In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A k-stack (respectively, k-queue, k-arch) layout of a graph consists of a total order of the vertices, and a partition of the edges into k sets of pairwise non-crossing (respectively, non-nested, non-disjoint) edges. Motivated by numerous applications, stack layouts (also called book embeddings) and queue layouts are widely studied in the literature, while this is the first paper to investigate arch layouts. Our main result is a characterisation of k-arch graphs as the almost (k + 1)-colourable graphs; that is, the graphs G with a set S of at most k vertices, such that G\S is (k + 1)-colourable. In addition, we survey the following fundamental questions regarding each type of layout, and in the case of queue layouts, provide simple proofs of a number of existing results. How does one partition the edges given a fixed ordering of the vertices? What is the maximum number of edges in each type of layout? What is the maximum chromatic number of a graph admitting each type of layout? What is the computational complexity of recognising the graphs that admit each type of layout? A comprehensive bibliography of all known references on these topics is included.

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Discrete Mathematics and Theoretical Computer Science
School of Linguistics and Language Studies

Dujmović, V, & Wood, D. (2004). On linear layouts of graphs. Discrete Mathematics and Theoretical Computer Science, 6(2), 339–358.