Genus, Treewidth, and Local Crossing Number

Abstract

We consider relations between the size, treewidth, and local crossing number (maximum crossings per edge) of graphs embedded on topological surfaces. We show that an n-vertex graph embedded on a surface of genus g with at most k crossings per edge has treewidth O( √ gkn) and layered treewidth O(gk), and that these bounds are tight up to a constant factor. As a special case, the k-planar graphs with n vertices have treewidth O( √ kn) and layered treewidth O(k), which are tight bounds that improve a previously known O(k 3/4 n 1/2 ) treewidth bound. Additionally, we show that for g < m, every m-edge graph can be embedded on a surface of genus g with O((m/g) log 2 g) crossings per edge, which is tight to within a polylogarithmic factor.

Crossing Patterns in Nonplanar Road Networks

“…However, there is no reason to expect any particular forbidden minors in road networks. Among the many generalizations of planar graphs that have been studied in the graph theory literature, another one seems more promising as a model for road networks: the 1-planar graphs [26] or more generally k-planar graphs [6,15,25]. A 1-planar graph is a graph in which every road segment has at most one crossing ( Figure 1).…”

“…Many of the sparsity properties of these graphs follow directly from planarization: if one replaces each crossing with a vertex, one obtains a planar graph in which the number of vertices has been blown up only by a factor of O(k). Based on this principle, it is known that these form sparse families of graphs, and in particular they obey a separator theorem like that for planar graphs but with a dependence on k as well as on the number of vertices in the size of the separator [6]. Although kplanar graphs are NP-hard to recognize from their graph structure alone [2,15], that is not problematic for their application to road networks, because in this case an embedding with few crossings would already be known: the actual embedding of the roads on the surface of the earth.…”

Stack and Queue Layouts via Layered Separators

“…The family of (g, k)-planar graphs is not closed under taking minors 1 even for g = 0, k = 1; thus the result of Blankenship and Oporowski [4,5], stating that proper minor-closed graph families have bounded stack-number, does not apply to (g, k)-planar graphs. Dujmović et al [12] showed that (g, k)-planar graphs have layered (4g + 6)(k + 1)-separators 2 . This and our Theorem 1 imply the following corollary.…”

“…The (g, 3)-map graphs are the graphs of Euler genus at most g [8], thus they are closed under taking minors. However, for every g ≥ 0 and d ≥ 4, the (g, d)-map graphs are not closed under taking minors [12], thus the result of Blankenship and Oporowski [4,5] does not apply to them. The (g, d)-map graphs have layered (2g +3)(2d+1)separators [12].…”

“…However, for every g ≥ 0 and d ≥ 4, the (g, d)-map graphs are not closed under taking minors [12], thus the result of Blankenship and Oporowski [4,5] does not apply to them. The (g, d)-map graphs have layered (2g +3)(2d+1)separators [12]. This and our Theorem 1 imply the following corollary.…”