Bar visibility graphs were introduced in the seventies as a model for some VLSI layout problems. They have been also studied since then by the graph drawing community, and recently several generalizations and restricted versions have been proposed. We introduce a generalization, witness-bar visibility graphs, and we prove that this class encompasses all the bar-visibility variations considered so far. In addition, we show that many classes of graphs are contained in this family, including in particular all planar graphs, interval graphs, circular arc graphs and permutation graphs.Ministerio de Economía y CompetitividadGeneralitat de CatalunyaEuropean Science FoundationMinisterio de Ciencia e Innovació

Cortés Parejo, María del Carmen

Díaz Báñez, José Miguel (Coordinador)

Garijo Royo, Delia (Coordinador)

Hurtado Díaz, Ferran

Márquez Pérez, Alberto

Márquez Pérez, Alberto (Coordinador)

Urrutia Galicia, Jorge (Coordinador)

Valenzuela Muñoz, Jesús

English

idUS. Depósito de Investigación Universidad de Sevilla

XV Spanish Meeting on Computational Geometry, June 26-28, 2013Witness Bar VisibilityCarmen Cortés∗1, Ferran Hurtado†2, Alberto Márquez‡1, and Jesús Valenzuela§11Departamento de Matemática Aplicada I, Universidad de Sevilla.2Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya.AbstractBar visibility graphs were introduced in the seventiesas a model for some VLSI layout problems. They havebeen also studied since then by the graph drawingcommunity, and recently several generalizations andrestricted versions have been proposed.We introduce a generalization, witness-bar visibil-ity graphs, and we prove that this class encompassesall the bar-visibility variations considered so far. Inaddition, we show that many classes of graphs are con-tained in this family, including in particular all planargraphs, interval graphs, circular arc graphs and per-mutation graphs.1 Introduction and preliminarydenitionsGiven a set S of disjoint horizontal line segments inthe plane (called bars hereafter) we say that G is abar-visibility graph if there is a bijection between Sand the vertices of G, and an edge between two ofthese if and only if there is a vertical segment (calledline of sight) between the corresponding bars thatdoes not intersect any other bar. We also say thatS is a bar visibility representation (or a bar visibilitydrawing) of G.Bar visibility graphs were introduced by Garey,Johnson and So [13] as a modeling tool for digitalcircuit design (see also [16]). These representationsare also a useful tool for displaying diagrams thatconvey visual information on relations among data,which is why many variations of these graphs havebeen considered by the graph drawing community[6, 7, 8, 9, 12, 14, 15].We need some denitions before we can pose pre-cisely our problem; we use standard terminology asin [5]. We call v-segment any vertical segment. We∗Email: ccortes@us.es.†Email: Ferran.Hurtado@upc.edu. Research supportedby projects MINECO MTM201230951, Gen. CatalunyaDGR 2009SGR1040, and ESF EUROCORES programme Eu-roGIGA, CRP ComPoSe: MICINN Project EUI- EURC-2011-4306, for Spain.‡Email: almar@us.es.§Email: jesusv@us.es.call ε-segment any axis aligned rectangle having widthε > 0 (intuitively, a thick vertical segment). Let s andt be two horizontal bars. We say that a v-segment con-nects s and t if its endpoints are in s and t. We saythat an ε-segment connects s and t if its horizontalsides are contained in s and t.Let S be a set of non-overlapping horizontal seg-ments (bars). Two bars s, t ∈ S are visible if, andonly if, there is a v-segment connecting s and t in-tersecting no other segment in S, and we say that sand t are ε-visible if, and only if there is an εsegmentconnecting s and t intersecting no other segment in S.With the preceding denition, bar visibility graphsas dened in the rst paragraph of this section takeas nodes a set of disjoint bars, and there is an edgebetween two nodes if and only if the correspondingbars are visible (this is also called a strong visibilityrepresentation of the graph [17]). If instead of visibil-ity we require ε-visibility, then we get bar ε-visibilitygraphs or, equivalently, an ε-visibility representationof the graph. The latter have been characterized asthose graphs that admit a planar embedding with allcutpoints on the exterior face [17, 18].A graph G is a weak bar visibility graph if its nodescan be put in bijection with a set of disjoint bars andthe nodes corresponding to every edge in G are ε-visible (note that not every ε-visibility need be anedge). This family of graphs is exactly the class of allplanar graphs [10].Finally, we say that G is a bar k-visibility graph ifthere is a bijection between a set of bars S and thevertices of G, and an edge between two of these if andonly if there is a v-segment joining the correspondingbars that intersects at most k other bars. This gener-alization has been introduced in recent years [8, 12].In this paper we introduce a stronger generaliza-tion, witness-bar visibility graphs, and we prove thatthis representation approach encompasses all the bar-visibility variations considered so far. In addition, weshow that many classes of graphs are contained inthis family, including in particular all planar graphs,interval graphs, circular arc graphs and permutationgraphs.For the denition of witness-bar visibility graphswe consider, in addition to the set S of bars that arein correspondence one-to-one with the vertices of the111Witness Bar Visibilitygraph being constructed, a set of green bars that fa-vor visibility, and a set of red bars that obstructvisibility. Green bars act as positive witnesses whilered bars correspond to negative witnesses. The barsfrom S neither favor nor obstruct visibilities.For the ease of description it is useful to consideralso purple bars that obstruct visibility in a slightlydierent way than red bars.Denition 1 Let S, SG, SP and SR be four setsof horizontal segments (bars, green-bars, purple-bars,and red-bars, respectively) such that any two elementsin S ∪ SG ∪ SR are disjoint. We dene:1. The green-bar visibility graph of S with respectto SG has one vertex for every element in S, andtwo bars s, t ∈ S are adjacent if and only if thereis an ε-segment connecting s and t that crossesat least one green bar.2. The purple-bar visibility graph of S with respectto SP has one vertex for every element in S, andtwo bars s, t ∈ S are adjacent if and only if thereis an ε-segment connecting s and t that does notcross any purple bar.3. The witness-bar visibility graph of S with respectto SG and SR has one vertex for every element inS, and two bars s, t ∈ S are adjacent if and onlyif there is an ε-segment connecting s and t thatcrosses strictly more green bars than red bars.The class of green, purple and witness-bar visibilitygraph are denoted, respectively, by GBG, PBG andWBG.An illustration of the three types of graphs is shownin Figure 1 (on a black and white printer, node-barsappear as thin lines, red bars as thick dark lines, pur-ple lines as thick lines colored light grey, and the greenlines are seen as thick striped lines).This work is devoted to the study of the classes ofgraphs that can be represented via green, purple orbar-visibility graphs and its properties. We start byconsidering the classes GBG and PBG, which will beproved to be subclasses of WBG. Then we will enu-merate classes of graphs that are contained in WBG,as well as properties of this class related to planarity.The terminology witness-bar visibility graphs is in-spired by the concept of witness proximity graphs,which focuses on deciding neighborliness relationsamong points in a nite set according to the pres-ence of some positive and/or negative witness points,a topic that has been studied in recent years [1, 2, 3,4, 11].2 The subclasses GBG and PBGIn this section we study the classes GBG and PBGand its relationships with other graph classes. Theabcdexyzabcdexyzabcdexyabc deabc deabc dezFigure 1: Examples of graphs in the families GBG(top), PBG (center) and WBG (bottom).interest of this is made clear by our rst result:Lemma 2 The class of graphs WBG contains strictlythe classes GBG and PBG.An interval graph is the intersection graph of a setof (closed) intervals on the real line.Theorem 3 Let G be a graph. If G is an intervalgraph, then G ∈ GBG. The reverse is in general false.The graph class inclusion in Theorem 3 is strict, asone can show that C4, which is not an interval graph,is in GBG. However, graphs with induced cycles oflength greater than 5 are not in GBG.Proposition 4 If the girth of a graph in GBG is -nite, then it is at most four.As a consequence of the previous result, it followsthat the green-bar visibility graph class does not con-tain any of the bar-visibility classes described in theintroduction of this paper, because Cn can be repre-sented as weak/ε/strong bar visibilty graph for everyn ≥ 3 [17].Note that even although one may think that theclasses GBG and PBG are related by complementa-tion, possibly by switching purple and green bar col-oring, but it is not the case. For example the union oftwo disjoint triangular cycles is in GBG, as seen in thepreceding section, but its complement is K3,3, whichis not in PBG, a fact that we will see below.On the positive side, let us see that interval graphsadmit a purple-bar visibility representation and provea useful lemma.112XV Spanish Meeting on Computational Geometry, June 26-28, 2013Theorem 5 If G is an interval graph, then G ∈PBG.Lemma 6 Let G be a triangle-free graph. If G ∈PBG then G is a planar graph.Proof. The idea is given in Figure 2. u1u2u3u4u5u6u7u8Figure 2: A purple-bar visibility representation of atriangle-free graph and the corresponding construc-tion of its planar embedding.Nevertheless the previous result cannot be extendedto a characterization of the class PBG:Theorem 7 1. K3,3 /∈ PBG.2. Kn ∈ PBG, ∀n.3. To admit a purple-bar visibility representation isnot inherited by subgraphs.Proposition 8 There are nonplanar graphs with tri-angles that do not admit a purple-bar visibility repre-sentation.An example of these graphs is given in Figure 3.abdcefg h ij k l mnopFigure 3: A nonplanar graph G with a triangle(△gjk), which does not admit a purple-bar visibilityrepresentation.The class PBG generalizes the classical bar-visibility representations:Theorem 9 Every graph G that can be representedas strong/ε/weak bar visibility graph admits as well apurple-bar visibility representation.Corollary 10 Every graph G that can be representedas strong/ε/weak bar visibility graph admits as well awitness-bar visibility representation.3 The class WBG of witness-barvisibility graphsWitness bar visibility also generalizes k-bar visibility:Theorem 11 Every graph G that can be representedas a bar k-visibility graph admits as well a witness-barvisibility representation.The idea of the proof is given in Figure 4.Figure 4: Assume we have to deal with bar 1-visibility,and consider a stack of 5 bars in a strip (left). Wesubdivide the strip into 7 slabs (right). In the secondbar we mimic 1-visibility using WBG-visibility for the3 lowest bars. In the fourth bar we do the same forthe 3 intermediate bars, and in the sixth slab for the3 highest bars.Theorem 12 The circular arc graphs and permuta-tion graphs are contained in WBGGiven a graph G, let G̃ be the graph resulting fromsubdividing once every edge in G.Lemma 13 Let G be a graph. If G̃ ∈ WBG then Gis a planar graph.Lemma 14 K̃3,3 /∈ WBG and K̃c3,3 ∈ WBG.As a consequence of the preceding lemma we im-mediately obtain the following result:Theorem 15 The class of the graphs that admit awitness-bar visibility representation is not closed un-der complementation.We conclude this section with another result on theclass WBG, that discards the possibility of character-izing the class by forbidden minors:Theorem 16 The property of admitting a witness-bar visibility representation is not inherited by sub-graphs.Proof. We know that K6 ∈ WBG from Theorem 7and Lemma 2. On the other hand K̃3,3 is a subdivi-sion of a subgraph ofK6, but we know from Lemma 14that K̃3,3 is not in WBG. This settles the claim. 113Witness Bar Visibility4 Concluding remarksLet us summarize the properties we have proved forthe class WBG of witness-bar visibility graphs:• Every graph G that can be represented asstrong/ε/weak bar visibility graph admits as wella witness-bar visibility representation.• Every graphG that can be represented as a bar k-visibility graph admits as well a witness-bar vis-ibility representation.• The class of interval graphs is contained in theclass WBG.• If G is a circular arc graph or a permutationgraph then G ∈ WBG.• The class of the graphs that admit a witness-bar visibility representation is not closed undercomplementation.• The property of admitting a witness-bar visibil-ity representation is not inherited by subgraphs,which discards the possibility of characterizingthe graph class WBG by forbidden minors.We conclude that the graph class WBG is very richand encompasses many other classes. 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