Abstract
A \emphk-stack layout (respectively, \emphk-queuelayout) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of non-crossing (non-nested) edges with respect to the vertex ordering. A \emphk-track layout of a graph consists of a vertex k-colouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. The \emphstack-number (respectively, \emphqueue-number, \emphtrack-number) of a graph G, denoted by sn(G) (qn(G), tn(G)), is the minimum k such that G has a k-stack (k-queue, k-track) layout.\par This paper studies stack, queue, and track layouts of graph subdivisions. It is known that every graph has a 3-stack subdivision. The best known upper bound on the number of division vertices per edge in a 3-stack subdivision of an n-vertex graph G is improved from O(log n) to O(log min\sn(G),qn(G)\). This result reduces the question of whether queue-number is bounded by stack-number to whether 3-stack graphs have bounded queue number.\par It is proved that every graph has a 2-queue subdivision, a 4-track subdivision, and a mixed 1-stack 1-queue subdivision. All these values are optimal for every non-planar graph. In addition, we characterise those graphs with k-stack, k-queue, and k-track subdivisions, for all values of k. The number of division vertices per edge in the case of 2-queue and 4-track subdivisions, namely O(log qn(G)), is optimal to within a constant factor, for every graph G. \par Applications to 3D polyline grid drawings are presented. For example, it is proved that every graph G has a 3D polyline grid drawing with the vertices on a rectangular prism, and with O(log qn(G)) bends per edge. Finally, we establish a tight relationship between queue layouts and so-called 2-track thickness of bipartite graphs. \par
Highlights
We consider undirected, finite, and simple graphs G with vertex set V (G) and edge set E(G)
We prove a refinement of the upper bound of Enomoto and Miyauchi [39], in which the number of division vertices per edge depends on the stack-number or queue-number of the given graph
In Theorem 2 we prove that queue-number is tied to 2-track thickness for bipartite graphs, and queue-number is tied to 2-track sub-thickness
Summary
Finite, and simple graphs G with vertex set V (G) and edge set E(G). The subgraph of G induced by a set of vertices A ⊆ V (G) is denoted by G[A]. For all A, B ⊆ V (G) with A ∩ B = ∅, we denote by G[A, B] the bipartite subgraph of G with vertex set A ∪ B and edge set {vw ∈ E(G) : v ∈ A, w ∈ B}. A subdivision of a graph G is a graph obtained from G by replacing each edge vw ∈ E(G) by a path with at least one edge whose endpoints are v and w. A graph H is a minor of G if H is isomorphic to a graph obtained from a subgraph of G by contracting edges. By α(G) we denote the function f : N → N, where f (n) is the maximum of α(G), taken over all n-vertex graphs G ∈ G. These notions were introduced by Gyarfas [51] in relation to near-perfect graph families for which the chromatic number is bounded by the clique-number
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