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 graph G is defined as 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 3-trees, which form a well-studied subclass of planar graphs. Prior to this work, it was known that the queue number of planar 3-trees is at most seven. In this work, we improve this upper bound to five. We also show that there exist planar 3-trees whose queue number is at least four. Notably, this is the first example of a planar graph with queue number greater than three.
Highlights
In a queue layout [19], the vertices of a graph are restricted to a line and its edges are drawn at different half-planes delimited by this line, called queues
The queue number of a graph is the smallest number of queues that are required by any queue layout of the graph
The most remarkable result in this area is due to Dujmović et al [9], who recently proved that planar graphs have constant queue number improving upon a series of results [2, 5, 7] and settling in the positive an old conjecture by Heath, Leighton and Rosenberg [18]
Summary
In a queue layout [19], the vertices of a graph are restricted to a line and its edges are drawn at different half-planes delimited by this line, called queues. The most remarkable result in this area is due to Dujmović et al [9], who recently proved that planar graphs have constant queue number (by the time of writing at most 49) improving upon a series of results [2, 5, 7] and settling in the positive an old conjecture by Heath, Leighton and Rosenberg [18]. It is worth noting that the authors of [9] use our main result to obtain the bound on the queue number of planar graphs, as paper [9] appeared after the conference version [1] of this work
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