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Non-Existence of Annular Separators in Geometric Graphs

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Non-Existence of Annular Separators in Geometric Graphs Non-existence of annular separators in geometric graphs Farzam Ebrahimnejad∗ James R. Lee† Paul G. Allen School of Computer Science & Engineering University of Washington Abstract Benjamini and Papasoglou (2011) showed that planar graphs with uniform polynomial volume growth admit 1-dimensional annular separators: The vertices at graph distance ' from any vertex can be separated from those at distance 2' by removing at most $ ' vertices. They asked whether geometric 3-dimensional graphs with uniform polynomial volume¹ º growth similarly admit 3 1 -dimensional annular separators when 3 7 2. We show that this fails in a strong sense: For¹ any− 3º > 3 and every B > 1, there is a collection of interior-disjoint spheres in R3 whose tangency graph has uniform polynomial growth, but such that all annular separators in have cardinality at least 'B. 1 Introduction The well-known Lipton-Tarjan separator theorem [LT79] asserts that any =-vertex planar graph has a balanced separator with $ p= vertices. By the Koebe-Andreev-Thurston circle packing ¹ º theorem, every planar graph can be realized as the tangency graph of interior-disjoint circles in the plane. One can define 3-dimensional geometric graphs by analogy: Take a collection of “almost 3 non-overlapping” bodies (E R : E + , where each (E is “almost round,” and the associated f ⊆ 2 g geometric graph contains an edge D,E if (D and (E “almost touch.” f g As a prototypical example, suppose we require that every point G R3 is contained in at most : 2 of the bodies (E , each (E is a Euclidean ball, and two bodies are considered adjacent whenever f g (D (E < . These are precisely the intersection graphs of :-ply systems of balls, studied by \ ; Miller, Teng, Thurston, and Vavasis [MTTV97]. Those authors also provide a generalization of the Lipton-Tarjan separator theorem: For : = $ 1 , such an intersection graph contains a balanced arXiv:2107.09790v1 [math.CO] 20 Jul 2021 ¹ º separator of size $ =1 1 3 . ¹ − / º Similarly, finite-element graphs associated to simplicial complexes with bounded aspect ratio can be viewed as subgraphs of geometric overlap graphs [MTTV98], and one again obtains balanced separators of size $ =1 1 3 . This covers a number of scenarios commonly arising in applications of ¹ − / º the finite-element method; we refer to the discussion of well-shaped meshes in [ST07, §6.2]. ∗[email protected] †[email protected] 1 We will not be too concerned with the particular notion of geometric graph used since our construction satisfies all these commonly employed sets of assumptions. Indeed, it can be cast as the tangency graph of a sphere packing, where adjacent spheres have uniformly comparable radii. It can also be cast as the 1-skeleton of a 3-dimensional simplicial complex whose simplices have uniformly bounded aspect ratio as studied. Such graphs were studied by, for instance, by Plotkin, Rao, and Smith [PRS94] in their work on shallow minors (see also the followup work [Ten98]). Annular separators. Note that the preceding results deal with global separators that separate the entire graph into two roughly equal pieces. In many settings, especially those arising in physical simulation, it useful to consider local separators. Let = +, be an undirected graph with path ¹ º metric 3, and define graph balls and graph spheres, respectively, by G,' := H + : 3 G,H 6 ' ¹ º 2 ¹ º ( G,' := H + : 3 G,H = ' . ¹ º 2 ¹ º Suppose that for some 7 1, we we want to separate ( G,' from ( G, ' by removing a ¹ º ¹ º small set of nodes * G, ' G,' . We refer to * as an annular separator. See Figure 1. ⊆ ¹ º n ¹ º (a) Separating ( G,' from ( G, 2' (b) A random triangulation1 of S2 ¹ º ¹ º Figure 1: Annular separators It is easy to see that even if is a planar graph, small annular separators don’t necessary exist. But in many cases, one can find annular separators of size $ ' . For instance, this is true for most ¹ º vertices at most scales in a uniformly random triangulation of the 2-dimensional sphere [Kri05] (a fact which extends experimentally to a variety of other models of random planar maps, e.g., those studied in [GHS20]). These models also have the properties that the cardinality of graph balls tends : to grow asymptotically like G,' ' (as ' , up to lower-order fluctuations), where the j ¹ ºj ∼ ! 1 exponent : depends on the model. For random triangulations, one has : = 4 [Ang03]. 1Depiction of a random triangulation is due to Nicolas Curien. 2 Benjamini and Papasoglou [BP11] give an explanation of this phenomenon as follows. Suppose that is an infinite planar graph and we assume, additionally, that has uniform polynomial growth of degree :: There exist numbers , : > 1 such that 1 : : − ' 6 G,' 6 ' , G +, ' > 1. j ¹ ºj 8 2 Then for every G + and ' > 1, there is a set * G, 2' G,' whose removal 2 ¹ º ⊆ ¹ º n ¹ º disconnects ( G,' from ( G, 2' in , and such that * 6 $ ' . This applies equally well to ¹ º ¹ º j j ¹ º finite graphs: Indeed, the authors actually show that if the graph metric restricted to G, 2' has ¹ º doubling constant , then one can find an annular separator of size at most '. We remark that there are rich families of planar graphs with uniform polynomial growth arising in a variety of contexts; see [BS01, BK02, EL20]. Indeed, one can obtain planar graphs with uniform polynomial growth of degree : for all real degrees : 7 1. Moreover, many models of random planar graphs have an almost sure asymptotic version of this property [DG20, GHS20]. The authors of [BP11] asked whether an analog of this phenomenon holds in higher dimensions. For instance, if is a graph with uniform polynomial growth that can be geometrically represented in R3, does it hold that has annular separators of size $ '2 ? We give examples showing that for ¹ º 3 > 3, this phenomenon fails in a strong way. Say that a graph is sphere-packed in R3 if is the tangency graph of a collection of interior- disjoint spheres in R3. Say that is "-uniformly sphere-packed in R3 if the collection of spheres can be taken such that the radii of any two tangent spheres lies in the interval " 1," , and that is » − ¼ uniformly sphere-packed in R3 if this holds for some " 5 . 1 Theorem 1.1 (Arbitrarily large annular separators). For every 3 > 3 and B > 1, there is a number 2 7 0 and a graph satisfying: 1. has uniform polynomial growth of degree $ B . ¹ º 2. is uniformly sphere-packed in R3. B 3. For every G + , there are at least 2' vertex-disjoint paths from ( G,' to ( G,' for any 2 ¹ º ¹ º ¹ 0º '0 7 ' > 1. Clearly if has uniform polynomial growth of degree 3, then one of the '-many spheres ( G,' 1 ,( G,' 2 ...,( G, 2' must be an annular separator of size $ '3 1 . We show that ¹ ¸ º ¹ ¸ º ¹ º ¹ − º the moment the growth degree exceeds 3, there are graphs sphere-packed in R3 that don’t have 3 1 -dimensional annular separators. ¹ − º Theorem 1.2 (Nearly-dimensional growth rate). For every 3 > 3 and 7 0, there are numbers 2, 7 0 and a graph satisfying: 1. has uniform polynomial growth of degree at most 3 . ¸ 2. is uniformly sphere-packed in R3. 3 1 3. For every G + , there are at least 2' vertex-disjoint paths from ( G,' to ( G,' for 2 ¹ º ¹ − º¸ ¹ º ¹ 0º any '0 7 ' > 1. 3 Note that the two preceding theorems refer to infinite graphs. A version for families of finite graphs appears in Theorem 2.7. Preliminaries. We will consider primarily connected, undirected graphs = +, , which we ¹ º equip with the associated path metric 3. We write + and , respectively, for the vertex and ¹ º ¹ º edge sets of . If * + , we write * for the subgraph induced on *. ⊆ ¹ º » ¼ For E +, let deg E denote the degree of E in . Let diam := sup 3 G,H denote 2 ¹ º ¹ º G,H + ¹ º the diameter of (which is only finite for finite and connected), and for a subset2 ( +, denote ⊆ diam ( := supG,H ( 3 G,H . For E + and A > 0, we use E,A = D + : 3 D,E 6 A to ¹ º 2 ¹ º 2 ¹ º f 2 ¹ º g denote the closed ball in . For subsets (, ) +, we write 3 (, ) := inf 3 B,C : B (, C ) . ⊆ ¹ º f ¹ º 2 2 g For two expressions and , we use the notation . to denote that 6 for some universal constant . The notation . ,, denotes that 6 , , where , , ··· ¹ · · · º ¹ · · · º denotes a number depending only on the parameters , , etc. We write for the conjunction . . ^ 2 Tilings of the unit cube Fix the dimension 3 > 2. Our constructions are based on tilings of subsets of R3 by axis-parallel hyperrectangles, a generalization of the planar constructions in [EL20]. A 3-dimensional tile is an 3 axis-parallel closed hyperrectangle R , i.e., a set of the form = 01, 11 02, 12 0 , 1 ⊆ » ¼ × » ¼ × · · · × » 3 3¼ for numbers satisfying 08 5 18 for each 8 = 1, 2, . , 3. We will encode such a tile as a 3 1 -tuple ? , ℓ1 , ℓ2 , . , ℓ , where ? := ¹ ¸ º ¹ ¹ º ¹ º ¹ º 3¹ ºº ¹ º 01, 02, . , 03 and ℓ8 := 18 08 is the length of the projection of along the 8th axis. ¹ º ¹ º − Ð 3 A tiling Z is a finite collection of interior-disjoint tiles. Denote Z := Z . If ' R , we say 2 ⊆ that Z is a tiling of ' if Z = '. We associate to a tiling its tangencyJ graphK Z with vertex set Z and ¹ º with an edge betweenJ twoK tiles , Z whenever has non-zero 3 1 -dimensional volume.
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