BBook Embeddings of Graph Products
Sergey [email protected]
Abstract A k -stack layout (also called a k -page book embedding) of a graph consists of atotal order of the vertices, and a partition of the edges into k sets of non-crossing edgeswith respect to the vertex order. The stack number (book thickness, page number) ofa graph is the minimum k such that it admits a k -stack layout. A k -queue layout isdefined similarly, except that no two edges in a single set may be nested.It was recently proved that graphs of various non-minor-closed classes are sub-graphs of the strong product of a path and a graph with bounded treewidth. Motivatedby this decomposition result, we explore stack layouts of graph products. We showthat the stack number is bounded for the strong product of a path and (i) a graphof bounded pathwidth or (ii) a bipartite graph of bounded treewidth and boundeddegree. The results are obtained via a novel concept of simultaneous stack-queuelayouts, which may be of independent interest. Embedding graphs in books is a fundamental problem in graph theory, which has beenthe subject of intense research since their introduction in 70s by Ollmann [25]. A bookembedding (also known as a stack layout ) of a graph G = ( V, E ) consists of a total order, σ , of V and an assignment of the edges to stacks ( pages ), such that no two edges in a singlestack cross ; that is, there are no edges ( u, v ) and ( x, y ) in a stack with u < σ x < σ v < σ y .The minimum number of pages needed for a book embedding of a graph G is called its stack number (or book thickness or page number ) and denoted by sn( G ).Book embeddings have been extensively studied for various families of graphs. In par-ticular, the graphs with stack number one are precisely the outerplanar graphs, while thegraphs with stack number at most two are the subgraphs of planar Hamiltonian graphs [6].The stack number of planar graphs is four [5], graphs of genus g have stack number O ( √ g ) [24], while for graphs of treewidth tw , it is at most tw + 1 [18]. More generally, allproper minor-closed graph families have a bounded stack number [7]. Non-minor-closedclasses of graphs have also been investigated. Bekos et al. proved that 1-planar graphshave bounded stack number [3]. Recall that a graph is k -planar if it can be drawn in theplane with at most k crossings per edge. Recently the result has been generalized to awider family of k -framed graphs that admit a drawing with a planar skeleton, whose faceshave degree at most k ≥ k -planar graphsis O (log n ) [12].We suggest to attack the problem of determining book thickness of non-planar graphsusing graph products. Formally, let A and B be graphs. A product of A and B is a graphdefined on a vertex set V ( A ) × V ( B ) = { ( v, x ) : v ∈ V ( A ) , x ∈ V ( B ) } ) . a r X i v : . [ c s . D M ] J u l a) P G (b) P × G (c) P (cid:2) G Figure 1: Examples of graph products of a path, P , and a cycle with an edge, G :(a) cartesian, (b) direct, (c) strong.A potential edge, ( v, x ) , ( u, y ) ∈ V ( A ) × V ( B ), can be classified as follows: • A -edge: v = u and ( x, y ) ∈ E ( B ), or • B -edge: x = y and ( v, u ) ∈ E ( A ), or • direct -edge: ( v, u ) ∈ E ( A ) and ( x, y ) ∈ E ( B ).The cartesian product of A and B , denoted by A B , consists of A -edges and B -edges.The direct product of A and B , denoted by A × B , consists of direct edges. The strongproduct of A and B , denoted by A (cid:2) B , consists of A -edges, B -edges, and direct-edges.Figure 1 illustrates examples of the defined graph products. Notice that all the productsare symmetric. In this paper, we study stack layouts of strong products of a path and abounded-treewidth graph (refer to Section 2 for a definition), focusing primarily on thefollowing question: Open Problem 1.
Is stack number of P n (cid:2) G , where P n is a path and G is a graph oftreewidth tw ≥ , bounded by f ( tw ) for some function f ? Our motivation for studying stack layouts of graph products comes from a recentdevelopment of decomposition theorems for planar and beyond-planar graphs [13, 15, 27].Dujmovi´c, Morin, and Wood [15] recently show the following:
Lemma 1 ([15]) . Every k -planar graph is a subgraph of the strong product of a path anda graph of treewidth O ( k ) . Notice that Lemma 1 together with an affirmative answer to Open Problem 1 wouldprovide a constant stack number for all k -planar graphs, thus resolving a long-standingopen problem listed in a recent survey on graph drawing of beyond-planar graphs [11].Furthermore, a similar decomposition exists for other classes of non-minor-closed familiesof graphs, such as map graphs, string graphs, graph powers, and nearest neighbor graphs,whose stack number is not known to be bounded by a constant; refer to [15] for exactdefinitions. Interestingly, a negative answer to Open Problem 1 would resolve anotherquestion in the context of queue layouts that remains unsolved for more than thirty years.A queue layout is a “dual” concept of a stack layout. For a graph G = ( V, E ), itconsists of a total order, σ , of V and an assignment of the edges to queues , such thatno two edges in a single queue nest ; that is, there are no edges ( u, v ) and ( x, y ) in aqueue with u < σ x < σ y < σ v . The minimum number of queues needed in a queuelayout of a graph is called its queue number and denoted by qn( G ) [20]. As with stack2
05 61 27 8 43 9a b c d ef g h i j (a) P (cid:2) P (b) 4-stack layout Figure 2: The strong product of P (cid:2) P and its 4-stack layout. The layout process iseasily extendable for P n (cid:2) P m with arbitrary values of n and m .layouts, the queue number is known to be bounded for many classes of graphs, includingplanar graphs [13], graphs with bounded treewidth [14, 30], and all proper minor-closedclasses of graphs [13, 15]. Queue layouts have been introduced by Heath, Leighton, andRosenberg [19,20], who tried to measure the power of stacks and queues to represent a givengraph. Despite a wealth of research on the topic, a fundamental question of what is more“powerful” remains unanswered. That is, Heath et al. [19] ask whether the stack numberof a graph is bounded by a function of its queue number, and whether the queue number ofa graph is bounded by a function of its stack number. In a study of queue layouts of graphproducts, Wood [31] shows that for a path P n and all graphs G , qn( P n (cid:2) G ) ≤ G ) + 1.This result together with a negative answer to Open Problem 1 would provide an exampleof a graph (namely, the strong product of a path and a bounded-treewidth graph) that hasa constant queue number but an unbounded stack number; thus, resolving one directionof the question posed by Heath et al. [19]. Results and Organization
In this paper we introduce and initiate an investigation of Open Problem 1. Our con-tribution is twofold. Firstly, we resolve the problem in affirmative for two subclasses ofbounded-treewidth graphs. Secondly, we provide an evidence that the most “natural”approach cannot lead to a positive answer of the problem.
Positive Results.
It is easy to verify that the stack number of P n (cid:2) G is bounded bya constant when G is a “simple” graph such as a path, a star, or a cycle. Notice that thestrong graph product consists of n copies of G , which are connected by inter-copy edges.A natural approach is to layout each copy independently using a constant number of stacksand then join individual results into a final stack layout. In order to be able to embedinter-copy edges in a few stacks, one has to alternate direct and reverse vertex orders forthe copies of G ; refer to Figure 2 for the process of embedding P n (cid:2) P m in four stacks.The above technique can be generalized using the concept of simultaneous stack-queuelayouts. Let σ be a total order of V for a graph G = ( V, E ). A simultaneous s -stack q -queue layout consists of σ together with (i) a partition of E into s stacks with respectto σ , and (ii) a partition of E into q queues with respect to σ . In such a layout every edgeof G is associated with a stack and with a queue. We stress the difference with so-called mixed layouts in which an edge belongs to a stack or to a queue [28].3n order to state the first main result of the paper, we use dispersable book embeddingsin which the graphs induced by the edges of each page are 1-regular; see Figure 7a. Theminimum number of pages needed in a dispersable book embedding of G is called its dispersable stack number , denoted dsn( G ); it is also known as matching book thickness [1,6]. Theorem 1.
Let H be a bipartite graph and G be a graph that admits a simultaneous s -stack q -queue layout. Then(i) sn( H G ) ≤ s + dsn( H ) ,(ii) sn( H × G ) ≤ q · dsn( H ) ,(iii) sn( H (cid:2) G ) ≤ q · dsn( H ) + s + dsn( H ) . What graphs admit simultaneous layouts for constant s and q ? We prove that graphsof bounded pathwidth (see Section 2 for a definition) have such a layout. Although it isknown that both the stack number and the queue number of pathwidth- p graphs is atmost p [14, 29], Lemma 2 (in Section 3) shows that the bounds can be achieved using acommon vertex order. As a direct corollary of the lemma, Theorem 1, and an observationthat dsn( P n ) = 2, we get the following result. Corollary 1.
Let G be a graph of pathwidth p . Then sn( P n (cid:2) G ) ≤ p + 2 . Notice that Corollary 1 combined with Lemma 1 implies an alternative proof of the O (log n ) upper bound for the stack number of k -planar graphs, since for every graph G ,pw( G ) ∈ O (tw( G ) · log n ) [8].Another corollary of Theorem 1 affirmatively resolves Open Problem 1 for the strongproduct of a path and a bounded-treewidth bipartite graph of bounded maximum vertexdegree. For that case we bound the dispersable stack number of a bipartite graph by afunction of its treewidth and the maximum vertex degree; see Lemma 3 in Section 3. Corollary 2.
Let G be a bipartite graph of treewidth tw with maximum vertex degree ∆ .Then sn( P n (cid:2) G ) ≤ tw + 1)∆ + 1 . Negative Results.
Next we investigate simultaneous stack-queue layouts. We provethat if a graph admits a simultaneous s -stack q -queue layout, then its pathwidth is boundedby a function of s and q . In other words, the class of O (1)-pathwidth graphs coincideswith the class of graphs admitting a simultaneous O (1)-stack O (1)-queue layout. Theorem 2.
Let G be a graph admitting a simultaneous s -stack q -queue layout. Then G has pathwidth at most · s · q . Corollaries 1 and 2 provide sufficient conditions for a graph G to imply a boundedstack number of P n (cid:2) G . Yet many relatively simple graphs of bounded treewidth (such astrees) have pathwidth Ω(log n ) and an unbounded vertex degree. A reasonable question iswhether the conditions are necessary. Next we study the aforementioned natural approach to construct stack layouts of graph products, and prove that it cannot lead to a constantnumber of stacks for graphs with an unbounded pathwidth. Formally, call a stack layoutof P n (cid:2) G separated if for at least two consecutive copies of G , G and G , all vertices of G precede all vertices of G in the vertex order. The next result shows that a separatedlayout of P n (cid:2) G with a constant number of stacks implies a bounded pathwidth of G . Theorem 3.
Assume P n (cid:2) G has a separated layout on s stacks. Then G admits asimultaneous s -stack s -queue layout, and therefore, pw( G ) ≤ s . Very recently, Dujmovi´c, Morin, and Yelle [16] independently proved a result asymptotically equivalentto Corollary 1; see Section 4 for a discussion.
Throughout the paper, G = (cid:0) V ( G ) , E ( G ) (cid:1) is a simple undirected graph. We denote apath with n vertices by P n . A vertex order , σ , of a graph G is a total order of its vertexset V ( G ), such that for any two vertices u and v , u < σ v if and only if u precedes v in σ . Let F be a set of k ≥ s i , t i ) , ≤ i ≤ k . If s < σ · · · < σ s k < σ t k < σ · · · < σ t , then F is a k -rainbow , while if s < σ · · · < σ s k < σ t < σ · · · < σ t k , then F is a k -twist . Two independent edges forminga 2-twist (2-rainbow) are called crossing ( nested ).A k -stack layout of a graph is a pair ( σ, {S , . . . , S k } ), where σ is a vertex order and {S , . . . , S k } is a partition of E ( G ) into stacks , that is, sets of pairwise non-crossing edges.Similarly, a k -queue layout is ( σ, {Q , . . . , Q k } ), where {Q , . . . , Q k } is a partition of E ( G )into sets of pairwise non-nested edges called queues . The minimum number of stacks(queues) in a stack (queue) layout of a graph is its stack number ( queue number ). Itis easy to see that a k -stack layout ( k -queue layout) cannot have a k -twist ( k -rainbow).Furthermore, a vertex order without a ( k +1)-rainbow corresponds to a k -queue layout [20].In contrast, a vertex order without a ( k + 1)-twist may not produce a k -stack layout butcorresponds to a f ( k )-stack layout; the best-known function f is quadratic [10].A tree decomposition of a graph G is given by a tree T whose nodes index a collection (cid:0) B x ⊆ V ( G ) : x ∈ V ( T ) (cid:1) of sets of vertices in G called bags such that: • For every edge ( u, v ) of G , some bag B x contains both u and v , and • For every vertex v of G , the set { x ∈ V ( T ) : v ∈ B x } induces a non-empty connectedsubtree of T .The width of a tree-decomposition is max x | B x |−
1, and the treewidth of a graph G , denotedtw( G ), is the minimum width of any tree decomposition of G .A path decomposition is a tree decomposition in which the underlying tree, T , is apath. Thus, it can be thought of as a sequence of subsets of vertices, called bags , such thateach vertex belongs to a contiguous subsequence of bags and each two adjacent verticeshave at least one bag in common. The pathwidth of a graph G , denoted pw( G ), is theminimum width of any path decomposition of G . We also use an equivalent definition ofthe pathwidth called the vertex separation number [8, 22]. Consider a vertex order σ of agraph G . The vertex cut in σ at a vertex v ∈ V ( G ) is defined to be C ( v ) = { x ∈ V ( G ) : ∃ ( x, y ) ∈ E ( G ) , x < σ v ≤ σ y } . The vertex separation number of G is the minimum, takenover all vertex orders σ of G , of a maximum cardinality of a vertex cut in σ . Theorem 1.
Let H be a bipartite graph and G be a graph that admits a simultaneous s -stack q -queue layout. Then(i) sn( H G ) ≤ s + dsn( H ) ,(ii) sn( H × G ) ≤ q · dsn( H ) ,(iii) sn( H (cid:2) G ) ≤ q · dsn( H ) + s + dsn( H ) . , σ G , σ r G , σ r G , σ Figure 3: An ( s + 4 q + 2)-stack layout of the strong product of P and a graph G thatadmits a simultaneous s -stack q -queue layout using vertex order σ . G , G , G , and G correspond to copies of G laid out by alternating σ and its reverse, σ r . Groups of stacksare colored differently. Proof.
For every pair of graphs, the set of edges of the strong product is the union of edgesof the cartesian product and the direct product of the graphs. Therefore, claim (iii) of thetheorem follows from claims (i) and (ii), which we prove next.Let π be a vertex order of H in the dispersable stack layout, and let σ and σ r be avertex order of G and its reverse in the simultaneous stack-queue layout. We call the partsof the bipartition of H white and black , and denote by 0 ≤ π ( v ) < n the index of vertex v ∈ V ( H ) in σ . To construct an order, φ , for the stack layout of a graph product, we startwith π and replace each white vertex of H with σ and each black vertex of H with σ r .Formally, for two vertices ( v, x ) and ( u, y ) of a product, let φ ( v, x ) < φ ( u, y ) if and only if(C1) v (cid:54) = u and π ( v ) < π ( u ), or(C2) v = u , v is white, and x < σ y , or(C3) v = u , v is black, and y < σ x .We emphasize that the same vertex order is utilized for all three graph products; seeFigure 2 and Figure 3 for illustrations.We first verify that sn( H G ) ≤ s + dsn( H ), thus proving claim (i) of the theorem.Since σ and σ r are vertex orders of an s -stack layout of G and different copies of G areseparated in φ , all G -edges are embedded in s stacks. Further, every edge of H is incidentto a white and a black vertex of H that correspond to σ and σ r . Thus, H -edges between apair of copies of G are non-crossing and can be assigned to the same stack. Since the edgesof H require dsn( H ) stacks and each stack consists of independent edges, all H -edges areembedded in dsn( H ) stacks.Next we show that direct-edges can be assigned to 2 q · dsn( H ) stacks, which we denoteby S ji for 1 ≤ i ≤ q and 1 ≤ j ≤ H ). To this end, partition all direct-edges into2 dsn( H ) groups and employ q stacks for each of the groups. A group of a direct-edge, e , with endpoints ( v, x ) and ( u, y ) is determined by the stack of ( v, u ) ∈ E ( H ) in thedispersable layout of H and by the relative order of x and y in σ . Specifically, • if x < σ y , ( v, u ) ∈ S j , and ( x, y ) ∈ Q i , then e ∈ S ji ; • if y < σ x , ( v, u ) ∈ S j , and ( x, y ) ∈ Q i , then e ∈ S j +1 i .Here {S , . . . , S dsn( H ) } is the partition of E ( H ) in the dispersable stack layout of H , and {Q , . . . , Q q } is the partition of E ( G ) in the q -queue layout of G .Let us verify that the direct-edges in a stack are non-crossing. For the sake of contra-diction, assume two edges, e with endpoints ( v , x ) and ( u , y ), and e with endpoints( v , x ) and ( u , y ), cross each other. We assume e and e belong to a group S ji forsome 1 ≤ i ≤ q, ≤ j ≤ dsn( H ); the other case is symmetric. Since e and e cross,6 b c d e f g h i j k (a) a b c d e f g h i j k (b) Figure 4: An illustration for Lemma 2: creating (a) a 2-stack and (b) a 2-queue layout fora graph with pathwidth 2 using its vertex separation order. π ( v ) = π ( v ) and φ ( v , x ) < φ ( v , x ) < φ ( u , y ) < φ ( u , y ). By (C2), we have x < σ x , and by (C3), we have y < σ y . Hence, two edges of G from the same queue,( x , y ) and ( x , y ), form a 2-rainbow in σ ; a contradiction.Therefore, H G admits an ( s + dsn( H ))-stack layout, H × G admits a (2 q · dsn( H ))-stack layout, and H (cid:2) G admits a (2 q · dsn( H ) + s + dsn( H ))-stack layout.The bounds of Theorem 1 can be improved for certain families of graphs. For example,the stack number of the strong product of two paths is at most 4, while the theorem yieldsan upper bound of 7; see Figure 2. However, for a complete graph on 2 k vertices, K k , itholds that sn( P n (cid:2) K k ) ≥ k − K k ) = qn( K k ) = k . Hence, the given bounds are asymptotically worst-case optimal.Next we explore simultaneous linear layouts of bounded-pathwidth graphs. While itis known that the stack number and the queue number of pathwidth- p graphs is at most p [14, 29], the existing proofs do not utilize the same vertex order for the stack and queuelayouts. We show that the bounds can be achieved in a simultaneous stack-queue layout. Lemma 2.
A graph of pathwidth p has a simultaneous p -stack p -queue layout.Proof. Consider a vertex order, σ , of the given graph, G , corresponding to its vertexseparation number, which equals to the pathwidth, p [8, 22]. We prove that σ yields a p -stack layout of G and a p -queue layout of G ; see Figure 4.Assume that edges of G form a rainbow of size greater than p with respect to σ .That is, let σ be such that u < σ · · · < σ u p (cid:48) < σ v p (cid:48) < σ · · · < σ v for some p (cid:48) > p and( u i , v i ) ∈ E ( G ) for all 1 ≤ i ≤ p (cid:48) . Then the vertex cut at v p (cid:48) has cardinality at least p (cid:48) ,as u , . . . , u p (cid:48) ∈ C ( v p (cid:48) ), which contradicts that the vertex separation is p . Therefore, thequeue number of G is at most p .To construct a p -stack layout, consider the vertices of G in the order v < σ v < σ · · · < σ v n . Let E i be the set of forward edges of v i , that is, E i = { ( v i , y ) ∈ E ( G ) : v i < σ y } .We process the vertices in the order and assign edges to p stacks while maintaining thefollowing invariant for every 1 < i ≤ n : • all edges E , . . . , E i − are assigned to one of p stacks, and • all edges from E j for every 1 ≤ j ≤ i − i = 2 by assigning E to a single stack. Suppose weobtained a stack assignment for all forward edges up to E i − ; let us process E i . Assumethat E i (cid:54) = ∅ , and observe that v i ∈ C ( v i +1 ) and | C ( v i +1 ) \ { v i }| ≤ p −
1. Edges of E i can cross only already processed edges incident to a vertex from C ( v i +1 ) \ { v i } . By theassumption of our invariant, such edges utilize at most p − E i , and thus maintaining the invariant.Corollary 1 follows from Theorem 1, Lemma 2, and an observation that dsn( P n ) = 2.In order to prove Corollary 2, we need the following auxiliary lemma.7 emma 3. Let G be a bipartite graph of maximum vertex degree ∆ that admits an s -stacklayout. Then dsn( G ) ≤ s · ∆ .Proof. Edges of every stack of the s -stack layout of G form an outerplanar graph. Since G is bipartite, the edges of each stack can be partitioned into at most ∆ subgraphs, whichare 1-regular. Thus the dispersable stack number of G is at most s · ∆.Corollary 2 follows from Theorem 1, Lemma 3, and the fact that the stack number ofa graph with treewidth tw is at most tw + 1 [18]. Notice that in order to apply Theorem 1,we set G to be a given path and H to be a given bipartite bounded-treewidth graph. Thebound of Corollary 2 can be reduced for low-treewidth graphs, whose dispersable stacknumber is lower than the one given by Lemma 3 [1]. In order to prove Theorem 2, we first consider the case when s = q = 1 and prove theexistence of a path decomposition of width 2 with a certain property. Lemma 4.
Let G be an n -vertex graph admitting a simultaneous -stack -queue layoutwith respect to a vertex order σ = ( v , v , . . . , v n ) . Then G has pathwidth at most .Furthermore, the corresponding path decomposition consists of n bags B , . . . , B n suchthat | B x | ≤ and v x ∈ B x for all ≤ x ≤ n .Proof. It is tempting to approach the lemma by arguing that for a vertex order, σ , thecorresponding vertex separation number is bounded. However, a simultaneous 1-stack1-queue layout of a star graph with its center at position (cid:98) n/ (cid:99) of σ has an unboundedvertex cut. Therefore, we explicitly construct a path decomposition of G to prove theclaim. We use induction on the number of vertices in G ; the base of the induction with n = 1 clearly holds.Consider the last vertex in the vertex order, v n ∈ V ( G ), and let d ≥ v n . If d = 0 then we inductively construct a path decomposition for the first n − { v n } . Thus we may assume d > v i ∈ V ( G ) be the smallest (with respect to σ ) neighbor of v n for some 1 ≤ i < n .Since σ corresponds to a simultaneous 1-stack 1-queue layout, no edges of G cross eachother and no edges of G nest each other. Thus, every vertex x ∈ V ( G ) with v i < σ x < σ v n is either (a) adjacent to v i , or (b) adjacent to v n , or (c) adjacent to both v i and v n , or(d) an isolated vertex. Otherwise edge ( v i , v n ) crosses or nests an edge ( x, y ) for some y ∈ V ( G ) \ { v i , v n } ; see Figure 5.In order to build a desired path decomposition, we inductively apply the argument toa subgraph of G induced by the vertices v , v , . . . , v i . Assume that the resulting pathdecomposition of the subgraph consists of bags { B j : 1 ≤ j ≤ i } . We extend it to a pathdecomposition of G by appending n − i bags. Namely, if d > { v i , v i +1 } , . . . , { v i , v h − } , { v i , v h , v n } , { v h +1 , v n } , . . . , { v n − , v n } , { v n } , where v h , i < h < n is the first neighbor of v n after v i in σ . Otherwise if d = 1 then weuse bags { v i , v i +1 } , . . . , { v i , v n − } , { v i , v n } . It is straightforward to verify that the constructed path decomposition of G satisfiesthe requirements of the lemma. 8 n v i v h x Figure 5: A graph admits a simultaneous 1-stack 1-queue layout if and only if its pathwidthis at most 2, since a non-neighbor of v i and v n , x , between the two vertices creates eithera crossing or a nested edge. Theorem 2.
Let G be a graph admitting a simultaneous s -stack q -queue layout. Then G has pathwidth at most · s · q .Proof. Consider a vertex order, σ , corresponding to the simultaneous s -stack q -queuelayout. Every edge of G belongs to a stack and to a queue of the simultaneous layout.Thus all the edges can be partitioned into s · q disjoint sets, denoted E i,j ⊆ E ( G ) with1 ≤ i ≤ s, ≤ j ≤ q , such that each set induces a simultaneous 1-stack 1-queue layoutwith vertex order σ . By Lemma 4, for every (possibly disconnected) subgraph G i,j =( V ( G ) , E i,j ) of G , there exists a path decomposition of width 2 whose bags are denoted by B i,jx for x ∈ V ( G ). Define a path decomposition of G to be {∪ i,j B i,jx : x ∈ V ( G ) } , wherethe union is taken over all 1 ≤ i ≤ s, ≤ j ≤ q . Next we verify that the construction isindeed a path decomposition of G : • Every edge of G belongs to some set of the edge partition; thus, there is a bag inthe corresponding path decomposition, which contains both endpoints of the edge. • For every 1 ≤ i ≤ s, ≤ j ≤ q , a vertex x ∈ V ( G ) is in a continuous interval ofbags of the path decomposition of G i,j . By Lemma 4, the interval contains bag B i,jx ;therefore, the union of such intervals taken over all path decompositions forms acontinuous interval.Finally we notice that for all 1 ≤ i ≤ s, ≤ j ≤ q , bag B i,jx , x ∈ V ( G ) consists ofvertex x and possibly two more vertices of G . Hence, | ∪ i,j B i,jx | ≤ · s · q + 1. That is, thewidth of the constructed path decomposition is at most 2 · s · q .Observe that the bound of the above theorem may not be tight when s > q > G , pw( G ) is linear in ( s + q ).Next our goal is to prove Theorem 3. To this end, we use an observation by Erd˝osand Szekeres [17] that for all a, b ∈ N , every sequence of distinct numbers of length a · b + 1 contains a monotonically increasing subsequence of length a + 1 or a monotonicallydecreasing subsequence of length b + 1. We start with an auxiliary lemma. Lemma 5.
Let G be a graph with n vertices and n independent edges ( u i , v i ) , ≤ i ≤ n .If G admits an s -stack layout in which u < u < · · · < u n < v i for all ≤ i ≤ n , thenthere exists a subgraph of G with at least r = (cid:100) n/s (cid:101) edges such that u j < u j < · · · k -twist in the stack layout of G . Hence,9 u u u u v v v v v u (cid:48) u (cid:48) u (cid:48) u (cid:48) u (cid:48) G G Figure 6: A large rainbow in a copy G of a separated layout of P n (cid:2) G yields a large twistformed by inter-copy edges between G and G . k ≤ s . Now we can apply the result of Erd˝os and Szekeres [17] for a = s and b = (cid:100) n/s (cid:101) − s ( (cid:100) n/s (cid:101) −
1) + 1 ≤ n for all integers n, s ≥
1. Thus, the permutation contains a de-creasing subsequence of length at least b + 1 = (cid:100) n/s (cid:101) , which completes the proof.Next we prove Theorem 3. Recall that a stack layout of P n (cid:2) G is separated if for twoconsecutive copies of G , denoted G and G , all vertices of G precede all vertices of G in the vertex order. Theorem 3.
Assume P n (cid:2) G has a separated layout on s stacks. Then G admits asimultaneous s -stack s -queue layout, and therefore, pw( G ) ≤ s .Proof. Suppose that σ is a vertex order corresponding to the separated layout of P n (cid:2) G ,and G = (cid:0) V ( G ) , E ( G ) (cid:1) , G = (cid:0) V ( G ) , E ( G ) (cid:1) are two copies of G separated in σ . Thatis, u < σ v for all u ∈ V ( G ) , v ∈ V ( G ).Consider a suborder of σ induced by the vertices of V ( G ) and denote it by σ . If thelargest rainbow formed by the edges of E ( G ) with respect to σ has the size at most s ,then σ corresponds to an s -queue layout [20]. In that case, G admits a simultaneous s -stack s -queue layout using σ as the vertex order, which proves the claim of the theorem.Therefore, we may assume that the largest rainbow in G is of size k > s ; see Figure 6.Let ( u i , v i ) ∈ E ( G ) , ≤ i ≤ k be such a k -rainbow with u < σ u < σ · · · < σ u k < σ v k < σ · · · < σ v < σ v . Consider vertices u (cid:48) , . . . u (cid:48) k ∈ V ( G ) that are corresponding copies of u , . . . u k in graph G . Since the stack layout of P n (cid:2) G is separated, we have u < σ · · · < σ u k < σ u (cid:48) i for all 1 ≤ i ≤ k . Hence, we may apply Lemma 5 for a graph induced by vertices u , . . . , u k , u (cid:48) , . . . , u (cid:48) k , which are connected by k independent edges in the strong product.Therefore, we find a subset of r = (cid:100) k/s (cid:101) > s edges in the graph such that u j < σ u j < σ · · · < σ u j r < σ u (cid:48) j r < σ · · · < σ u (cid:48) j < σ u (cid:48) j , where 1 ≤ j , . . . , j r ≤ k . Finally, we observe that ( v i , u (cid:48) i ) , ≤ i ≤ k are direct-edges inthe strong product, and vertices v j r < σ v j r − < σ · · · < σ v j < σ u (cid:48) j r < σ · · · < σ u (cid:48) j < σ u (cid:48) j , form an r -twist in the s -stack layout of G . This contradicts to our assumption that thelargest rainbow in G is of size greater than s .The bound on the pathwidth of G follows from Theorem 2.10 a) a dispersable 2-stack layout (b) a strict 2-queue layout Figure 7: Examples of a dispersable stack layout and a strict queue layout.
Although there exists numerous works on stack and queue layouts of graphs, layoutsof graph products received much less attention. Wood [31] considers queue layouts ofvarious graph products, and shows that the queue number of a product of graphs H and G is bounded by a function of the strict queue number of H and the queue number of G .Here a queue layout with an order σ is strict if for no pair of edges, ( u, v ) and ( x, y ), itholds that u ≤ σ x < σ y ≤ σ v ; see Figure 7b. Specifically, it is shown that for all H and G , qn( H (cid:2) G ) ≤ H ) · qn( G ) + sqn( H ) + qn( G ), where sqn( H ) is the strict queuenumber of H . Similar bounds are given for the cartesian and direct products of H and G . It follows that qn( P n (cid:2) G ) ≤ G ) + 1. We stress that the result combined with adecomposition theorem for planar graphs [13, 27] (such as one given by Lemma 1) and thefact that the queue number of planar 3-trees is bounded by a constant [2], yield a constantupper bound on the queue number of planar graphs.Stack layouts of graph products have also been studied [6, 9, 21, 26], though most ofthe results are less complete as the problem is notoriously more difficult. Bernhart andKainen [6] introduce the concept of dispersable (also known as matching ) book embeddingsin which the graphs induced by the edges of each page are 1-regular; see Figure 7a. Theminimum number of pages needed in a dispersable book embedding of G is called its dispersable stack number , denoted dsn( G ); it is also known as matching book thickness [1].Clearly for every graph G of maximum vertex degree ∆, we have dsn( G ) ≥ ∆. Theauthors of [6] observed that for every path, every tree, every cycle of an even length, everycomplete bipartite graph, and every binary hypercube, it holds that dsn( G ) = ∆. Thatmade them conjecture that the equation holds for every regular bipartite graph. Theconjecture was disproved in 2018 for every ∆ ≥ G ) = ∆ forevery 3-connected 3-regular bipartite planar graph [1].Bernhart and Kainen [6] show that for a bipartite graph H and all graphs G , it holdsthat sn( H G ) ≤ dsn( H ) + sn( G ); see [26] for an alternative proof. Our Theorem 3generalizes the result. Several subsequent papers study book embeddings of cartesianproducts for special classes of graphs [9, 21, 23]; for example, when H is a path and G isa tree. However to the best of our knowledge, no results on stack layouts of direct andstrong products of graphs have been published.We remark that very recently, Dujmovi´c, Morin, and Yelle [16] independently proveda result equivalent to Corollary 1. Specifically, they study stack layouts of graphs withbounded layered pathwidth. That is, a path decomposition with a layering of a graph (amapping (cid:96) : V ( G ) → Z such that | (cid:96) ( u ) − (cid:96) ( v ) | ≤ u, v ) ∈ E ( G )) in which the sizeof the intersection of a bag and a layer is bounded by a constant. It is shown that everygraph of layered pathwidth p has stack number at most 4 p . Since the strong product ofa path and a pathwidth- p graph has layered pathwidth p + 1, the result of [16] implies(asymptotically) Corollary 1. We emphasize that neither our work nor [16] provides atight bound on the stack number of the class of graphs.11 Conclusion
In this paper we initiated the study of book embeddings of strong graph products. Asexplained in Section 1, resolving Open Problem 1 would either provide a constant upperbound on the stack number of several families of non-planar graphs, or it would answer afundamental question of Heath et al. [19] on the relationship of stack and queue layouts.Theorem 3 indicates that solving the open problem might be a challenging task. Thuswe suggest to explore the problem for natural subclasses of bounded-treewidth graphs.
Open Problem 2.
Is stack number of P n (cid:2) G bounded by a constant when G is(i) a tree (having an unbounded maximum degree)?(ii) an outerplanar (1-stack) graph?(iii) a planar graph with a constant treewidth, tw( G ) ≥ ?(iv) a bipartite graph with a constant treewidth, tw( G ) ≥ ? Notice that by the result of Wood [31], the queue number of P n (cid:2) G is a constant forall the aforementioned graph families. References [1] J. M. Alam, M. A. Bekos, M. Gronemann, M. Kaufmann, and S. Pupyrev. On dis-persable book embeddings. In
International Workshop on Graph-Theoretic Conceptsin Computer Science , pages 1–14. Springer, 2018.[2] J. M. Alam, M. A. Bekos, M. Gronemann, M. Kaufmann, and S. Pupyrev. Queuelayouts of planar 3-trees.
Algorithmica , pages 1–22, 2020.[3] M. A. Bekos, T. Bruckdorfer, M. Kaufmann, and C. N. Raftopoulou. The bookthickness of 1-planar graphs is constant.
Algorithmica , 79(2):444–465, 2017.[4] M. A. Bekos, G. Da Lozzo, S. Griesbach, M. Gronemann, F. Montecchiani, andC. Raftopoulou. Book embeddings of nonplanar graphs with small faces in few pages.In
Symposium on Computational Geometry , volume 164 of
LIPIcs , pages 16:1–16:17,2020.[5] M. A. Bekos, M. Kaufmann, F. Klute, S. Pupyrev, C. Raftopoulou, and T. Ueck-erdt. Four pages are indeed necessary for planar graphs.
Journal of ComputationalGeometry , 2020.[6] F. Bernhart and P. C. Kainen. The book thickness of a graph.
Journal of Combina-torial Theory, Series B , 27(3):320–331, 1979.[7] R. Blankenship.
Book Embeddings of Graphs . PhD thesis, Louisiana State University,2003.[8] H. L. Bodlaender. A partial k -arboretum of graphs with bounded treewidth. Theo-retical Computer Science , 209(1-2):1–45, 1998.[9] Y. Chen. Layout of planar products.
Journal of Mathematical and ComputationalScience , 6(2):216–229, 2016.[10] J. Davies and R. McCarty. Circle graphs are quadratically χ -bounded. arXiv:1905.11578 , 2019. 1211] W. Didimo, G. Liotta, and F. Montecchiani. A survey on graph drawing beyondplanarity. ACM Computing Surveys (CSUR) , 52(1):1–37, 2019.[12] V. Dujmovi´c and F. Frati. Stack and queue layouts via layered separators.
Journalof Graph Algorithms and Applications , 22(1):89–99, 2018.[13] V. Dujmovi´c, G. Joret, P. Micek, P. Morin, T. Ueckerdt, and D. R. Wood. Planargraphs have bounded queue-number. In
Foundations of Computer Science , pages862–875. IEEE, 2019.[14] V. Dujmovi´c, P. Morin, and D. R. Wood. Layout of graphs with bounded tree-width.
Journal on Computing , 34(3):553–579, 2005.[15] V. Dujmovi´c, P. Morin, and D. R. Wood. Graph product structure for non-minor-closed classes. arXiv:1907.05168 , 2020.[16] V. Dujmovi´c, P. Morin, and C. Yelle. Two results on layered pathwidth and linearlayouts. arXiv:2004.03571 , 2020.[17] P. Erd¨os and G. Szekeres. A combinatorial problem in geometry.
Compositio Math-ematica , 2:463–470, 1935.[18] J. L. Ganley and L. S. Heath. The pagenumber of k-trees is O ( k ). Discrete AppliedMathematics , 109(3):215–221, 2001.[19] L. S. Heath, F. T. Leighton, and A. L. Rosenberg. Comparing queues and stacksas machines for laying out graphs.
Journal on Discrete Mathematics , 5(3):398–412,1992.[20] L. S. Heath and A. L. Rosenberg. Laying out graphs using queues.
Journal onComputing , 21(5):927–958, 1992.[21] Y. Jiao, S. Ze-Ling, and L. Zhi-Guo. Embedding cartesian product of some graphsin books.
Communications in Mathematical Research , 34(03):253–260, 2018.[22] N. G. Kinnersley. The vertex separation number of a graph equals its path-width.
Information Processing Letters , 42(6):345–350, 1992.[23] X. L. Li.
Book Embedding of Graphs . PhD thesis, Zhengzhou University, 2002.[24] S. M. Malitz. Genus g graphs have pagenumber O ( √ g ). Journal of Algorithms ,17(1):85–109, 1994.[25] L. T. Ollmann. On the book thicknesses of various graphs. In
Southeastern Conferenceon Combinatorics, Graph Theory and Computing , volume 8, page 459, 1973.[26] S. B. Overbay.
Generalized book embeddings . PhD thesis, Colorado State University,1998.[27] M. Pilipczuk and S. Siebertz. Polynomial bounds for centered colorings on properminor-closed graph classes. In
Symposium on Discrete Algorithms , pages 1501–1520.SIAM, 2019.[28] S. Pupyrev. Mixed linear layouts of planar graphs. In
International Symposium onGraph Drawing and Network Visualization , pages 197–209. Springer, 2017.1329] M. Togasaki and K. Yamazaki. Pagenumber of pathwidth- k graphs and strongpathwidth- k graphs. Discrete Mathematics , 259(1-3):361–368, 2002.[30] V. Wiechert. On the queue-number of graphs with bounded tree-width.
ElectronicJournal of Combinatorics , 24(1):P1.65, 2017.[31] D. R. Wood. Queue layouts of graph products and powers.