A note on intersection dimensions of graph classe

Ales Kubena

Petr Hlineny

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Comment.Math.Univ.Carolinae 36,2 (1995)255{261 255

summary:The intersection dimension of a graph $G$ with respect to a class $\Cal A$ of graphs is the minimum $k$ such that $G$ is the intersection of some $k$ graphs on the vertex set $V(G)$ belonging to $\Cal A$. In this paper we follow [\,Kratochv'\i l J., Tuza Z.: {\sl Intersection dimensions of graph classes\/}, Graphs and Combinatorics 10 (1994), 159--168\,] and show that for some pairs of graph classes $\Cal A$, $\Cal B$ the intersection dimension of graphs from $\Cal B$ with respect to $\Cal A$ is unbounded

Hliněný, Petr

Kuběna, Aleš

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaePetr Hliněný; Aleš KuběnaA note on intersection dimensions of graph classesCommentationes Mathematicae Universitatis Carolinae, Vol. 36 (1995), No. 2, 255--261Persistent URL: http://dml.cz/dmlcz/118754Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1995Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czComment.Math.Univ.Carolin. 36,2 (1995)255–261 255A note on intersection dimensions of graph classesPetr Hliněný, Aleš KuběnaAbstract. The intersection dimension of a graph G with respect to a class A of graphs isthe minimum k such that G is the intersection of some k graphs on the vertex set V (G)belonging to A. In this paper we follow [Kratochv́ıl J., Tuza Z.: Intersection dimensionsof graph classes, Graphs and Combinatorics 10 (1994), 159–168 ] and show that for somepairs of graph classes A, B the intersection dimension of graphs from B with respect toA is unbounded.Keywords: intersection graph, intersection dimensionClassification: 05C10, 05C301. IntroductionIn this paper we consider finite undirected graphs without loops or multipleedges. Classes of graphs are understood to be closed under isomorphism. Thevertex set (edge set) of a graph G is denoted V (G) (E(G)), and we write G =(V (G), E(G)). Given a set V , we denote byKV the complete graph and byDV thediscrete graph (graph with no edges) on the vertex set V . For some set V , |V | = n,we also denote by Kn (Dn) the graph KV (DV ). The complement −G of a graphG is the graph (V (G),(V2)−E(G)). As the intersection (union) of two graphs G,H on the same vertex set we understand the graph (V (G) = V (H), E(G)∩E(H))((V (G) = V (H), E(G) ∪ E(H))). The Zykov sum G ⊕ H of two graphs G, H isdefined as their disjoint union plus all edges between the graphs.We call a set of vertices U ⊆ V (G) in a graph G independent if there is noedge between the vertices of U . Similarly we call U ⊆ V (G) a clique if there areall edges between them in G. A chromatic number of a graph G (denoted byχ(G)) is the minimum number of colours needed to colour the vertices of G suchthat no edge has end vertices of the same colour. It is easy to see that for everytwo graphs G, H , χ(G ∪ H) ≤ χ(G) · χ(H) (we colour vertices of G ∪ H by pairsof colours composed from colours of proper colourings of G and H). A graph ofchromatic number 2 is called bipartite.Several types of graph dimensions that could be seen as intersection dimensionswith respect to some special graph classes (boxicity as intersection dimensionwith respect to interval graphs, circular dimension – circular-arc graphs, overlapdimension – circle graphs, see [1], [3]) have been previously studied. The notion ofintersection dimension (with respect to graphs having some property) was actuallyintroduced by Cozzens and Roberts in [2]. We define it in a slightly different wayas in [8]:256 P.Hliněný, A.KuběnaDefinition 1.1. Given a class A of graphs and a graph G = (V, E), the intersec-tion dimension of G with respect to A (otherwise called the A-dimension of G)isdimAG = min{ k | ∃E1, . . . , Ek ⊆(V (G)2)s.t. (V (G), Ei) ∈ A for each i and E =k⋂i=1Ei} .Definition 1.2. Given classes A, B of graphs, the intersection dimension of Bwith respect to A (the A-dimension of B ) isdimAB = supG∈BdimAG(we write ∞ if the dimension is unbounded).The intersection graph of some set family M is a graph isomorphic to thegraph, whose vertices are sets fromM and two vertices are adjacent iff these twosets have nonempty intersection. We define following special classes of graphs:Definition 1.3. Intervals graphs (denoted by INT) are intersection graphs ofintervals on a line, circular-arc graphs (CA) are intersection graphs of intervals(arcs) on a circle, circle graphs (CI) are intersection graphs of chords of a circle,permutation graphs (PER) are intersection graphs of straight line segments withend points on two parallel lines, line segment graphs (SEG) are intersection graphsof line segments in the plane, function graphs (FUN) are intersection graphs ofcontinuous functions on a closed interval, chordal graphs (CHOR) are graphs inwhich each cycle of length greater than 3 has a chord, split graphs (SP) are graphswhose vertices can be divided into a clique and an independent set.Note that for each above defined class A and any graph G the A-dimensionof G is well defined, because each of these classes contains all complete graphsand all complete graphs minus an edge. There are some clear inclusions betweenthem, e.g. INT ⊆ CA, PER ⊆ CI, SP ⊆ CHOR, INT ⊆ CHOR, INT ⊆ SEG,CI ⊆ SEG. For characterizations of the classes INT, PER and FUN see [4].We will use here the fact that function graphs are exactly the complements ofcomparability graphs. As was proved in [7], for each SEG graph we have linesegment representation where no two segments are parallel.We include the table of dimensions computed in [8], the number dimAB isplaced in the row A and the column B. We add the class SEG to the table and fillits known dimensions — the numbers marked by ′ follow from PER ⊆ CI ⊆ SEG,the one marked by ∗ follow from INT ⊆ SEG and the ones marked by ◦ areconsequences of ∗. The items marked by (1) are proved here in Section 2, the onesA note on intersection dimensions of graph classes 257marked by (2) are proved in Section 3 and for the ones marked by (3) this papergives a different proof than [8]. The number dimSEGFUN =∞ answers a questionwhether FUN ⊆ SEG asked in [6]. There are still two remaining unknown itemsmarked by ? in the table.INT CA CI PER SEG SP CHOR FUNINT 1 ∞ ∞ ∞ ∞′ ∞(3) ∞(3) ∞(3)CA 1 1 ∞(2) ∞(2) ∞(2) ∞(1) ∞(1) ∞(3)CI ∞ ∞ 1 1 ∞◦ ∞(1) ∞(1) ∞(3)PER ∞ ∞ ∞ 1 ∞◦ ∞(1) ∞(1) ∞(3)SEG 1∗ ? 1′ 1′ 1 ∞(1) ∞(1) ∞(1)SP ∞ ∞ ∞ ∞ ∞′ 1 ∞ ∞CHOR 1 ∞ ∞ ∞ ∞′ 1 1 ∞FUN 1 2 2 1 ? 2 2 12. Small and large classesThe main idea of this section is to compute how “large” are graph classes (howmany graphs on a fixed n-element set they contain). We prove that if a class Bis “much larger” than a class A, then there exists a graph G ∈ B with arbitraryhigh A-dimension.Definition 2.1. For any class A of graphs and any integer n > 0 we denote thesetAn = {G ∈ A | V (G) = {1, 2, . . . , n}} .Lemma 2.2. Let A, B be classes of graphs such that A contains all completegraphs and all complete graphs minus an edge (i.e. the A-dimension of any graphis well defined). ThendimAB ≥ supn∈N,n>1log |Bn|log |An|.Proof: If dimAB = ∞, there is nothing to prove. Let us suppose that thereexist d, n0 ∈ N for which d = dimAB <log |Bn0 |log |An0 |, that means |Bn0 | > |An0 |d.Due to the definition of the A-dimension of B, each graph G ∈ Bn0 should be anintersection of at most d graphs fromA. We may add complete graphsKn0 so thatG is the intersection of exactly d graphs (not necesarily distinct) H1, . . . , Hd ∈ A.But from Definition 2.1 we have that H1, . . . , Hd ∈ An0 and we can choose dgraphs from An0 in at most |An0 |d ways, so |Bn0 | ≤ |An0 |d, a contradiction. 258 P.Hliněný, A.KuběnaTo use the previous Lemma 2.2, we need to compute the aproximate numberof graphs in the considered classes. For this purpose we use the notation f(n) =Θ(g(n)), which means that there exist constants b > a > 0 such that a · g(n) <f(n) < b · g(n) for almost all n. We divide the graph classes into two groupsINT, CA, CI, PER, SEG of aproximate size eΘ(n logn) and SP, CHOR, FUN ofaproximate size eΘ(n2). Although it is enough to prove here the upper boundfor the first group and the lower bound for the second group, the proof of tightaproximate size is not difficult.Lemma 2.3. For every integer n, |CAn| ≤ (2n)2n.Proof: For n arcs on a circle there are at most 2n distinct end points of thesearcs, and the intersection graph of these arcs if fully determined by their circularorder. So we can construct all CAn graphs by taking 2n distinct points on a circleand for each of n arcs we choose from these 2n possibilities the start point andthe end point. Therefore |CAn| ≤ (2n)2n. Lemma 2.4. For every integer n, |SEGn| ≤ n6n ·(n2)n.Proof: Firstly we determine how many possibilities there are of placing n linesdenoted by p1, p2, . . . , pn into the plane such that no two of them are paralleland two configurations are distinct if some line crosses the other lines in differentorders. This is exactly the dual problem to the problem to compute the numberof distinct order types of simple numbered configurations of n points in the plane.The upper bound of at most n6n distinct configurations for that problem is provedin [5].Each graph from SEGn we can represent (see [7]) as an intersection graph ofline segments, where no two segments are parallel. Such representation is fullydetermined by the configuration of n lines that the segments lie on, and by theposition of each segment on its line with respect to the crossing points with otherlines. The number of possibly distinct configurations of lines we determined above,and for each line that has at most n − 1 crossings with other lines, we have atmost(n2)distinct possibilities to choose the line segment. Thus the number ofdistinct line segment graphs with n labelled vertices is|SEGn| ≤ n6n ·(n2)n.Lemma 2.5. For every integer n, |SPn| ≥ 2(n−1)24 .A note on intersection dimensions of graph classes 259Proof:A⌊n/2⌋ ⌈n/2⌉Figure 1We may divide the vertices {1, 2, . . . , n} into two parts of size ⌊n/2⌋ and ⌈n/2⌉.Then we add all edges inside the first part (see Figure 1), no edge inside the secondpart and an arbitrary choice of edges between them, and we get 2⌊n/2⌋·⌈n/2⌉ ≥2(n−1)24 distinct graphs from SPn. Lemma 2.6. For every integer n, |FUNn| ≥ 2(n−1)24 .Proof: We use nearly the same construction as in Lemma 2.5, the only differenceis that we add all edges inside both parts (and arbitrary choice of edges betweenthem). Constructed graphs are in FUN because their complements are bipartite,hence comparability graphs. Theorem 2.7. For any arbitrary pair A from INT, CA, CI, PER, SEG and Bfrom SP, CHOR, FUNdimAB =∞ .Proof: From Lemma 2.3 and INT ⊆ CA we have log |INTn| ≤ log |CAn| ≤Θ(n logn). Lemma 2.4 implies (we know PER ⊆ CI ⊆ SEG) log |PERn| ≤log |CIn| ≤ log |SEGn| ≤ Θ(n logn). On the other hand, we know from Lemmas2.5, 2.6 that log |CHORn| ≥ log |SPn| ≥ Θ(n2) and also log |FUNn| ≥ Θ(n2).Therefore for arbitrary A from INT, CA, CI, PER, SEG and B from SP, CHOR,FUN we havedimAB = supn∈N,n>1Θ(n2)Θ(n logn)=∞ .3. The CA-dimensionIn this section we give arguments that the CA-dimension of permutation graphsand hence also circle and line segment graphs is unbounded, although these classeshave aproximately the same number of graphs.260 P.Hliněný, A.KuběnaLemma 3.1. Suppose that graphs G, H are such that −G is not bipartite andG ⊕ H ∈ CA. Then H is a complete graph.Proof:AByAxFigure 2Let us suppose that there are two non-adjacent vertices x, y in the graph H .Consider now the CA-representation of the graph G⊕H and denote by A, B twopoints on the circle that lie between arcs representing vertices x, y (see Figure 2).Then for every vertex v ∈ V (G) (which is adjacent to both x and y) the arcrepresenting v must cross the point A or the point B, and we can divide thevertices of G into two sets (not necessarily disjoint) V (G) = VA∪VB , such that allarcs representing the vertices from VA cross the point A and all arcs representingthe vertices from VB cross the point B. But this means that sets VA, VB areindependent in −G and hence the graph −G is bipartite, which is a contradiction.Lemma 3.2. Let F be a non-complete graph such that dimCAF ≥ d, and letn = 2d + 1. Then dimCAF ⊕ Dn ≥ d+ 1.Proof: Let us denote V = V (F ), G = F ⊕ DW where |W | = n, then V (G) =V ∪ W , V ∩ W = ∅. For a contradiction let us suppose that dimCAG ≤ d, i.e.there exist graphs Hi ∈ CA for i = 1, 2, . . . , d such that V (Hi) = V (G) andG =d⋂i=1Hi (for dimCAG < d we add complete graphs). Because G contains alledges between vertices from V andW , each Hi should also contain all these edgesand we can write Hi = H1i ⊕H2i where V (H1i ) = V , V (H2i ) =W andd⋂i=1H1i = F ,d⋂i=1H2i = DW .If there exist j ∈ {1, . . . , d} such that −H2j is not bipartite, we can useLemma 3.1 for the graph Hj = H1j ⊕ H2j and thus H1j is a complete graph,which is not possible when d = 1 and for d > 1 we can writeE(F ) =⋂i∈{1,...,j−1,j+1,...,d}E(H1i ), i.e. dimCAF ≤ d − 1, a contradiction.A note on intersection dimensions of graph classes 261Hence −H2i is bipartite for every i. Butd⋂i=1H2i = DW , therefored⋃i=1−H2i =KW and n = χ(KW ) ≤ χ(−H21 ) · . . . ·χ(−H2d) ≤ 2d = n− 1, a contradiction. Theorem 3.3. dimCAPER = dimCACI = dimCASEG =∞.Proof: Define G1 = D2 and Gd+1 = Gd ⊕ D2d+1 by means of recursion. Sinceevery discrete graph is a permutation graph, and the Zykov sum of two permuta-tion graphs is again a permutation graph, Gd ∈ PER for every d. On the otherhand, it follows by induction from Lemma 3.2 that dimCAGd+1 ≥ d + 1 for ev-ery d. Hence dimCAPER =∞.References[1] Cozzens M.B., Roberts F.S., Computing the boxicity of a graph by covering its complementby cointerval graphs, Discrete Appl. Math. 6 (1983), 217–228.[2] , On dimensional properties of graphs, Graphs and Combinatorics 5 (1989), 29–46.[3] Feinberg R.B., The circular dimension of a graph, Discrete Math. 25 (1979), 27–31.[4] Golumbic M.C., Algorithmic Graph Theory and Perfect Graphs, Academic Press, NewYork, 1980.[5] Goodman J.E., Pollack R., Upper bounds for configurations and polytopes in Rd, DiscreteComputational Geometry 1 (1986), 219–227.[6] Janson S., Kratochv́ıl J., Thresholds for classes of intersection graphs, Discrete Math. 108(1992), 307–326.[7] Kratochv́ıl J., Matoušek J., Intersection graphs of segments, J. Combin. Theory Ser. B 62(1994), 289–315.[8] Kratochv́ıl J., Tuza Z., Intersection dimensions of graph classes, Graphs and Combinatorics10 (1994), 159–168.Department of Applied Mathematics, Faculty of Mathematics and Physics, CharlesUniversity, Malostranské nám. 25, 118 00 Praha 1, Czech Republic(Received October 25, 1994)

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A note on intersection dimensions of graph classes

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Comment.Math.Univ.Carolinae 36,2 (1995)255{261 255

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Institute of Mathematics AS CR, v. v. i.