Book Embeddings of Graph Products
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A k-stack layout (also called a k-page book embedding) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of non-crossing edges with respect to the vertex order. The stack number (book thickness, page number) of a graph is the minimum k such that it admits a k-stack layout. A k-queue layout is defined similarly, except that no two edges in a single set may be nested. It was recently proved that graphs of various non-minor-closed classes are subgraphs of the strong product of a path and a graph with bounded treewidth. Motivated by this decomposition result, we explore stack layouts of graph products. We show that the stack number is bounded for the strong product of a path and (i) a graph of bounded pathwidth or (ii) a bipartite graph of bounded treewidth and bounded degree. The results are obtained via a novel concept of simultaneous stack-queue layouts, which may be of independent interest.
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