On Linear Layouts of Graphs

Abstract

International audience 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

Drawing a Graph in a Hypercube

“…A queue in σ is a set of edges Q ⊆ E(G) such that no two edges in Q are nested in σ. A k-queue layout of G consists of a vertex ordering σ of G, and a partition of E(G) into k queues in σ. Heath et al [13,14] introduced queue layouts; see Dujmović and Wood [7] for references and a summary of known results. Heath and Rosenberg [14] proved that it is N P-complete to determine if a given graph has a 1-queue layout.…”

Improved bounds for colouring circle graphs

“…Circle graphs are also deeply related to planar graphs; the fundamental graphs of planar graphs are exactly the class of bipartite circle graphs [16]. Direct applications of colouring circle graphs include finding the minimum number of stacks needed to obtain a given permutation [14], solving routing problems such as in VLSI physical design [33], and finding stack layouts of graphs, which also has a number of additional applications of its own (see [12]).…”

An Online Framework to Interact and Efficiently Compute Linear Layouts of Graphs

“…Linear layouts of graphs have been fruitful subjects of intense research over the years, both from a combinatorial and from an algorithmic point of view, as they play an important role in various fields; see, e.g., [16]. Formally, a linear layout of graph G = (V, E) consists of a linear order of its vertices (that is, a bijective function σ : V → {1, .…”

“…There exists a plethora of theoretical results for each of the aforementioned types of linear layouts; in the following, we overview existing results for planar graphs, which is mainly the focus of our work. For a more detailed overview, we point the reader to [16].…”