree of a graph G taken over all vertices in V (G).The graph distance between two vertices u, v ∈ V (G) is denoted by d G (u, v).We use K n to denote a clique on n vertices, K n,m to denote a complete bipartite graph with bags of size n and m, and P n to denote a path on n vertices.For two graphs G and H, we write G ∼ = H to mean that G is isomorphic to H, and for two sets S, K we let S△K = (S \ K) ∪ (K \ S) denote their symmetric difference.
Graph Products.Let G and H be any two graphs, and let } be the product of their vertex sets.Note that, to make proofs easier to follow, we will use letters from {u, v, w} for vertices from G, and letters from {i, j, k} for vertices from H. We consider several graph products G ⋆ H in this paper, all will have the vertex set V (G) × V (H) but the edge sets will depend on the product ⋆ and the graphs G and H. See the relevant sections for definitions of the different products.
Trigraphs, contraction sequences, and twin-width of a graph.Following the notation introduced in Bonnet et al. (2022e), a trigraph G has vertex set V (G), a set E(G) of black edges, and a set R(G) of red edges (which are thought of as 'error edges'), with E(G) and R(G) being disjoint.In a trigraph G, the neighbourhood N G (v) of a vertex v ∈ V (G) consists of all the vertices adjacent to v by a black or red edge.A d-trigraph is a trigraph G such that the red graph (V (G), R(G)) has degree at most d.In that instance, we also say that the trigraph has red degree at most d.
A contraction in a trigraph G consists of merging/contracting two (non-necessarily adjacent) vertices u and v into a single vertex w, and updating the edges of G as follows.Every vertex of the symmetric difference N G (u)△N G (v) is linked to w by a red edge, and every vertex x of the intersection N G (u) ∩ N G (v) is linked to w by a black edge if both ux ∈ E(G) and vx ∈ E(G), and by a red edge otherwise.All other edges in the graph (those not incident to u or v) remain unchanged.The vertices u and v, together with any edges incident to these vertices, are removed from the trigraph.To make proofs easier to follow, we will sometimes use v to refer to the vertex w if v has been removed as part of contracting u and v into w.That is, one can think of the new vertex w as having both the label v and the label u.
A d-sequence is a sequence of d-trigraphs G n , G n−1 , . . ., G 1 , where G n = G, G 1 = K 1 is the graph on a single vertex, and G i−1 is obtained from G i by performing a single contraction of two (non-necessarily adjacent) vertices.Note that G i has precisely i vertices, for every i ∈ [n].The twin-width of G, denoted by tww(G), is the minimum integer d such that G admits a d-sequence.
We say that G is a trigraph over a graph We also use the following useful fact.3 Bounds on Twin-width of Standard Products

Cartesian Product
The Cartesian product G H has vertex set V (G) × V (H) and Theorem 3.1.For any graphs G and H we have Proof: Let G = G n , . . ., G 1 = K 1 be a tww(G)-sequence for G, and label the m vertices of H by [m] in any order.For a fixed i ∈ [m], we call i-th copy of G, the subgraph of G H induced by vertices of the form (v, i) for every v ∈ V (G).We continue to refer to all vertices (v, i) as the i-th copy of G even if contractions have been made.We contract G H to K 1 H by following the series G n , . . ., G 1 of contractions in each copy of G.That is, we reduce G n to G n−1 in the first copy, then the second, etc. until we have reduced G n to G n−1 in the m-th copy, and then we return to the first copy to reduce G n−1 to G n−2 etc.We proceed like this until we are left with a trigraph over K 1 H, which is isomorphic to H. We then reduce this trigraph using a tww(H)-sequence for H.
Observe that at any stage in the contraction sequence of G H described above there can be at most tww(G) red edges of the form (v, i)(u, i), for some fixed v ∈ V (G), in the current trigraph.This follows since we do not make any contractions between two vertices (u, i) and (v, j) where j = i until there is only a single vertex left in each copy of G. Now observe that there are at most 2∆(H) edges of the form (v, i)(u, j), where j = i, for any v ∈ V (G) at any given time.To see this note that if (u, i) and (v, i) are being contracted to (w, i) in the i-th copy of G but (u, j) and (v, j) have not yet been contracted in some other copies of G where j = i, then the red edges (w, i)(u, j) and (w, i)(v, j) will be formed for each such j.There can be at most 2∆(H) such edges and so it follows that there is a tww(G) + 2∆(H)-sequence transforming G H into a trigraph over K 1 H ∼ = H.This trigraph admits a tww(H)+∆(H)-sequence by Lemma 2.1.Thus tww(G H) ≤ max{tww(G) + 2∆(H), tww(H) + ∆(H)} as claimed.
As K 2 K 2 ∼ = C 4 , which has twin-width 0, Theorem 3.1 does not always give a tight bound.However the following result shows that Theorem 3.1 is tight for Cartesian products of two non-trivial cliques where at least one has more than two vertices.
Proposition 3.2 (Rook Graphs).For any n, m ≥ 1 we have Thus as we have tww(C 4 ) and tww(K n ) = 0, we establish the first case.We can now assume that n, m ≥ 2 and (n, m) = (2, 2).
For the upper bound observe that as and also the bound . By symmetry we have b((u, i), (u, j)) ≥ 2(n − 1) for any i = j and u ∈ V (G).
Case 2 [v = u and i = j]: For any v = u and i = j we have For integers d, k ≥ 1, let H(d, k) be the Hamming graph with k d vertices, defined inductively by Hamming graphs are a generalisation of Rook graphs and have applications in coding theory and distributed computing.Arguably the most well known Hamming graph is the hypercube H(d, 2).Our next result determines the twin-width of all Hamming graphs and is proved by applying Theorem 3.1 iteratively.Similarly to Rook graphs, the twin-width of Hamming graphs matches the bound in Theorem 3.1.
Proposition 3.3 (Hamming Graphs).For any d, k ≥ 1 we have Applying Theorem 3.1 iteratively to H(d, k), for d ≥ 3, and using the fact that tww .
We now consider the lower bound for d, k ≥ 2. The lower bound will follow Lemma 2.
It follows that u and v differ at two coordinates i and j where i = j.Thus if w ∈ V (H(d, k)) is adjacent to both u and v it must be equal to u at all but one coordinate and equal to v at all but one coordinate, so Thus by Lemma 2.2 we have tww(H(d, k)) ≥ t(d, k) as claimed.

Tensor Product
The tensor product G × H (also called the direct product, Kronecker product, weak product, or conjunction) has vertex set V (G) × V (H) and Theorem 3.4.For any graphs G and H we have Proof: Given a tww(G)-sequence G n , . . ., G 1 of contractions for G the first stage is to apply one contraction at a time in each copy (as we did in the proof of Theorem 3.1), which leaves a trigraph over H.
The second stage, as in Theorem 3.1, is to reduce this trigraph using a tww(H)-sequence for H.
To begin observe that there are no red edges of the form (v, i)(u, i) created in the first stage.This follows since there are no edges of the form (u, i)(v, i) in E(G × H), we only perform contractions between two vertices (u, i) and (v, i), and edges of this form cannot be created from contractions.We claim that the largest red degree in the first stage is at most (tww(G) + 2) • ∆(H).First consider the red degree of a vertex (w, i) directly after a contraction of (u, i) and (v, i) into (w, i).After the equivalent contraction in G the vertex w has at red degree at most tww(G), and all these edges are of the form wz where z / ∈ {u, v}, thus these edges correspond to tww(G) • ∆(H) red edges of the form (w, i)(z, j).However there may be up to 2∆(H) additional other red edges of the form (w, i)(u, j) and (w, i)(v, j) created by the contraction of (u, i) and (v, i).This occurs if there was an edge uv ∈ E(G) and the pair u, v we merged in i-th copy before they were merged in some other copy j, where ij ∈ E(H).Thus there is a For a graph G we let G denote its complement and note that for any graph G we have tww(G) = tww(G).For the special case of cliques, observe that K n K m = K n × K m .Thus we obtain the following corollary to Proposition 3.2.
Corollary 3.5 (Rook Complement Graphs).For any n, m ≥ 1 we have For integers d, k ≥ 1, let T(d, k) be the weak power of a clique, defined inductively by Weak powers of cliques have been studied in the contexts of colourings & independent sets Greenwell and Lovász (1974); Alon et al. (2004); Ghandehari and Hatami (2008) and isoperimetric inequalities Brakensiek (2017).Using Theorem 3.4 we can determine the twin-width for these graphs.

Proof:
The following observation holds by the definition of tensor product: To begin, if k = 1 or d = 1 then the graph is either a set of isolated vertices or a single clique, respectively, so the result holds.For the case k = 2 we claim that for any d ≥ 1 the graph T(d, 2) is a complete matching; thus tww(T(d, 2)) = 0, in agreement with the statement.To see this note that, by the observation above, any vertex x = (x 1 , . . .x d ) ∈ V (T(d, 2)) = {0, 1} d only has one neighbour given by x = (1 − x 1 , . . . 1 − x d ).
Thus from now on we can assume that k ≥ 3 and d ≥ 2.
We now prove the upper bound tww(T(d, k)) ≤ 2(k − 1) d−1 by induction in d.Observe that for any k ≥ 3 and d = 2 we have tww(T(2, k)) = 2(k − 1) by Corollary 3.5, which establishes the base case.For the inductive step observe that T(d, k) is a regular graph of degree (k − 1) d since, for any fixed vertex x, each coordinate of of a neighbouring vertex y ∈ [k] d can take one of k − 1 values which differ from the corresponding coordinate of x.Recall also that tww(K k ) = 0, thus by Theorem 3.4, for any k ≥ 3, we have For the lower bound, observe that any two vertices u, v ∈ V (T(d, k))), where u = v, must differ at i ∈ [d] coordinates.Thus for such a pair we have N (u) ∩ N (v) = (k − 2) i (k − 1) d−i as for a vertex w to be adjacent to both u and v it must differ from both at all coordinates, so there are k − 2 options for each of the coordinates where u and v differ and k − 1 options for coordinates which are shared.Now, since uv ∈ E(T(d, k)) if and only if i = d, it follows that for any u Thus by Lemma 2.2 we have tww(T(d, k)) ≥ 2(k − 1) d−1 .
Although the previous result shows that Theorem 3.4 is tight, one of the graphs in the tensor product has twin-width zero and so it is not clear the tww(G) • ∆(H) term in Theorem 3.4 is needed.However, by a very similar proof to that of Proposition 3.9, one can show that for any c, d ≥ 2 the twin-width of the tensor product of c and d dimensional hypercubes satisfies This shows that the tww(G) • ∆(H) term in Theorem 3.4 is necessary.
The following result shows that if one of the graphs in the tensor product is a star graph, where K 1,n denotes a star on n + 1 vertices, then we do not incur the full cost of the max degree term.Proposition 3.7 (Tensor with a Star).Let G be any graph, then for any integer n ≥ 1 Proof: Let s be the centre of the star K 1,n .We do the following series of contractions to reduce G × K 1,n to G × K 2 .In each phase we choose a pair of vertices i, j ∈ V (K 1,n )\{s} and then one by one we contract pairs (u, i) and (u, j) into a single vertex (u, k) until there are no vertices (u, i) or (u, j) for any u ∈ V (G) remaining.We then start the next phase, continuing like this until there are is only one vertex i = s remaining.At this point we are left with a trigraph over G × K 2 .We claim that during this sequence of contractions no red edges are created, the final trigraph has no red edges.
To see why this holds observe that by the definition of the tensor product, for any vertex (u, i) we have N (u, i) = {(v, s) : uv ∈ E(G)} as K 1,n is a star.It follows that for any pair i, j = s of distinct leaves in S n and any u ∈ V (G) we have N (u, i) = N (u, j).Thus the pair (u, i), (u, j) are twins and can be contracted without creating red edges.Finally the inequality tww(G × K 2 ) ≤ tww(G) + 2 holds by Theorem 3.4.

Strong Product
The strong product The following bound on the twin-width of a strong product is proved in Bonnet et al. (2021a).
Theorem 3.8 ((Bonnet et al., 2021a, Theorem 2.7)).Let G and H be two graphs.Then Unlike for tensor products this bound is not tight for strong products of complete graphs as K n ⊠K m ∼ = K n•m , and thus tww(K n ⊠K m ) = 0, for any n, m ≥ 1. However the following result shows that Theorem 3.8 is tight for the strong product of two hypercubes.
and for the final case, where w.l.o.g.we can assume c = 2 and d ≥ 3 by symmetry, we have this completes the proof of the upper bound.
We now consider the lower bound which will follow by Lemma 2.2.Let N G (v) = N G (v) ∪ {v} be the closed neighbourhood of a vertex v ∈ V (G).It is known (Hammack et al., 2011, Exercise 4.6), that for any graphs G, H and vertices u, v ∈ V (G) and i, j ∈ V (H) we have Thus, we have Observe also that for any (u, i), (v, j) ∈ V (G ⊠ H) we have (2) Let B((u, i), (v, j)) = (N S(c,d) ((u, i))△N S(c,d) ((v, j)))\{(u, i), (v, j)} and for any integer k ≥ 1 and denote N k (u) = N H(k,2) (u) for ease of reading.Then, by ( 1) and ( 2), for any (u, i), (v, j) we have By Lemma 2.2, the twin-width of S(c, d) is lower-bounded by the smallest value of |B((u, i), (v, j))| over all these cases.We now establish a very simple claim.
Claim.For any d ≥ 2 and u, v ∈ V (H(d, 2)) where u = v, we have Proof of Claim: It will help to consider H(d, 2) as follows: we identify each vertex as a string in {0, 1} d where two vertices are connected if and only if their strings differ in one exactly coordinate.We have three cases: Case 1 [uv ∈ E(H(d, 2))]: Recall that the hypercube H(d, 2) is bipartite, thus u and v cannot have any common neighbours.Hence, It follows that u and v differ at exactly two coordinates i and j where i = j.
) is adjacent to both u and v it must be equal to u at all but one coordinate and equal to v at all but one coordinate, so Having established the claim we can now check, using (3), that for the following three cases, which cover all possible pairs of vertices (u, i) and (v, j).
Case 1 [u = v, i = j]: Observe that in this case, by the Claim above, we have ) and so by ( 3) This establishes the lower bound by Lemma 2.2.

Lexicographic Product
The lexicographic product of G and H, written We note that the following theorem is a corollary of a more general result (Bonnet et al., 2022d, Lemma 9) which determines the twin-width of a modular partition.However, we give an alternative proof of the result here for completeness.
Theorem 3.10 ( (Bonnet et al., 2022d, Lemma 9)).For any graphs G, H we have Proof: First, note that as there are induced subgraphs of G • H that are isomorphic to both G and H, we get that tww (G • H) ≥ max{tww(G), tww(H)} from Proposition 2.3.It remains to prove that max{tww(G), tww(H)} is an upper bound, which we show by giving a contraction sequence for G • H. Let H m , . . ., H 1 be a tww(H)-sequence for H, and label the vertices of G with [n] in an arbitrary order.For a fixed u ∈ [n], we call the u-th copy of H the subgraph of G • H induced by the vertex set {(u, i) : i ∈ V (H)}.We contract G • H to G • K 1 by contracting each copy of H in turn by following the sequence H m , . . ., H 1 .That is, we first contract the first copy of H to K 1 by following H m , . . ., H 1 and then start with the second copy of H.
For any pair (u, i) and Thus, for any two vertices of the form (u, i) and (u, j) (i.e., two vertices in the same copy of H), if there is some (v, k) is in the same copy of H as (u, i) and (u, j)).This means that as we contract each copy of H to K 1 , any red edges must be in the copy of H, and so no vertex ever has a red degree greater than tww(H).Lastly, we are left with G • K 1 , which is isomorphic to G, with no red edges, and so we can contract G according to a tww(G)-sequence, and it follows that the maximum red degree of the whole sequence, and an upper bound on tww(G • H), is given by max{tww(G), tww(H)}.
Remark 3.11.Since cographs are precisely the graphs of twin-width zero, this result gives another proof of the fact that the class of cographs is closed under lexicographic products.

Modular Product
The modular product of G and H, written G ⋄ H, has vertex set V (G) × V (H) and for u, v ∈ V (G), where u = v and i, j ∈ V (H) where i = j, we have Recall that P n denotes a path on n vertices.
Proof: We prove this by showing that for any pair of distinct vertices in P n ⋄ P n , contracting the pair gives a trigraph with red degree at least n + 1.
We will work with P n ⋄ P ′ n , where P n and P ′ n are both paths on n vertices.The extra prime marker serves to distinguish the two graphs and makes the proof easier to follow.Label the vertices of each of P n and P ′ n with the elements of [n] such that, for i ∈ [n − 1], vertices i and i + 1 are adjacent.Let (u, i) and (v, j) be the first pair of vertices contracted by any contraction sequence.Without loss of generality, we can say that u ≤ n 2 and u < v.We note that the case u = v is, by symmetry of P n and P ′ n , absorbed by Case 1 below.
Case 1 [i = j]: As (u, i) and (v, j) must be distinct to be contracted, it must hold that u = v, and so there must exist some vertex, say w, in P n such that u = w, uw ∈ E(P n ) but vw ∈ E(P n ).For each k ∈ V (P ′ n ) \ {i}, (u, i) is adjacent to (w, k) in P n ⋄ P ′ n if and only if (v, j) is not adjacent to (w, k), adding n − 1 red edges incident with the new contracted vertex.
Additionally, let ℓ be a neighbour of i in , leading to two more red edges incident with the new contracted vertex.Alternatively, if uv ∈ E(P n ), then for each k ∈ V (P ′ n ) that is not a neighbour of i, and satisfying k = i, the edge (u, i)(v, k) is in For n ≥ 5, and for any choice of i, there are at least two choices for k leading to at least two more red edges incident with the new contracted vertex.In either case, we have at least n + 1 red edges adjacent to the new vertex.
Case 2 [i = j]: Again there must exist a vertex, say w, in P n such that u = w, uw ∈ E(P n ) but vw ∈ E(P n ).For each k ∈ V (P ′ n ) \ {i, j}, (u, i) is adjacent to (w, k) in P n ⋄ P ′ n if and only if (v, j) is not adjacent to (w, k), adding n − 2 red edges incident with the new contracted vertex.
Similarly, there must exist a vertex, say k, in n if and only if (v, j) is not adjacent to (x, k), adding a further n − 2 red edges incident with the new contracted vertex.
For n ≥ 5, such a contraction must result in at least 2n − 4 ≥ n + 1 red edges incident with the new vertex, concluding the result.A computational search using a SAT encoding for twin-width Schidler and Szeider (2022) has shown that tww(P 6 ⋄ P 6 ) = 9, proving that this bound is not tight.However, since for any n ≥ 1 the path P n has twin-width at most one and degree at most two, we obtain the following corollary to Theorem 3.12.by a constant or removed entirely.If G = K 3 and H = K 1 then tww(G) = 0, tww(H) = 0, and tww(G H) = 2, so the 2 is necessary in the maximum.If G is the Paley graph P (9) and H = K 1 then tww(G) = 4, tww(H) = 0, and tww(G H) = 5, so tww(G) + 1 is necessary in the maximum.The l-corona product, a repeated product defined as G 1 H = G H and for an integer ℓ ≥ 2, G ℓ H = (G ℓ−1 H) H was recently introduced Furmańczyk and Zuazua (2022).For this repeated product, we derive the following from repeated application of Theorem 3.14.
Corollary 3.15.For any graphs G and H and integer ℓ ≥ 1 we have

Rooted Product
The rooted product of a graph G and a rooted graph H with root r, written G ♯ H, has vertex set Theorem 3.16.For a graph G and a rooted graph H with root vertex r, where H ′ is the induced subgraph of H on vertex set V (H) \ {r} and d H (r) is the degree of vertex r in H.
Proof: We show this by giving a contraction sequence that obtains this bound.For each v ∈ V (G), let the v'th copy of H be the subgraph of G ♯ H induced by vertices of the form {(v, i) : i ∈ V (H)}, and let the v'th copy of H ′ be the induced subgraph of G ♯ H on vertices of the form {(v, i) | i ∈ V (H) and i = r}.
We first contract each copy of H ′ to K 1 , then follow a tww(G)-sequence of G interspersed with contractions of the leaf vertices resulting from contractions of the copies of H ′ .
Note that any edge in G ♯ H that leaves the v'th copy of H ′ must be incident with (v, r), so as we contract each copy of H ′ the maximum red degree of any vertex within the copy of H ′ must be at most tww(H ′ ) + 1.At the same time, the maximum red degree of (v, r) is at most d H (r), as no vertex of the form (u, r), for any u ∈ V (G), is involved in a contraction.
After these contractions, we are left with G ♯ P 2 where each edge of the form (v, i)(v, j) is potentially red.Let G n , . . ., G 1 be a tww(G)-sequence for G, and let {v n−1 , u n−1 }, . . ., {v 1 , u 1 } be a sequence of pairs of vertices such that for k ∈ in the same copy of H.For k ∈ [n − 1] decreasing, first contract the vertices (u k , i) and (v k , i), and then contract the vertices (v k , r) and (u k , r).Contracting (u k , i) and (v k , i) will create a vertex of degree at most 2 (and hence red degree at most 2), and the subsequent contraction of (u k , r) and (v k , r) will have red degree at most tww(G) + 1.
Since most bounds on tww(G ⋆ H) in this paper, for some product ⋆, depend on both tww(G) and tww(H) it might appear odd that the bound above does not depend on tww(H).However, recall that |V (H)| = ∆(G) and so we have tww(H) ≤ ∆(G) − 1.The following lower bound is almost immediate.Proof: Each cloud in G r H induces a copy of H, so the bound follows from Proposition 2.3.We now give an example which shows Theorem 4.2 is tight up-to a constant factor.Let F q be the field with q elements.Then for q = 1 mod 4, the Paley graph P(q) has vertex set F q and edge set (a, b) : a − b = c 2 where c ∈ F q .The finite field F q contains (q − 1)/2 distinct squares, thus P(q) is q−1 2 -regular.It is known by (Ahn et al., 2022, Theorem 1.4.) that tww(P(q)) = (q − 1)/2.For the construction, take K q+1 r P (q) where q satisfies q = 1 mod 4, K q+1 is a clique on q + 1 vertices, and P (q) is the Paley graph on q vertices.Since tww(P(q)) = (q − 1)/2 by (Ahn et al., 2022, Theorem 1.4.),Lemma 4.3 shows that Theorem 4.2 is tight up-to a constant factor (just slightly bigger than 2) for the graph K q+1 r P (q).
We now prove a less trivial lower bound.An interesting consequence of this bound is that the twinwidth of the replacement product of any two (non-trivial) graphs is always at least two.We now consider the first contraction in a tww(G r H)-sequence for G r H where two vertices from different clouds are contracted.Let this be the τ -th contraction and observe that such a time must exist as |V (G)| > 1.Up to this point the only contractions that could have occurred are between pairs of vertices in the same cloud.It follows that if there are any contractions before time τ then the vertices remaining have a red degree which is at least the number of vertices that have been contracted into them (since each vertex in a cloud leads to a distinct vertex in another cloud).Let x, y be the two vertices, where x ∈ C u and y ∈ C v and u = v, to be contracted at time τ .After this contraction all edges from each of x and y to vertices within their clouds will be red.Observe that if the maximum red degree of a vertex in x ′ s cloud at time τ is i, then x is adjacent to at least ⌈∆(H)/i⌉ vertices in its own cloud C u at time τ .Similarly, if the maximum red degree in C v at time τ is j then y is adjacent to at least ⌈∆(H)/j⌉ vertices in C v at time τ .When we contract x and y to w the red degree of w is at least ⌈∆(H)/i⌉ + ⌈∆(H)/j⌉.
We thus obtain the bound tww(G r H) ≥ 2 • ∆(H) by minimising over all i and j.
Theorem 4.4.For any regular graph G and any ∆(G)-vertex regular graph H we havetww(G r H) ≥ 2 • ∆(H) .Proof: Let C v = {(v, i) : i ∈ [∆(G)]} be the cloud of ∆(G) vertices at the vertex v ∈ V (G).Observe that if the cloud is contracted to a single vertex w before there are any contractions between two vertices x, y ∈ N (C v ) \ C v then ∆(G) ≥ ∆(H) red edges must be adjacent to w.