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On Graph Thickness, Geometric Thickness, and Separator Theorems

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On Graph Thickness, Geometric Thickness, and Separator Theorems CCCG 2009, Vancouver, BC, August 17{19, 2009 On Graph Thickness, Geometric Thickness, and Separator Theorems Christian A. Duncan∗ Abstract fined to be the smallest number of forests whose union is G [25, 26, 27]. In other words, the minimum number We investigate the relationship between geometric of edge colors such that the subgraph induced by each thickness and the thickness, outerthickness, and ar- color is a forest of trees. In linear arboricity the result- boricity of graphs. In particular, we prove that all ing colored subgraphs must be collections of paths [2, 3]. graphs with arboricity two or outerthickness two have In outerthickness, the resulting colored subgraphs must geometric thickness O(log n). The technique used can be outerplanar graphs [16, 17]. Outerplanar graphs are be extended to other classes of graphs so long as a stan- analogous to graphs with book thickness one; however, dard separator theorem exists. For example, we can outerthickness and book thickness are not identical. apply it to show the known boundp that thickness two Another way to look at the problem is to divide the graphs have geometric thickness O( n), yielding a sim- edges of the graphs into different layers and draw the ple construction in the process. layers independently as planar graphs such that the ver- tex positions are identical in each layer. In this case, 1 Introduction the problem is to minimize the number of layers. A common related application of this problem is in VLSI In many applications of graph visualization and graph design where wires are placed in various layers to avoid theory, it is often useful to draw the edges of a graph crossings, see for example [1]. Minimizing the number with multiple colors or in multiple layers. The gen- of layers reduces cost and improves performance of the eral class of thickness problems deals with determining created circuits. the minimum number of these colors needed under vari- Another related problem is the area of simultaneous ous conditions. Traditionally, the (graph) thickness of a embeddings. In simultaneous embedding problems, the graph G is defined to be the minimal number of planar edges are already assigned to various layers, and one subgraphs whose union forms G. From an edge coloring must determine a placement of vertices to realize each perspective, we can also define the thickness to be the drawing without crossings, if possible [9]. smallest number of edge colors needed so that we can draw the graph in the plane with no intersections be- 1.1 Related Work tween two edges having the same color. In this variant, the only constraint on how a graph's edges are drawn Our work is motivated by and related to recent results is continuity. Essentially, the thickness of a graph is in characterizing the geometric thickness of graphs. In a minimum edge coloring of the graph such that each particular, Eppstein [14] characterizes a class of graphs color represents a planar graph. Initially, the notion of having arboricity three and thickness three, for which thickness derived from early work on biplanar graphs, the geometric thickness grows as a function of n, the graphs with thickness two [18, 20, 28]. Generalized by number of vertices. The proof relies on Ramsey theory Tutte [29], the research in graph thickness problems is and so the lower bound on the geometric thickness is too rich to summarize here. The interested reader is a very slow growing function of n. Eppstein [13] also referred to the survey by Mutzel et al. [24]. characterizes a class of graphs with arboricity two and By adding the constraint that all edges must be rep- geometric thickness two having a book thickness that resented by straight-line segments, we arrive at the ge- grows as another slow function of n. ometric thickness problem [10, 21]. If we further con- Using results on the simultaneous embedding of two strain the problem such that the vertices must lie in con- paths [9], Duncan et al. [12] proved that graphs with vex position, the edge coloring number is known as the linear arboricity two have geometric thickness two. book thickness [8]. Although the values of graph, geo- Arboricity-two graphs are not as simple. Geyer et metric, and book thickness are related, there are graphs al. [15] show two trees which cannot be embedded si- that have different values for each [13, 14]. multaneously. In the context of geometric thickness, We can also constrain the graph thickness in other this would imply that one cannot simply take a graph ways. For example, the arboricity of a graph G is de- of arboricity two, decompose it into two forests arbi- ∗Department of Computer Science, Louisiana Tech University, trarily, and then embed the two graphs simultaneously. Ruston, LA 71270, USA, [email protected] However, because the union of the two trees described 21st Canadian Conference on Computational Geometry, 2009 does have geometric thickness two, it is still in fact open that states for any graph G 2 G with n = jV (G)j > c, as to whether arboricity two implies geometric thickness there exists a cut set C such that G1;G2 2 G and jVij ≤ two. f(n) for i = 1; 2 and jCj ≤ g(n). Our primary result In [23], Malitz characterizesp the book thickness of uses the following key lemma: graphs with E edges to be O( E). This result imme- Lemma 1 Let G be a graph in G with n vertices. There diately impliesp that thickness-t graphs have geometric thickness O( tn). There has also been work on charac- exists an assignment of colors to e 2 E(G) in the range terizing the geometric thickness of a graph in terms of 1 to F(n), and unique x-coordinate values to v 2 V (G), its degree. In particular, graphs with degree less than in the range 1 to n, such that for any assignment of y- two trivially have geometric thickness one, graphs with coordinates to v, with all vertex points being in general degree three and four have geometric thickness two [12], position, no two edges with the same color assignment intersect, except at common endpoints, when drawn as and there exist graphsp with degree δ ≥ 9 having geomet- ric thickness at least c δn1=2−4/δ− for sufficiently large straight-line segments from their respective endpoints. n and constants c and > 0 [7]. Here F(n) is defined by the following recurrence rela- Dujmovi´cand Wood [11] discuss the relationship be- tion: tween geometric thickness and graph treewidth. In par- c if n ≤ c F(n) = ticular, they show that graphs with treewidth k have ge- F(f(n)) + g(n) otherwise. ometric thickness at most dk=2e. This is complementary to our work as even planar graphs can have arbitrarily Proof. In our arguments, we shall also color each ver- large treewidth. For example, the n × n grid graph has tex as c(v) so that an edge e = (u; v) 2 E(G) has color treewidth n. However, the treewidth is known for many either c(u) or c(v). types of graphs. We prove this lemma inductively. If n ≤ c, we simply (arbitrarily) assign the vertices v 2 V (G) with increas- 1.2 Our Results ing x-coordinates from 1 to n and the colors c(v) as 1 to n. We then assign each edge (u; v) 2 G with u; v 2 V (G) In this paper, we provide further analysis into the re- as color c(u), the choice of u or v is arbitrary. This pro- lationship between geometric thickness and thickness. cess requires the assignment of at most c colors. Since In particular, we show that graphs with arboricity two edges with similar colors also share a common endpoint, and outerthickness two have a geometric thickness of the only color crossings possible are between adjacent O(log n). We show these by providing a more gen- edges but if the points are in general position, the only eralized solution for graphs that can be decomposed intersection is at the endpoints. Therefore, our lemma into two subgraphs having some separation property. holds for the base case. This allows for further relations between the two thick- Assume now that the lemma holds for all graphs with ness measures. For example, we can also show that size less than n. Let G have size n. Since G belongs to graphsp with thickness two have a geometric thickness of the class G with the separator property, we can partition O( n), which is also immediately implied by the results V (G) into the three sets V1, V2, C. Let n1 = jV1j, from [23]. Additionally, if the graph can be decomposed n2 = jV2j, nc = jCj. into two Kh-minor free graphs, then the graph has ge- We then compute color and x-coordinate assignments 3=2p ometric thickness O(h n). separately for G1 and G2. From our inductive assump- tion, both G1 and G2 can be assigned values indepen- 2 Using a Separator Theorem dently without any invalid crossings in their respective graphs. A cut set for a graph G = (V; E) is a set of vertices To combine the two assignments and provide assign- C ⊂ V (G) such that the subgraph of G induced by ments for the remaining vertices and edges, we proceed the removal of the vertices in C consists of at least two as follows. Let V1 have x-coordinate assignments rang- connected components G1 and G2. Note, if the cut set ing from 1 to n1 and V2 have x-coordinate assignments produces more than two connected components, we can ranging from 1 to n2.
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