Queue Layouts of Planar 3-Trees

Abstract

A queue layout of a graph G consists of a linear order of the vertices of G and a partition of the edges of G into queues, so that no two independent edges of the same queue are nested. The queue number of G is the minimum number of queues required by any queue layout of G.In this paper, we continue the study of the queue number of planar 3trees. As opposed to general planar graphs, whose queue number is not known to be bounded by a constant, the queue number of planar 3-trees has been shown to be at most seven. In this work, we improve the upper bound to five. We also show that there exist planar 3-trees, whose queue number is at least four; this is the first example of a planar graph with queue number greater than three.

Planar Graphs of Bounded Degree Have Bounded Queue Number

“…Hence, mv(v 2 ) − mv(v 1 ) = |M u2,v2 | = (∆ − 1) , i.e., the difference between the minimum nesting-value of edges in M u2,v2 and the minimum nesting-value of edges in M u1,v1 is (∆ − 1) . This however, by (1) gives g(v 2 ) = g(v 1 ) + 1, which contradicts the fact that v 2 ≺ v 1 according to Condition C.2. So far, we constructed a queue layout L 1 of graph G 1 with 2∆ − 2 queues.…”

“…Moreover, the set of matching edges in G 2 incident to the leaves of T u is given by M u,v , and the same holds for T v . This implies that mv min{nesting-value of e | e ∈ M u,v } and consequently g(u) = g(v) by (1). Now suppose for the sake of contradiction that level edge (…”

“…The best-known upper bound is due to Dujmović [10], who showed that the queue number of planar graphs on n vertices is O(log n) improving upon a previous bound by Di Battista et al [7]; recently, Bannister et al [2] improved the constant factor in the result of Dujmović [10]. On the other hand, the best-known lower bound is due to a family of planar 3-trees that require four queues [1].…”

“…Trees are arched-level planar and therefore have queue number one [25]. Outerplanar graphs have queue number at most two [24], Halin graphs and series-parallel graphs have queue number at most three [23,30], and planar 3-trees have queue number at most five [1]. However, it is still unknown whether the queue number of planar graphs is bounded (see also [6]).…”

On the Queue Number of Planar Graphs

“…1c. Since their introduction, queue layouts of graphs have been a fruitful subject of intense research with several important milestones over the years [1,2,4,13,16,20,21].…”