An Online Framework to Interact and Efficiently Compute Linear Layouts of Graphs
Michael A. Bekos, Mirco Haug, Michael Kaufmann, Julia Männecke
AAn Online Framework to Interact and EfficientlyCompute Linear Layouts of Graphs
Michael A. Bekos, Mirco Haug, Michael Kaufmann, Julia M¨annecke
Institut f¨ur Informatik, Universit¨at T¨ubingen, T¨ubingen, Germany { bekos,mk } @informatik.uni-tuebingen.de { mirco.haug,julia.maennecke } @student.uni-tuebingen.de Abstract.
We present a prototype online system to automate the pro-cedure of computing different types of linear layouts of graphs underdifferent user-specific constraints. The system consists of two main com-ponents; the client and the server sides. The client side is built uponan easy-to-use editor, which supports basic interaction with graphs, en-riched with several additional features to allow the user to define andfurther constraint the linear layout to be computed.The server side, which is available to multiple clients through a well-documented API, is responsible for the actual computation of the linearlayout. Its algorithmic core is an extension of a SAT formulation [5] thatis known to be robust enough to solve non-trivial instances in reason-able amount of time. However, it has also several known limitations andpotential improvements that we address in this work (e.g., limited appli-cability to a particular type of linear layouts, no support for additionalconstraints, limited extendability e.t.c.).As a proof of concept, we present our findings for a sketch of a proof ofan important result in the field that was proposed by Yannakakis [31]back in 1986 (whose details, however, have not been published so far).
Linear layouts of graphs have been fruitful subjects of intense research over theyears, both from a combinatorial and from an algorithmic point of view, asthey play an important role in various fields; see, e.g., [16]. Formally, a linearlayout of graph G = ( V, E ) consists of a linear order of its vertices (that is,a bijective function σ : V → { , . . . , | V |} ), and a partition of its edges into aparticular number of sets. Different constraints on the edges that may reside inthe same set, give rise to different types of linear layouts; see, e.g., [1,8,23,27,32].In particular, there is a rich body of research on two specific types of linearlayouts; the stack and the queue layouts.In a stack ( queue ) layout of a graph, no two edges of the same set, called stack ( queue ) in this context, are allowed to cross (nest, respectively) [23,32],where two edges ( u, v ) and ( w, z ), such that σ ( u ) < σ ( v ) and σ ( w ) < σ ( z ), cross if σ ( u ) < σ ( w ) < σ ( v ) < σ ( z ), and nest if σ ( u ) < σ ( w ) < σ ( z ) < σ ( v ); see, e.g.,Fig. 1. The stack-number ( queue-number ) of a graph is the minimum number a r X i v : . [ c s . D M ] M a r M. A. Bekos, M. Haug, M. Kaufmann, J. M¨annecke
111 32 6 78 94 510 (a) (b) (c)
Fig. 1:
Illustration of: (a) the Goldner-Harary graph [20], (b) a 3-stack layout of it,and (c) a 2-queue layout of it. of stacks (queues) required by any of its stack layouts (of its queue layouts,respectively). Note that stack layouts are widely known also as book embeddings . Known Results . There exists a plethora of theoretical results for each of theaforementioned types of linear layouts; in the following, we overview existingresults for planar graphs, which is mainly the focus of our work. For a moredetailed overview, we point the reader to [16].– For stack layouts, the most notable result is to due Yannakakis, who back in1986 showed that every planar graph admits a 4-stack layout [31,32] improvinga series of earlier results [9,21,24]. However, a fundamental question, whichstill remains open in the field, is whether every planar graph admits a 3-stacklayout or whether there exists one which requires four stacks in any of its stacklayouts. Note that several subfamilies of planar graphs allow for layouts withfewer than four stacks; see, e.g., [4,6,11,12,17,21,25,26,28].– For queue layouts, a breakthrough result is due to Dujmovi´c et al. [15], whorecently showed that every planar graph admits a 49-queue layout, improvingprevious (poly-)logarithmic bounds [3,14,13]; the best-known correspondinglower bound is four [2], which implies that the current gap between the twobounds is still very large. Again, several subfamilies of planar graphs allow forlayouts with significantly fewer than 49 queues; see, e.g., [2,19,22,23,28].
Motivation . The primary motivation of this project stems from the aforemen-tioned paper by Yannakakis, which had appeared at STOC in 1986 [31] andcontained a sketch of a proof for the existence of a planar graph that does notadmit a 3-stack layout (we provide an outline of this sketch in Section 3.1).The details of this proof, however, never appeared in a paper (in particular, theproof-sketch was not part of the subsequent journal version [32] of the extendedabstract that had appeared at STOC [31]), and thus the problem of determiningwhether there exists a planar graph that requires four stacks in any of its stacklayouts still remains unsolved, and clearly forms the most intriguing open prob-lem in the field. Note that the arguments in the proof-sketch by Yannakakis [31]seem to be sound (apart from the fact that some of the gadget-graphs that arecentral in the proof have not been properly defined), and potentially may giverise to a formal proof. n Online Framework to Interact and Compute Efficiently Linear Layouts 3
So, in this work we wanted to dig the details of this proof-sketch, so tounderstand if (and more importantly where) the proof fails. In addition, wewere interested in finding the claimed gadget-graphs, which are not given in theproof. To this end, we decided to continue working on a SAT formulation [5], thatwe proposed few years ago (for the problem of finding a stack layout of a givengraph with a certain number of stacks), which we had to extend with severalnew features that are of independent value, even for future considerations.
Contribution . The main contribution of this work is a novel system, which isavailable online ( algo.inf.uni-tuebingen.de/linearlayouts ) and automatesthe procedure of computing different types of linear layouts of graphs (i.e., stacklayouts, queue layouts or mixtures of these). Besides the description of the sys-tem and its functions, we deem important to stress how the extensions that wepropose stem from realistic problems that we encountered, while trying to checkthe different parts of the proof-sketch by Yannakakis [31]. The first major issuethat we encountered was the need to define additional constraints on the layoutsto be computed (e.g., on the relative order of the vertices, or on the edges toappear at specific pages e.t.c.), that is, without, e.g., writing lines of code tai-lored to each of these constraints. A second major issue was the need to interactwith the graph, that we were investigating, as efficiently as possible. To this end,we featured the system with an easy-to-use graph editor, which supports basicinteraction with graphs, and simultaneously provides the necessary functionalityto define and further constraint the linear layouts to be computed. Overall, webelieve that the real value of the system is that, at its current state, it automatesseveral of the standard procedures that a domain expert needs when interactingwith linear layouts of graphs.
Paper Organization . In Section 2, we describe in details the extended featuresthat we have implemented in the system together with some insights on theSAT formulation. Section 3 serves as a proof of concept for our system; we firstpresent an outline of the proof-sketch by Yannakakis mentioned above, and thenwe present our findings for the different parts of the proof-sketch. We concludein Section 4 with further considerations and plans.
Our system, as introduced in Section 1, consists of two main components, theclient and the server sides, and introduces a series of innovations over its previousimplementation [29]. The actual code of the system is available to the communityat a github repository ( github.com/linear-layouts/SAT ). In the following, wedescribe in details the extended features that are currently supported both inthe client (Section 2.1) and the server side (Section 2.2).
The client side is web-based, developed with standard tools (e.g.,
HTML , jQuery and yFiles [30], which is free for academic usage) and operates in two modes; M. A. Bekos, M. Haug, M. Kaufmann, J. M¨annecke
Fig. 2:
A screenshot of the system in editing mode. the editing and the view (see Figs. 2 and 4). In the editing mode, the user createsthe graph and specifies the constraints of the linear layout to be computed (ifany). The graph is created through a graph editor supporting basic interactionwith graphs (e.g., creation and deletion of vertices and edges, navigation over thegraph, panning, zooming, e.t.c.), that we configured appropriately to meet ourneeds. The graph editor is accompanied with a configuration panel (refer to thebottom part of Fig. 2), where the user can configure the type of the layout, aswell as, define different constraints on it. More precisely, the following functionsare currently supported:
Specification of the type of the linear layout . Through the configurationpanel, the user defines the number of available pages (i.e., stacks or queues) ofthe linear layout to be computed. The current implementation supports up tofour pages, in total. In a subsequent step, the user defines the type (i.e., stack orqueue) of each of the pages of the linear layout. In this way, the system providessupport both for stack and queue layouts, but also for mixed layouts , in whichsome of the pages are stacks, while some others are queues; see, e.g, [27].
Specification of structural constraints . In a stack (queue) layout of a graph,the subgraphs induced by the edges of each of its stacks (queues) are outerpla-nar [6] (arched-level planar [23], respectively). Besides the number and the typeof the available pages of the linear layout, the user can also impose additionalstructural constraints on these subgraphs. Currently, for each of the available n Online Framework to Interact and Compute Efficiently Linear Layouts 5
Fig. 3:
A snapshot illustrating the options available on selected vertices and edges. pages, the user can choose one of the following options; not to impose any re-striction on the subgraph induced by the edges of a particular page (apart fromthose that are necessarily imposed by the type of the page), or to restrict the sub-graph induced by the edges of a particular page to be either a matching or a tree.
Imposition of restrictions on the linear order . There exist several ways toimpose restrictions on the linear order of vertices of the linear layout (see Fig. 3).The constraints are created through the editor and are stored in a separatecomponent of the configuration panel (so that the user is able to remove them).R.1
Specification of successors and predecessors of vertices : By selectingtwo vertices of the graph and by right-clicking on one of them, the user isable to set one of the selected vertices successor or predecessor of the other.R.2
Specification of vertices to be consecutive : In the same way, the useris able to require two selected vertices to appear consecutively in the lin-ear order of the vertices. Note that this constraint does not restrict therelative order of them. However, this can be easily achieved by combiningRestrictions R.1 and R.2.R.3
Specification of required and forbidden partial orders.
By clickingon two or more vertices of the graph, while keeping the ctrl button of thekeyboard pressed, the user is able to select multiple vertices in a specificorder. Then, by right-clicking on one of them the system provides supportto restrict the relative order of the selected vertices to be the one (or not tobe the one), in which the vertices were clicked on.
Imposition of restrictions on the edge assignments.
Besides the restric-tions on the linear order of the vertices, the user is also able to assign specificedges to particular pages. This can be achieved through the following functions.R.4
Assignment of edges to the same or to different pages : By selectingtwo or more edges of the graph and by right-clicking on one of them, theuser is able to instruct the system to assign the selected edges to the same orto different pages of the linear layout. Note that the latter option becomesunavailable, when the number of selected edges is greater than the numberof available pages of the linear layout.R.5
Assignment of edges to specific pages : In the same way as above, theuser is able to assign selected edges of the graph to one of a set of specific
M. A. Bekos, M. Haug, M. Kaufmann, J. M¨annecke pages of the linear layout. In contrast to Restriction R.4, the user here hasto specify the exact pages, to which the selected pages will be assigned.R.6
Assignment of edges incident to vertices to specific pages : The useris also able to assign edges to specific pages through a selection of vertices;in particular, the edges incident to these vertices. In the special case, inwhich the selected vertices are only two, say u and v , then in the linearorder of the vertices, u and v define two intervals; the one between u and v ,and the remaining one. In this particular case, the user is also able to applythe constrain only to the edges from u and v that end to only one of thesetwo intervals. Additional features . The editing mode is equipped with several additionalfeatures; in the following, we name few. In order to facilitate the definition ofthe constraints on the different elements of the graph, the user may restrict theselection mode of the graph editor only to vertices or only to edges, depending onthe type of constraints that she wish to introduce. The user is able to save boththe graph and its associated constraints in a
GraphML file for future considera-tions (see, e.g., graphml.graphdrawing.org ). There exist also two options thatare currently supported for exporting the constructed graph; one in
PDF and onein
PNG format. As side features, the editing mode is also equipped with standardlayout algorithms (such as, spring-based, orthogonal and radial), while there isalso support for querying the graph for standard properties (e.g., connectivity,acyclicity and planarity).Once the creation of the graph and the definition of the constraints on itslinear layout have been completed, the user may request to compute the actuallinear layout (if any). At this point, the created graph and its constraints arepassed to the server side, which is responsible for the actual computation. Oncethe computation has been completed, the system enters the view mode, wherethe computed linear layout is presented to the user. In contrast to the editingmode, in the view mode the user can partially interact with the computed linearlayout, e.g., (i) change the common color and the placement (i.e., above or belowthe line, on which the vertices reside) of the edges assigned to each page, (ii) saveor export the final layout to a file, and (iii) navigate, pan or zoom over the layout.If the user seeks in further editing the graph and its constraints, then shehas to return back to the editing mode. In this transition, the user may chooseto work either with the original layout that had constructed before or with thecomputed linear layout (the user’s configurations are also restored).
The server side of the system has been developed in
Python , while for solvingthe SAT instances, we used the
Lingeling solver ( fmv.jku.at/lingeling );the source code of the server side is also contained in the github repositorymentioned earlier. All requests that arrive to the server are stored in a database(
SQLite ), which is responsible for associating each of them with a unique id.Once the processing of a request is finished, the computed layout is stored to the n Online Framework to Interact and Compute Efficiently Linear Layouts 7
Fig. 4:
A screenshot of the system in view mode. database, so to be available to the client at any time. Note that the server sidebecomes available to multiple clients through a well-documented API available atthe server’s interface ( alice.informatik.uni-tuebingen.de:5555 ). In otherwords, any client, that complies with the developed interface, may communicatewith the server side and request a linear layout of a given graph under a set ofadditional constraints that are currently supported at the server.The algorithmic core of the server side is an extension of the SAT formula-tion [5] mentioned in the introduction, featured with additional functions to over-come known limitations; in particular to support different types of linear layoutsand different types of user-specific constraints. Even though SAT formulationsare of limited applicability, and therefore not so common, in graph drawing (withfew notable exceptions; e.g., [7,10,18]), in this particular scenario the formula-tion is robust enough to solve non-trivial instances in reasonable amount of time(and, as expected, its performance increases when additional constraints are im-posed). In the remainder of this section, we give a short overview of the originalformulation followed by a high-level description of the extensions that we made.The original formulation makes use of three types of variables σ , φ and χ withthe following meanings: (i) for a pair of vertices u and v , variable σ ( u, v ) is true if and only if u precedes v in the linear order, (ii) for an edge e and a page ρ ,variable φ ρ ( e ) is true if and only if edge e is assigned to page ρ of the layout,and (iii) for a pair of edges e and e (cid:48) , variable χ ( e, e (cid:48) ) is true if and only if e and M. A. Bekos, M. Haug, M. Kaufmann, J. M¨annecke e (cid:48) are assigned to the same page. So, there exist O ( n + m + pm ) variables,where n denotes the number of vertices of the graph, m its number of edges, and p the number of available pages. A set of O ( n + m ) clauses ensure that theunderlying order is linear, and that the layout is valid; for details, refer to [5].To support different types of layouts, each page ρ is associated with a type τ ( ρ ) ∈ { stack , queue } . If τ ( ρ ) = stack , then we employ the same set of con-straints as in the original formulation to avoid crossings in page ρ . Otherwise, τ ( ρ ) = queue holds, in which case we need to guarantee that no two edges ofpage ρ nest. This can be ensured by introducing the following constraint for everypair of edges ( u, v ) and ( z, w ), such that u , v , z and w are pairwise different. φ ρ ( u, v ) ∧ φ ρ ( z, w )) → ¬ ( σ ( u, z ) ∧ σ ( z, w ) ∧ σ ( w, v )) ∧¬ ( σ ( u, w ) ∧ σ ( w, z ) ∧ σ ( z, v )) ∧¬ ( σ ( v, z ) ∧ σ ( z, w ) ∧ σ ( w, u )) ∧¬ ( σ ( v, w ) ∧ σ ( w, z ) ∧ σ ( z, u ))To ensure Restrictions R.1–R.3, we introduce further constraints. Recall thatR.1 and R.2 apply on a pair of vertices, denoted by u and v in the following,while R.3 applies on an ordered set of vertices, denoted by { u , . . . , u k } .R.1: σ ( u, v ), if u must be the predecessor of vσ ( v, u ), otherwiseR.2: σ ( x, u ) ∨ σ ( v, x ), ∀ x / ∈ { u, v } R.3: ( σ ( u , u ) ∧ . . . ∧ σ ( u k − , u k )), if the order must be preserved ¬ ( σ ( u , u ) ∧ . . . ∧ σ ( u k − , u k )), if the order must be avoidedRestrictions R.4–R.6 are ensured in a similar fashion. Recall that R.4 and R.5apply on an set of edges, denoted by { e , . . . , e (cid:96) } in the following, while R.6 onan set of vertices, denoted by { u , . . . , u k } .R.4: χ ( e, e (cid:48) ) , ∀ e, e (cid:48) ∈ { e , . . . , e (cid:96) } , if the edges must be in the same page ¬ χ ( e, e (cid:48) ) , ∀ e, e (cid:48) ∈ { e , . . . , e (cid:96) } , if the edges must be in different pagesR.5: ¬ φ ρ ( e ) , ∀ e ∈ { e , . . . , e (cid:96) } , for each page ρ that is not selectedR.6: ¬ φ ρ (( u, · )) , ∀ u ∈ { u , . . . , u k } , for each page ρ that is not selected In this section, we demonstrate how the system can be used by applying it tothe proof-sketch by Yannakakis [31] that we mentioned in the introduction ofthe paper. At first, we give a short overview of this proof-sketch (Section 3.1),and then we described how we exploited the different functions of the systemthat we presented in Section 2 to get meaningful insights of this proof-sketch(Section 3.2). We note at this point that the planar graph that Yannakakisclaimed not to admit a 3-stack layout is a tremendously large graph, since it isconstructed by composing several smaller graphs, each of which has a certainproperty. Hence, testing via SAT (or via another such approach) whether thisfinal graph admits a 3-stack layout becomes realistically an impossible task. n Online Framework to Interact and Compute Efficiently Linear Layouts 9 AB x x n (a) Step 1 AB y y k (b) Step 2 AB y i y i +1 a i b i (c) Step 3 Fig. 5:
Illustrations for the different steps of proof: (a) the skeleton graph, (b) thestellated-skeleton, and (b) the operation of attaching three copies of graph Q along theedges of the subgraph of the stellated-skeleton induced by A , y i − , a i , b i , y i and B . In this section, we give an overview of the aforementioned proof-sketch [31] byYannakakis that appeared as an extended abstract of [32] at STOC back in1986. We deem important to stress at this point that since several parts of theproof-sketch have been omitted in [31], the description of the main ideas of itare subject to the best of our understanding of the line of the arguments.The proof-sketch consists of three steps, each of which assumes the existenceof a particular planar graph with a specific property. The graph of the firststep, which we call skeleton , consists of two designated vertices A and B thatare connected to all the vertices of a long path x → x → . . . → x n , whichwe call skeleton path ; for an illustration, refer to Fig. 5a. The property of thisgraph is essentially the following: if n is chosen large enough (e.g., n = 1000),then in any 3-stack layout of the skeleton graph at least one pair of consecutivevertices of the skeleton path will appear between A and B . The observation thatgives rise to the second step of the proof-sketch is that if one triples the lengthof the skeleton path, then one may assume two pairs of consecutive verticesof the skeleton path to lie between A and B , and so on. In other words, if n ischosen large enough, then there exists a particular number of pairs of consecutivevertices of the skeleton path, say (cid:104) y , y (cid:105) . . . (cid:104) y k − , y k (cid:105) , that lie between A and B in any 3-stack layout of the skeleton graph.With this observation in mind, in the second step of the proof-sketch, theskeleton graph gets further augmented by stellating each of its 2 n triangularfaces, that is, by adding a vertex in its interior and by connecting it to the threevertices of its boundary (refer to the gray-colored vertices in Fig. 5a). Now, focuson the subgraph of the resulting graph, which we call stellated-skeleton , thatis induced by A , B , y , . . . , y k and the vertices a , . . . , a k and b , . . . , b k thatstellate the faces (cid:104) A, y , y (cid:105) . . . (cid:104) A, y k − , y k (cid:105) and (cid:104) B, y , y (cid:105) . . . (cid:104) B, y k − , y k (cid:105) ,respectively; see Fig. 5b. The property of the stellated-skeleton is essentially thefollowing. If ( n is chosen sufficiently large such that) k is large enough, thenthere exists a 4-tuple (cid:104) y i − , a i , b i , y i (cid:105) of vertices that appear in this orderbetween A and B in any 3-stack layout of the graph constructed so far. The third step of the proof-sketch yields another augmentation to the graph,which is based on an internally-triangulated planar graph, denoted by Q in [31],whose outer face is bounded by a 4-cycle ( s, a, t, b ). In particular, three copiesof graph Q are “attached” along each edge ( u, v ) of the graph that has beenconstructed so far, such that vertices s and t of each of the three copies areidentified with the endvertices u and v of the edge ( u, v ); see Fig. 5c for anillustration of this operation along the edges of the subgraph of the stellated-skeleton induced by A , y i − , a i , b i , y i and B .Note that graph Q is not specified in the proof-sketch; Yannakakis only de-scribes the properties for this graph, which are as follows. Graph Q does notadmit a 3-stack layout such that (P.1) a and b lie between s and t in the linearorder of the vertices, and (P.2) none of the edges from s and t to the verticesthat lie between s and t in the linear order of the vertices (i.e., including a and b ) can be assigned to the third stack of the layout.The three steps described so far formed the difficult part in the proof-sketch;the remainder of it is relatively easier. More precisely, assuming that the 4-tuple (cid:104) y i − , a i , b i , y i (cid:105) of the second step of the proof-sketch is guaranteed, andthat the graph Q of the corresponding third step (with the claimed Properties P.1and P.2) exists, the rest of the proof is sound (and more importantly describedin the sketch). Hence, putting all the pieces of this proof together reduces in de-termining whether the 4-tuple (cid:104) y i − , a i , b i , y i (cid:105) as well as the graph Q exist. As already mentioned in the introduction, most of the extensions that we imple-mented in our system are motivated by our efforts to check the different parts(in particular, the three main steps) of the proof-sketch by Yannakakis. This willbecome clear in this section, where we report our findings and describe how thedeveloped system helped us to obtain meaningful insights of this proof.
Our findings on the first step of the proof-sketch . The task here wasrather clear; we had to check whether two consecutive vertices of the skeletonpath will inevitably appear between the two designated vertices A and B of theskeleton graph in any of its 3-stack layouts, when the length of the skeleton pathis large enough. To check this, we created an instance of the skeleton graph withapproximately 100 vertices. Then, using restriction R.1, we instructed our systemto constrain the layout to be computed, such that: (i) vertex A (vertex B) is apredecessor (successor, respectively) of every odd-indexed vertex of the skeletonpath, and (ii) vertex A (vertex B) is a successor (predecessor, respectively) ofevery even-indexed vertex of the skeleton path. If the system could not reporta solution under these two constraints, then the claim would follow. However,this was not the case, as the system could easily report a solution, which was sosymmetric that we were able to generalize it to all skeleton graphs; see Fig. 6. Our findings on the second step of the proof-sketch . Even though ourfindings on the first step of the proof-sketch were a bit discouraging, we found itnecessary to continue digging the details of the proof-sketch. Our idea was that if n Online Framework to Interact and Compute Efficiently Linear Layouts 11
A 6 8 n n -2 3 n -1 7 5 n -34 Fig. 6:
A 3-stack layout of the skeleton graph in which no two consecutive vertices ofthe skeleton path lie between A and B . our findings on the second step were more promising, then we could try to modify(e.g., somehow augment) the skeleton graph, and eventually manage to guaranteeits property. So, we decided to proceed to check the second step of the proof-sketch, under the assumption that somehow we can guarantee the property of thefirst step, that is, several pairs of consecutive vertices (cid:104) y , y (cid:105) , . . . , (cid:104) y k − , y k (cid:105) ofthe skeleton path appear between the two designated vertices A and B .Under this assumption, we had to check whether two stellating vertices a i and b i of two faces (cid:104) A, y i − , y i (cid:105) and (cid:104) B, y i − , y i (cid:105) of the stellated-skeleton willinevitably appear between y i − and y i , for some i ∈ { , . . . , k } . To check this,we created an instance of the stellated skeleton with approximately 100 vertices(that is, k ≈ i = 1 , . . . , k , the partial order (cid:104) y i − , a i , b i , y i (cid:105) is forbidden, and(iv) the same holds for the partial order (cid:104) y i − , b i , a i , y i (cid:105) . As it was the casewith the first step of the proof-sketch, the system could report a solution, whichwas again symmetric and we were able to generalize it to all stellated skeletons;see Figure 7. Our findings on the third step of the proof-sketch . Finding a graph withthe properties of the graph Q in [31] was definitely a challenging task. In ourefforts, we crafted several internally-triangulated planar graphs with a quad-rangular outer face, which contained substructures that are known to require athird stack, such as planar 3-trees [21], the Goldner-Harary graph [20] and planargraphs with several separating triangles [25]. We also tested approximately 4.000randomly generated planar graphs with 100 to 150 vertices, which we created byfirst triangulating an evenly distributed point set within a quadrangular region,and then by connecting four vertices at the corners of this region to all verticeson the (internally triangulated) point set without introducing crossings. Eachof these graphs had by construction a quadrangular outer face ( s, a, t, b ) andwas tested for the property of graph Q in [31] using restriction R.3 and R.6 as A y b b B y y k a k y a y b k y k − a Fig. 7:
A 3-stack layout of the stellated-skeleton in which there exist no index i ∈{ , . . . , k } , such that the vertices y i − , a i , b i , y i appear in this order between A and B . follows: (i) vertices s , a and t appear in this order, (ii) vertices s , b and t appearin this order, and (iii) the edges from s and t that end in vertices between s and t are assigned to the first two pages of the layout.To put it briefly: we did not manage to find a graph with the claimed prop-erties of graph Q in [31]. As a matter of fact, in the layouts that we computedwe observed that most of the times very few vertices were eventually between s and t (besides a and b ), which is an indication that the restrictions may not bestrong enough. It is worth noting at this point that, in the original proof-sketch,it is mentioned that graph Q is the endproduct of a series of constructions ofsmaller gadget-graphs Q , Q and so forth (that are not described; actually, noteven their properties are described), which is another indication that if a graphwith the properties of graph Q exists, then it might be tremendously large. In this paper, we presented a novel system equipped with several features thatautomate most of the standard procedures that a domain expert needs for com-puting different types of (constrained) linear layouts of graphs. As a proof ofconcept, we used the developed system to check the different parts of a proof-sketch by Yannakakis [31], and we managed to gain valuable insights. Our find-ings, of course, were mostly negative. The proposed system, however, might beuseful in finding a suitable graph with the properties of graph Q in [31] and someother graph (than the one Yannakakis proposed) yielding the 4-tuple describe inSection 3.1. And so we are still hopeful that the proof-sketch can be completed.More in general, we believe that the developed system is extremely usefulfor proving lower bounds (e.g., to narrow the current wide gap between theupper and the lower bound on the queue number of planar graphs), which isthe main reason to have it available online. We further plan to equip it withadditional features, such as adding support for searching and filtering the layouts,highlighting the restrictions and more advanced definitions of restrictions. n Online Framework to Interact and Compute Efficiently Linear Layouts 13 References
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