# On Linear Layouts of Graphs

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1 School of Computer Science Ottawa 2 SOCS - School of Computer Science Quebec 3 KAM - Department of Applied Mathematics Prague

Abstract : In a total order of the vertices of a graph, two edges with no endpoint in common can be \emphcrossing, \emphnested, or \emphdisjoint. A \emphk-stack respectively, \emphk-queue, \emphk-arch \emphlayout of a graph consists of a total order of the vertices, and a partition of the edges into k sets of pairwise non-crossing non-nested, non-disjoint edges. Motivated by numerous applications, stack layouts also called \emphbook embeddings and queue layouts are widely studied in the literature, while this is the first paper to investigate arch layouts.\par Our main result is a characterisation of k-arch graphs as the \emphalmost 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.\par 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?\par A comprehensive bibliography of all known references on these topics is included. \par

Keywords : graph layout graph drawing stack layout queue layout arch layout book embedding queue-number stack-number page-number book-thickness

Author: ** Vida Dujmović - David Wood - **

Source: https://hal.archives-ouvertes.fr/