Bekos et al.  recently proved that planar graphs with bounded (maximum) degree have bounded queue-number. We extend this result by showing that graphs with bounded degree and bounded genus have bounded queue-number.
First we introduce queue layouts and give the background to the above results. For a graph and integer , a -queue layout of consists of a linear ordering of and a partition of , such that for , no two edges in are nested with respect to . Here edges and are nested if . The queue-number of a graph is the minimum integer such that has a -queue layout. These definitions were introduced by Heath et al. [12, 13] as a dual to stack layouts (also called book embeddings).
Heath et al.  conjectured that every planar graph has bounded queue number. This conjecture has remained open despite much research on queue layouts [18, 3, 10, 12, 13, 11, 16, 17, 5, 8, 7, 14, 6]. Dujmović and Wood  observed that every graph with edges has a -queue layout using a random vertex ordering. Thus every planar graph with vertices has queue-number . Di Battista et al.  proved the first breakthrough on this topic, by showing that every planar graph with vertices has queue-number . Dujmović  improved this bound to with a simpler proof.
Dujmović et al.  established (poly-)logarithmic bounds for more general classes of graphs111The Euler genus of a graph is the minimum integer such that embeds in the orientable surface with handles (and is even) or the non-orientable surface with cross-caps. Of course, a graph is planar if and only if it has Euler genus 0; see  for more about graph embeddings in surfaces. A graph is a minor of a graph if a graph isomorphic to can be obtained from a subgraph of by contracting edges. . For example, they proved that every graph with vertices and Euler genus has queue-number , and that every graph with vertices excluding a fixed minor has queue-number .
Recently, Bekos et al.  proved a second breakthrough result, by showing that planar graphs with bounded degree have bounded queue-number.
Theorem 1 ().
Every planar graph with maximum degree has queue-number at most .
Note that bounded degree alone is not enough to ensure bounded queue-number. In particular, Wood  proved that for every integer and all sufficiently large , there are graphs with vertices, maximum degree , and queue-number .
The first contribution of this paper is to extend the result of Bekos et al. , by showing that graphs with bounded Euler genus and bounded degree have bounded queue-number.
Every graph with Euler genus and maximum degree has queue-number at most .
We remark that using well-known constructions [5, 7, 9], Theorem 2 implies that graphs with bounded Euler genus and bounded degree have bounded track-number, which in turn can be used to prove linear volume bounds for three-dimensional straight-line grid drawings of the same class of graphs. These results can also be extended for graphs with bounded degree that can be drawn in a surface of bounded Euler genus with a bounded number of crossings per edge (using [10, Theorem 6]). We omit all these details in this note.
The proof of Theorem 2 uses Theorem 1 as a ‘black box’. Starting with a graph of bounded Euler genus and bounded degree, we construct a planar subgraph of . We then apply Theorem 1 to obtain a queue layout of , from which we construct a queue layout of . This approach suggests a direct connection between the queue-number of graphs with bounded Euler genus and planar graphs, regardless of degree considerations. The following theorem establishes this connection. A class of graphs is hereditary if it is closed under taking induced subgraphs.
Let be a hereditary class of graphs, such that every planar graph in has queue-number at most . Then every graph in with Euler genus has queue-number at most .
Theorem 2 is an immediate corollary of Theorems 3 and 1, where is the class of graphs with maximum degree at most . Theorem 3, where is the class of all graphs, implies the following result of interest:
If every planar graph has queue-number at most , then every graph with Euler genus has queue-number at most .
2 The Proof
The following definition is a key concept in our proofs (and that of several other papers on queue layouts [6, 5, 8]). A layering of a graph is a partition of such that for every edge , if and , then . If is a vertex in a connected graph and for all , then is called a bfs layering of .
Let be a connected graph with Euler genus . For every bfs layering of , there is a set with at most vertices in each layer , such that is planar.
Fix an embedding of in a surface of Euler genus . Say has vertices, edges, and faces. By Euler’s formula, . Let be a bfs layering of rooted at some vertex . Let be the corresponding bfs spanning tree. Let be the graph with , where for each edge of , if and are the faces of with on their boundary, then there is an edge in . (Think of as the spanning subgraph of consisting of those edges that do not cross edges in .) Note that and . Since is a tree, is connected; see [6, Lemma 11] for a proof. Let be a spanning tree of . Let . Thus . Say . For , let be the union of the -path and the -path in , plus the edge . Let be . Say has vertices and edges. Since consists of a subtree of plus the edges in , we have .
We now describe how to ‘cut’ along the edges of to obtain a new graph ; see Figure 1. First, each edge of is replaced by two edges and in . Each vertex of incident with no edges in is untouched. Consider a vertex of incident with edges in in clockwise order. In replace by new vertices , where is incident with , and all the edges incident with clockwise from to (exclusive). Here means and means . This operation defines a cyclic ordering of the edges in incident with each vertex (where is followed by in the cyclic order at ). This in turn defines an embedding of in some orientable surface. (Note that if is embedded in a non-orientable surface, then the edge signatures for are ignored in the embedding of .)
Say has vertices and edges, and the embedding of ’ has faces and Euler genus . Each vertex in with degree in is replaced by vertices in . Each edge in is replaced by two edges in , while each edge of is maintained in . Thus
and . Each face of is preserved in . Say new faces are created by the cutting. Thus . Since is connected, it follows that is connected. By Euler’s formula, . Thus , implying . Hence , implying . Since and , we have and . Therefore is planar.
Note that is a subgraph of , and is planar. By construction, each path has at most two vertices in each layer . Thus has at most vertices in each . ∎
We need the following lemma of independent interest.
If a graph has a -queue layout, and is a layering of , then has a -queue layout using ordering .
Say is the edge-partition and is the ordering of in a -queue layout of . For , let be the set of edges with for some ; let be the set of edges with and and for some ; and let be the set of edges with and and for some . Then is a partition of .
Let be the ordering of where each is ordered by . No two edges in some set are nested in , as otherwise the same two edges would be in and would be nested in . Suppose that for some edges . So and for some , and and . Now by the definition of . Hence , which is a contradiction since both and are in . Thus no two edges in are nested in . By symmetry, no two edges in are nested in . Hence is the ordering in a -queue layout of . ∎
We now prove our main result.
Proof of Theorem 3.
Let be a graph in with Euler genus . Since the queue-number of equals the maximum queue-number of the connected components of , we may assume that is connected. Let be a bfs layering of . By Lemma 5, there is a set with at most vertices in each layer , such that is planar. Since is hereditary, , and by assumption has a -queue layout. Note that is a layering of . By Lemma 6, has a -queue layout using ordering . Recall that for all . Let be the ordering
of . where each set is ordered arbitrarily, and each set is ordered according to the above -queue layout of . Edges of inherit their queue assignment. We now assign edges incident with vertices in to queues. For
and odd, put each edge incident with the -th vertex in in a new queue . For and even , put each edge incident with the -th vertex in (not already assigned to a queue) in a new queue . Suppose that two edges and in are nested, where . Say and and and . By construction, . Since is an edge, . At least one endpoint of is in for some odd , and one endpoint of is in for some odd . Since are distinct, . Thus . This is a contradiction since . Thus is a queue. Similarly is a queue. Hence this step introduces new queues. We obtain a -queue layout of . ∎
Whether the result of Bekos et al.  can be generalised for arbitrary excluded minors is an interesting question. That is, do graphs excluding a fixed minor and with bounded degree have bounded queue-number? It might even be true that graphs excluding a fixed minor have bounded queue-number.
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