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Upper Bound Constructions for Untangling Planar Geometric Graphs∗

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Upper Bound Constructions for Untangling Planar Geometric Graphs∗ Upper bound constructions for untangling planar geometric graphs∗ Javier Canoy Csaba D. T´othz Jorge Urrutiax Abstract For every n 2 N, we construct an n-vertex planar graph G = (V; E), and n distinct points p(v), v 2 V , in the plane such that in any crossing-free straight-line drawing of G, at most :4948 O(pn ) vertices v 2 V are embedded at points p(v). This improves on an earlier bound of O( n) by Goaoc et al. [10]. 1 Introduction V A graph G = (V; E) is defined by a vertex set V and an edge set E ⊆ 2 .A straight-line 2 drawing of a graph G = (V; E) is a mapping p : V ! R of the vertices into distinct points in the plane, which induces a mapping of the edges fu; vg 2 E to line segments p(u)p(v) between the corresponding points. A straight-line drawing is crossing-free if no two edges intersect, except perhaps at a common endpoint. By F´ary'stheorem [9], a graph is planar if and only if it admits a crossing-free straight-line drawing. 2 Suppose we are given a planar graph G = (V; E) and a straight-line drawing p : V ! R (in which some edges may cross each other). Since G is planar, we can obtain a crossing-free straight- 0 2 line drawing p : V ! R by moving some of the vertices to new positions in the plane. The process 2 0 2 of changing a drawing p : V ! R to a crossing-free drawing p : V ! R is called an untangling 0 of (G; p). A vertex v 2 V is fixed in the untangling if p(v) = p (v). Denote by f(n), n 2 N, the maximum integer such that every straight-line drawing of every n-vertex planar graph can be untangled while keeping at least f(n) vertices fixed. In this paper, we study the asymptotic growth rate of f(n). The first question on untangling planar graphs was posed by Mamoru Watanabe in 1998: Is it true that every polygon P with n vertices can be untangled in at most n steps, for some absolute constant < 1, where in each step, we move a vertex of P to a new location? Pach and Tardos [16] n gave a negative answer to Watanabe's question. They showed that every polygonp with vertices (i.e., the straight-line drawing of the cycle Cn) can be untangled in at most n − n moves, and there are n-vertex polygons where no more than O((n log n)2=3) vertices can be fixed. Recently, Cibulka [6] proved that every n-vertex polygon can be untangled such that Ω(n2=3) vertices are fixed. et al. f n ≤ p The problem of untangling planar graphs was studied by Goaoc [10]. They proved ( ) n + 2 by constructing drawings of the planar graph P2 ∗ Pn−2 with n vertices such that any ∗Preliminary results have been presented at the 19th Symposium on Graph Drawing (Eindhoven, 2011) [4]. yPosgrado en Ciencia e Ingenier´ıade la Computaci´on,Universidad Nacional Aut´onomade M´exico,D.F. M´exico, j [email protected] zDepartment of Mathematics, California State University Northridge, Los Angles, CA, and Department of Com- puter Science, Tufts University, Medford, MA, USA, [email protected] xInstituto de Matem´aticas,Universidad Nacional Aut´onomade M´exico,D.F. M´exico, [email protected] 1 p untangling fixes at most n + 2 vertices. Here Pk denotes a path with k vertices; and for two graphs, G and H, the join G ∗ H consists of the vertex-disjoint union of G and H and all edges V G V H et al. et al. between ( ) and ( ), see Fig. 1. Bose [3], Kang [15] and Ravsky and Verbitskyp [17] explored drawings of several families of n-vertex planar graphs in which any untangling fixes O( n) vertices. Cibulka [6] showed that every 3-connected planar graph with n vertices admits a drawing for which any untangling fixes O((n log n)2=3) vertices. Bose et al. [3] devised an algorithm that untangles every geometric graph with n vertices while n= 1=4 f n ≥ n= 1=4 fixing ( 3) vertices, which proves ( ) ( 3) . p In this paper, we improve the upper bound for f(n) from O( n) to O(n:4948). We construct an n-vertex planar graph G = (V; E) and arrange the vertices in the plane by an injective mapping 2 2=(7−3λ)+" p : V ! R such that any untangling of (G; p) fixes O(n ) vertices for every " > 0, where λ is the shortness exponent of the family of 3-connected cubic planar graphs (i.e., polyhedral graphs). The exact value of the shortness exponent λ is not known. The currently known best upper bound is λ ≤ log23 22 ≈ 0:9858 by Gr¨unbaum and Walther [11]. Any improvement on the upper bound for λ would immediately improve our upper bound for f(n). Our argument crucially depends on a correspondence between the shortness exponent of cubic polyhedral graphs and crossing-free straight-line drawings of the dual graph in which a line stabs many faces. We define the stabbing number stab(G) of a polyhedral graph G = (V; E) to be the maximum number of faces that intersect a line L in any crossing-free straight-line drawing of G. We prove that stab(G) is the size of the maximum cycle in the dual graph G∗. Previously, Biedl et al. [1] proved a similar but weaker result for polyline drawings (rather than straight-line drawings). Organization. In Section 2, we discuss two key ingredients of our construction: (i) the shortness exponent of cubic polyhedral graphs, and (ii) permutations with certain special properties related to the Erd}os-Szekeres Theorem. In Section 3, we present a family of planar geometric graphs and prove f(n) 2 O(n2=(7−3 log23 22)+"). We conclude in Section 4 by establishing a correspondence between the shortness exponent of cubic polyhedral graphs and the stabbing number of triangulations. 2 Preliminaries Dual graphs of triangulations. The value of f(n) is attained for edge-maximal planar graphs with n vertices, since by augmenting a planar graph with new edges, the set of its crossing-free straight-line drawings decreases or remains the same. The edge-maximal planar graphs are called triangulations. By Euler's formula, a triangulation with n ≥ 3 vertices has exactly 3n−6 edges and 2n − 4 faces (including the outer face). Note that in every crossing-free drawing of a triangulation, every face (including the outer face) is bounded by three edges. It follows that every triangulation with n ≥ 4 vertices is 3-connected [7][Lemma 4.4.5]. The 3-connected planar graphs are also called polyhedral graphs. By Whiteley's theorem, every polyhedral graph has a topologically unique crossing-free drawing, apart from the choice of the outer face. More precisely, in every crossing-free drawing of a 3-connected graph G, the face boundaries are precisely the nonseparating chordless cycles of G [7][Proposition 4.2.7]. Hence, G has a well-defined dual graph G∗ (independent of the crossing-free drawings of G): the vertices of G∗ correspond to the faces (i.e., nonseparating chordless cycles) of G, and two vertices of G∗ are adjacent if and only if the corresponding faces share an edge. If G is a triangulation with n ≥ 4 vertices, then G∗ is a cubic polyhedral graph with 2n − 4 nodes and 3n − 6 edges. 2 Stabbing triangulations and dual cycles. The following observation is crucial for our con- struction. Observation 1 Let T be a polyhedral graph. Suppose that a line L stabs the faces f1; : : : ; fk (in this order) in a crossing-free straight-line drawing of T , and these faces correspond to the vertices ∗ ∗ ∗ ∗ ∗ ∗ f1 ; : : : ; fk , respectively, in the dual graph T . Then (f1 ; : : : ; fk ) is a simple cycle in T . In Section 3, we will construct a planar graph G from two triangulations, S and T . Specifically, we plug a copy of S in each face of T . We then draw G in the plane such that the vertices of every copy of S lie on a line L. If the dual graph T ∗ is not Hamiltonian, then in any crossing-free straight-line drawing of G, the line L will miss at least one face of T . If L misses a face f of T , then none of the vertices can be fixed in the copy of S plugged into f. In the next few paragraphs, we review the currently known best bounds on the maximum cycles in the dual graphs of triangulations. In Section 4, we establish a somewhat surprising converse of Observation 1, and show that ∗ ∗ ∗ if (f1 ; : : : ; fk ) is a simple cycle in the dual graph T of a polyhedral graph T , then T admits a crossing-free straight-line drawing in which a line L stabs the corresponding faces f1; : : : ; fk of T in this order. Maximum cycles in cubic polyhedral graphs. In an attempt at proving the Four Color Theorem, Tait [18] conjectured in 1884 that every cubic polyhedral graph is Hamiltonian. In 1946, Tutte [20] found a counterexample with 44 vertices. The smallest known counterexample, due to Barnette, Bos´ak,and Lederberg, has 38 vertices, and it is known that there is no counterexample with 36 or fewer vertices [12]. Using the smallest known counterexample to Tait's conjecture, one can build a cubic polyhedral graph with Θ(n) vertices for every n 2 N in which every cycle has at most O(nlog37 36) ⊂ O(n0:9925) vertices.
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