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Linear Size Universal Point Sets for Classes of Planar Graphs⋆

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Linear Size Universal Point Sets for Classes of Planar Graphs⋆ Lines: 400 Linear size universal point sets for classes of planar graphs? Stefan Felsner[0000−0002−6150−1998], Hendrik Schrezenmaier[0000−0002−1671−9314], Felix Schr¨oder[0000−0001−8563−3517], and Raphael Steiner[0000−0002−4234−6136] Institute of Mathematics, Technische Universit¨atBerlin, Germany ffelsner,fschroed,schrezen,[email protected] 2 1 Abstract. A finite point set P ⊆ R is n-universal with respect to a 2 class G if every n-vertex-graph G 2 G admits a crossing-free straight-line 3 drawing with vertices being placed at points of P . A popular task in 4 graph drawing is to identify small universal point sets. 5 For the class of all planar graphs the best known upper bound on the 6 size of a universal point set is quadratic and the best known lower bound 7 is linear in n. One of the classical results in the area is that every set 8 of n points in general position (no three collinear) is n-universal for out- 9 erplanar graphs. While some other classes are known to admit universal 10 point sets of near linear size, we are not aware of truly linear bounds for 11 interesting classes beyond outerplanar graphs. 12 In this paper we study a specific ordered point set H (the exploding 13 double chain) and show that all planar graphs G on n ≥ 2 vertices which 14 are subgraphs of a planar graph admitting a one-sided Hamiltonian cycle 15 have a straight-line drawing on the initial piece Hn of size 2n − 2 in H. 0 16 Let H be the class of all subgraphs of planar graphs admitting a one- 17 sided Hamiltonian cycle. It had been conjectured that all 4-connected 0 18 triangulations belong to H . While the conjecture has been disproved, it 19 is still true that Hn is n-universal for a large class H of planar graphs. We 20 show that all bipartite plane graphs and all cubic plane graphs belong 0 0 21 to H ⊆ H. To our surprise, however, not all planar 2-trees are in H . Keywords: universal point set · one-sided Hamiltonian · 2-book em- beddings. 22 1 Introduction 2 23 Given a family G of planar graphs and a positive integer n, a point set P ⊆ R is 24 called an n-universal point set for the class G or simply n-universal for G if for 25 every graph G 2 G on n vertices there exists a straight-line crossing-free drawing 26 of G such that every vertex of G is placed at a point of P . It is a widely studied ? This research started at the 4th Workshop within the collaborative DACH project Arrangements and Drawings, February 24{28, 2020, in Malchow. Supported by the German Research Foundation DFG Project FE 340/12-1. 2 S. Felsner et al. 27 and fundamental open problem in geometric graph theory (compare also the 28 entry [14] in the Open Problem Garden) to determine, given a class of graphs G, 29 (the asymptotics of) the minimum size fG(n) of an n-universal point set for G. 30 If G is the class of all planar graphs we simply write f(n) := fG(n). 31 Schnyder [18] showed that for n ≥ 3 the regular (n − 2) × (n − 2)-grid forms 32 an n-universal point set for planar graphs, even if the combinatorial embedding 2 2 33 of the planar graph is prescribed. This shows that f(n) < n = O(n ). Asymp- 34 totically, the quadratic upper bound on f(n) remains the state of the art up 35 until today. However, the multiplicative constant in this bound was improved 1 2 36 in follow-up papers, see [4,5]. The current upper bound is f(n) ≤ 4 n + O(n) 37 by Bannister et al. [4]. For several subclasses G of planar graphs, better upper 38 bounds are known: A classical result by Pach et al. [16] is that every outerplanar 39 n-vertex graph embeds straight-line on any set of n points in general position, 40 and hence fout-pl(n) = n. Near-linear upper bounds of fG(n) = O(n polylog(n)) 41 are known for 2-outerplanar graphs, simply nested graphs, and for the classes 42 of bounded pathwidth [3,4]. Finally, for the class G of planar 3-trees (also 43 known as apollonian networks or stacked triangulations), an upper bound of 3=2 44 fG(n) = O(n log n) has been proved by Fulek and T´oth[12]. 45 As for lower bounds, the trivial bounds n ≤ fG(n) ≤ f(n) hold for all n 2 N 46 and all planar graph classes G. The best lower bound f(n) ≥ 1:293n − o(n) 47 from [17] has been shown using planar 3-trees, we refer to [6,13,8,9] for earlier 48 work on lower bounds. 49 It seems that in order to improve the quadratic upper bound on f(n) to 2 50 o(n ), the considered point sets should be not too uniformly distributed. Indeed, 51 Choi, Chrobak and Costello [7] recently proved that point sets chosen uniformly 2 52 at random from the unit square must have size Ω(n ) in order to be universal 53 for n-vertex planar graphs, with high probability. 54 In this paper we study a specific ordered point set H (the exploding double 55 chain) and let Hn be the initial piece of size 2n−2 in H (for n ≥ 2). Throughout 56 the paper, denote by H be the class of all planar graphs G which have a plane 57 straight line drawing on the point set Hn with n = jV (G)j. That is, Hn forms 58 an n-universal point set for H. 59 A graph is a POSH (partial one-sided Hamiltonian) if it is a spanning sub- 60 graph of a planar graph admitting a one-sided Hamiltonian cycle. Our main 0 61 result (Theorem 1) is that every POSH is in H. We let H = fG : G is POSHg. 0 62 Theorem 1 motivates the study of H . This class of planar graphs seems 63 to be quite large, e.g., the smallest 4-connected triangulation which is known 64 not to be POSH has 113 vertices [2]. On the positive side we show that every 65 bipartite plane graph is POSH (Section 3). In Section 4 we use the construction 66 for bipartite graphs to show that cubic plane graphs are POSH. Section 5 has a 67 negative result, namely that not all 2-trees are POSH. We conclude with some 68 conjectures and open problems. Linear size universal point sets 3 69 2 The point set and the embedding strategy 0 70 In this section we introduce an ordered point point set H and a class H of 71 planar graphs and show that for every n ≥ 2 the initial part Hn of size 2n − 2 0 72 is n-universal for the class H . 73 A sequence Y = (yi)i≥0 of real numbers satisfying y1 = 0, y2 = 0,p and 74 yi+1 > 2yi + yi−1 for all i ≥ 2 is called exploding. Note that if α > 1 + 2, i−3 75 then y1 = y2 = 0 and yi = α for i ≥ 3 is an exploding sequence. Given an 76 exploding sequence Y let P (Y ) = (pi)i≥0 be the set of points with pi = (i; yi) ¯ 77 and let P (Y ) = (qi)i≥0 be the set of points with qi = (i; −yi), i.e., the reflected ¯ 78 point set, and note that p1 = q1 and p2 = q2. Let H(Y ) = P (Y ) [ P (Y ) and 79 Hn(Y ) = fpi; qij1 ≤ i ≤ ng so that jHn(Y )j = 2n−2. Figure 1 illustrates H6(Y ). Let H = H(Y ) for some exploding sequence Y . For two points p and q let H(p; q) be the set of points of H in the open right half-plane of the directed line −!pq. Note that1 8 (p ) [ (p ) [ (q ) if i > j <> k k≤j k k>i ` `<j H(pi; qj) = (pk)k<i [ (q`)`<i if i = j > :(pk)k<i [ (q`)`≤i [ (q`)`>j if i < j 83 Moreover, if i < j then H(qi; qj) = H(pi; qj) n fqig and if i > j then H(pi; pj) = 84 H(pi; qj) n fpjg. These sidedness informations characterize the order type of H. 85 A point set A = fpi; qiji ≥ 1g is an exploding double chain if it has the order 86 type of H. p6 q6 p5 q5 p4 q4 p3 q3 x p2 q2 y p1 q1 87 Fig. 1. An example of a point set H6 in a rotated coordinate system. 88 A plane graph G has a one-sided Hamiltonian cycle with reverse edge vu if 89 it has a Hamiltonian cycle (v1; v2; : : : ; vn) such that u = v1, v = vn and uv is 90 incident to the outer face. Moreover, for every j = 2; : : : ; n in the restriction 91 G[v1; : : : ; vj; vj+1] of G the edges vj−1vj and vj+1vj are consecutive in the rota- 92 tion of vj. A more pictural reformulation of the second condition is that in the 1 80 In cases where i or j are in f1; 2g the following may list one of the two points defining 81 the halfspace with its second name as member of the halfspace. For correctness such 82 listings have to be ignored.
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