Improved product structure for graphs on surfaces

Dujmovi\'c, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every graph $G$ with Euler genus $g$ there is a graph $H$ with treewidth at most 4 and a path $P$ such that $G\subseteq H \boxtimes P \boxtimes K_{\max\{2g,3\}}$. We improve this result by replacing"4"by"3"and with $H$ planar. We in fact prove a more general result in terms of so-called framed graphs. This implies that every $(g,d)$-map graph is contained in $ H \boxtimes P\boxtimes K_\ell$, for some planar graph $H$ with treewidth $3$, where $\ell=\max\{2g\lfloor \frac{d}{2} \rfloor,d+3\lfloor\frac{d}{2}\rfloor-3\}$. It also implies that every $(g,1)$-planar graph (that is, graphs that can be drawn in a surface of Euler genus $g$ with at most one crossing per edge) is contained in $H\boxtimes P\boxtimes K_{\max\{4g,7\}}$, for some planar graph $H$ with treewidth $3$.

The motivation for this work is the following question: what is the global structure for graphs embeddable in a fixed surface?Dujmović et al. (2020b) answered this question for planar graphs (i) in terms of products (ii)  of graphs of bounded treewidth (iii) (iv) .
Theorem 1 ( (Dujmović et al., 2020b)).Every planar graph is contained in H P K 3 for some planar graph H with treewidth 3 and for some path P .
Theorem 2 ( (Dujmović et al., 2020b)).Every graph with Euler genus g is contained in H P K max{2g,3} for some apex graph H with treewidth 4 and for some path P .
This paper improves this bound on the treewidth of H from 4 to 3.
Theorem 3. Every graph with Euler genus g is contained in H P K max{2g,3} for some planar graph H with treewidth 3 and for some path P .
The bound on the treewidth of H in Theorem 3 is optimal since Dujmović et al. (2020b) showed that for every integer 0 there is a planar graph G such that if G is contained in H P K , then H has treewidth at least 3.
We in fact prove a more general result in terms of so-called framed graphs.Let G be a multigraph embedded in a surface Σ without crossings, where each face is bounded by a cycle.For any integer d 3, let G (d) be the multigraph embedded in Σ obtained from G as follows: for each face F of G bounded by a cycle C of length at most d, for all distinct non-adjacent vertices v, w in C, add an edge vw across F to G (d) .We say that G We prove the following theorem.
Theorem 4. For all integers g 0 and d 3, every (g, d)-framed multigraph is contained in H P K for some planar graph H with treewidth 3 and for some path P , where = max{2g d 2 , d + 3 d 2 − 3}.Framed graphs (for g = 0) were introduced by Bekos et al. (2020) and are useful because they include several interesting graph classes, as shown by the following three examples.
First, every graph with Euler genus g is a subgraph of a (g, 3)-framed multigraph.Thus Theorem 4 with d = 3 implies Theorem 3. Now consider map graphs.Start with a graph G embedded in a surface Σ without crossings, with each face labelled a 'nation' or a 'lake', where each vertex of G is incident with at most d nations.Let M be the graph whose vertices are the nations of G, where two vertices are adjacent in G if the corresponding faces in G share a vertex.Then M is called a (Σ, d)-map graph.If Σ has Euler genus at most g, then M is called a (g, d)-map graph.Graphs embeddable in Σ are precisely the (Σ, 3)-map graphs (Dujmović et al., 2017).So map graphs are a natural generalisation of graphs embeddable in surfaces.
We show that every (Σ, d)-map graph is a spanning subgraph of G (d) for some multigraph G embedded in Σ without crossings; see Lemma 11.Thus Theorem 4 implies that (g, d)-map graphs have the following product structure. (v) The Euler genus of a surface with h handles and c cross-caps is 2h + c.The Euler genus of a graph G is the minimum integer g 0 such that there is an embedding of G in a surface of Euler genus g; see Mohar and Thomassen (2001) for more about graph embeddings in surfaces.A triangulation of a surface Σ is a graph embedded in Σ with no crossings, such that every face is a triangle.
Theorem 5. Every (g, d)-map graph is contained in H P K for some planar graph H with treewidth 3 and for some path P , where = max{2g d 2 , d + 3 d 2 − 3}.A graph is k-planar if it has an embedding in the plane where each edge is involved in at most k crossings.This definition has a natural extension for other surfaces Σ.A graph is (Σ, k)-planar if it has an embedding in Σ where each edge is involved in at most k crossings.A graph is (g, k)-planar if it is (Σ, k)-planar for some surface Σ with Euler genus at most g.In the planar setting (g = 0), these graphs have been extensively studied; see Kobourov et al. (2017); Didimo et al. (2019) for surveys.

Proofs
Undefined terms and notation can be found in Diestel's text (Diestel, 2018).A partition of a graph G is a set P of non-empty sets of vertices in G such that each vertex of G is in exactly one element of P. Each element of P is called a part.The quotient of P is the graph, denoted by G/ P, with vertex set P where distinct parts A, B ∈ P are adjacent in G/ P if and only if some vertex in A is adjacent in G to some vertex in B. An H-partition of G is a partition P = (A x : x ∈ V (H)) where H ∼ = G/ P. For simplicity, we sometimes abuse notation and say J ∈ P where J is a subgraph of G with V (J) ∈ P.
If T is a tree rooted at a vertex r, then a non-empty path P in T is vertical if the vertex of P closest to r in T is an end-vertex of P .If T is a rooted spanning tree in a graph G, then a tripod in G (with respect to T ) consists of up to three pairwise vertex-disjoint vertical paths in T whose lower end-vertices form a clique in G.
A layering of a graph G is an ordered partition L := (L 0 , L 1 , . . . ) of V (G) such that for every edge vw ∈ E(G), if v ∈ L i and w ∈ L j , then |i − j| 1.A layered partition (P, L) of a graph G consists of a partition P and a layering L of G. Observation 7 (Dujmović et al. (2020b)).For all graphs G and H, G is contained in H P K for some path P if and only if G has a layered H-partition (P, L) with width at most .
We need the following lemma of Dujmović et al. (2019), which is a special case of their Lemma 24 (which is an extension of Lemma 17 from (Dujmović et al., 2020b)).
Lemma 8 ( (Dujmović et al., 2019)).Let G + be a plane multigraph in which each face of G + is bounded by a cycle with length in {3, . . ., d}.Let T be a spanning tree of G + rooted at some vertex r on the boundary of the outer-face of G + .Assume there is a vertical path P in T with end-vertices p 1 and p 2 such that the cycle C obtained from P by adding the edge p 1 p 2 is a subgraph of G + − r.Let G be the plane graph consisting of all the vertices and edges of G + contained in C and the interior of C. Then G (d) has an H-partition P such that P ∈ P and each part S i ∈ P \{P } has a partition {X i , Y i } where |X i | d − 3 and Y i is the union of at most three vertical paths in T , and H is planar with treewidth at most 3.
The next lemma is the heart of our proof.
Lemma 9. Let G be a connected multigraph embedded in a surface of Euler genus g without crossings, where each face of G is bounded by a cycle.Then for every spanning tree T of G and every integer d 3, G (d) has an H-partition P such that one part Z ∈ P is the union of at most 2g vertical paths in T and each part S i ∈ P \{Z} has a partition {X i , Y i } where |X i | d − 3 and Y i is the union of at most three vertical paths in T , and H is planar with treewidth at most 3.
Proof: We start by following the proof of (Dujmović et al., 2020b, Lemma 21), which is the heart of the proof of Theorem 2. Near the end of our proof we follow a different strategy to obtain the stronger result.
If g = 0, then the claim follows from Lemma 8 by considering an appropriate supergraph G + of G. Now assume that g 1. Say G has n vertices, m edges, and f faces.By Euler's formula, n − m + f = 2 − g.Let D be the multigraph with vertex-set the set of faces in G, where for each edge e of E(G) \ E(T ), if f 1 and f 2 are the faces of G with e on their boundary, then there is an edge joining f 1 and f 2 in D. (Think of D as the spanning subgraph of the dual graph consisting of those edges that do not cross edges in T .)(Dujmović et al., 2017, Lemma 11) for a proof.Let T * be a spanning tree of D.
. ., a g b g } be the set of edges in G dual to the edges in E(D) \ E(T * ).Let r be the root of T , and for i ∈ {1, 2, . . ., g}, let Z i be the union of the a i r-path and the b i r-path in T , plus the edge a i b i .Let Z := Z 1 ∪ Z 2 ∪ • • • ∪ Z g .By construction, Z is a connected subgraph of G; see Figure 1 for an example.In fact, since r is contained in each of the 2g vertical paths, T [V (Z)] is connected.Say Z has p vertices and q edges.Since Z consists of a subtree of T plus the g edges in Q, we have q = p − 1 + g.
We now describe how to 'cut' along the edges of Z to obtain a new embedded graph G; see Figure 2. First, each edge e of Z is replaced by two edges e and e in G.Each vertex of G that is not contained in V (Z) is untouched.Consider a vertex v ∈ V (Z) incident with edges e 1 , e 2 , . . ., e d in Z in clockwise order.In G replace v by new vertices v 1 , v 2 , . . ., v d , where v i is incident with e i , e i+1 and all the edges incident with v clockwise from e i to e i+1 (exclusive).Here e d+1 means e 1 and e d+1 means e 1 .This operation defines a cyclic ordering of the edges in G incident with each vertex (where e i+1 is followed by e i in the cyclic order at v i ).This in turn defines an embedding of G in some orientable surface (vi) .Let Z be the set of vertices introduced in G by cutting through vertices in Z.
We now show that G is connected.Consider vertices x 1 and x 2 of G. Select faces f 1 and f 2 of G respectively incident to x 1 and x 2 that are also faces of G. Let P be a path joining f 1 and f 2 in the dual tree T * .Then the edges of G dual to the edges in P were not split in the construction of G. Therefore an x 1 x 2 -walk in G can be obtained by following the boundaries of the faces corresponding to vertices in P .Hence G is connected. (vi) If G is embedded in a non-orientable surface, then the edge signatures for G are ignored in the embedding of G.
and m = m + q = m + p − 1 + g.Each face of G is preserved in G. Say s new faces are created by the cutting.Thus Since g 0, we have s 1.Since g 1, by construction, s 1.Thus s = 1 and g = 0. Hence G is plane and all the vertices in Z are on the boundary of a single face, F , of G.Moreover, the boundary of F is a cycle C F and V (C F ) = Z .Consider F to be the outer-face of G.
Now construct a supergraph G + of G by adding a vertex r + in F and edges from r + to each vertex in Z .Then G + is a plane multigraph where each face of G + is bounded by a cycle.
We now depart from the proof of Dujmović et al. (2020b, Lemma 21).Let P + be an arbitrary path such that V (P + ) = V (C F ) and let v + ∈ V (P + ) be an end-vertex of P + .Let T + be the following spanning tree of G + rooted at r + .Initialise T + to be the path P + plus the edge r Now every vertical path in T + contained in V (G) \ V (Z) corresponds to a vertical path in T .Every maximal vertical path in T + consists of the edge r + v + , a subpath of P + , some edge v i w (where w ∈ V (G) \ V (Z)), followed by a path in T − V (Z) from w to a leaf in T .Since every vertical path P in T + is contained in some maximal vertical path in T + , it follows that P ∩ (V (G) \ V (Z)) is a vertical path in T .Thus every vertical path in Triangulate every face in G + whose facial cycle has length greater than d.Since r + is on the boundary of the outer-face of G + , V (P + ) = V (C F ), every facial cycle has length in {3, . . ., d} and P + is a vertical Theorem 4 is an immediate consequence of Observation 7 and the next lemma.
Lemma 10.Let G be a multigraph embedded in a surface of Euler genus g without crossings, where each face is bounded by a cycle.Then G (d) has a layered H-partition (P, L) with width at most max{2g d 2 , d+ 3 d 2 − 3}, such that H is planar with treewidth at most 3.
Proof: Since each face of G is bounded by a cycle, G is connected.Let T be a BFS-spanning tree of G with corresponding BFS-layering (vii)  d) has an H-partition P such that one part Z ∈ P is the union of at most 2g vertical paths in T and each part S i ∈ P \{Z} has a partition {X i , Y i } where |X i | d − 3 and Y i is the union of at most three vertical paths in T , and H is planar with treewidth at most 3.It remains to adjust the layering of G to obtain a layering of G d) with width at most max{2g d 2 , d + 3 d 2 − 3}, as required.We conclude by showing that (Σ, d)-map graphs and (Σ, 1)-planar graphs are contained in framed graphs. (vii) If G is a connected graph and T is a spanning tree of G rooted at vertex r, then An analogous proof works for arbitrary surfaces, which we include for completeness.Together with Theorem 4, this implies Theorem 6.
Lemma 12. Every (Σ, 1)-planar graph G with at least three vertices is contained in G (4) 0 for some multigraph G 0 embedded in Σ with no crossings where each face of G 0 is bounded by a cycle.
Proof: We may assume that G is embedded in Σ with at most one crossing on each edge, such that no two edges of G incident to a common vertex cross, since such a crossing can be removed by a local modification to obtain an embedding of G in which the two edges do not cross.
Initialise G := G. Add edges to G to obtain an edge-maximal multigraph embedded in Σ such that each edge is in at most one crossing, no two edges incident to a common vertex cross, and no face is bounded by two parallel edges.The final condition ensures that G is well-defined, since it follows from Euler's formula that if G has k crossings, then |E(G )| 3(|V (G)| + k + g − 2) − 2k.
Consider crossing edges e 1 = vw and e 2 = xy in G .So v, w, x, y are distinct.Since e 1 is the only edge that crosses e 2 and e 2 is the only edge that crosses e 1 , by the edge-maximality of G , there is a cycle C = (v, x, w, y) in G that bounds a disc whose interior intersects no edge of G except e 1 and e 2 .
Let G 0 be the embedded multigraph obtained from G by deleting each pair of crossing edges.Thus the above-defined cycle C bounds a face of G 0 .By the edge-maximality of G , every other face of G 0 (that is, not arising from a pair of deleted crossing edges) is a triangular face of G .Thus, G 0 is a multigraph embedded in Σ with no crossings, such that each face of G 0 is bounded by a 3-cycle or a 4-cycle, and G is contained in G (4) 0 .
Layered partitions were introduced byDujmović et al. (2020b) who observed the following connection to strong products (which follows directly from the definitions).

Fig. 1 :
Fig. 1: Example of the construction in the proof of Lemma 9, where brown edges are in T , red edges are in Q, and blue edges are in T and in Z − E(Q).Say G has n vertices and m edges, and the embedding of G has f faces and Euler genus g .Each vertex with degree d in Z is replaced by d vertices in G.Each edge in Z is replaced by two edges in G, while each edge of E(G) − E(Z) is maintained in G. Thus

Fig. 2 :
Fig. 2: Cutting the blue edges in Z at each vertex.

Fig. 3 :
Fig.3: Example of the spanning tree T + in the graph G + , where the edges in E(P + ) ∪ {r + v + } are red and the edges that are either in E(T − V (Z)) or of the form viw are orange.path of T + , Lemma 8 is applicable.Let P be the H-partition of G(d) given by Lemma 8. Therefore, H is planar with treewidth at most 3, where P + ∈ P and each part in S i ∈ P \ {P + } has a partition {X i , Y i } where |X i | d − 3 and Y i is the union of at most three vertical paths in T .Let P be the partition of G(d)  obtained by replacing P + by Z. Since V (P + ) = V (Z ) and all the split vertices of G are in Z, we have G(d) /P ∼ = G(d) /P ∼ = H.Hence P is also an H-partition where H is planar with treewidth at most 3.In addition, since each vertical path in T + that is disjoint from V (Z ) ∪ {r + } is a vertical path in T , each part S i ∈ P \{Z} has a partition {X i , Y i } where |X i | d − 3 and Y i is the union of at most three vertical paths in T , as required.