Crossing Minimization for 1-page and 2-page Drawings of Graphs with Bounded Treewidth

We investigate crossing minimization for 1-page and 2-page book drawings. We show that computing the 1-page crossing number is fixed-parameter tractable with respect to the number of crossings, that testing 2-page planarity is fixed-parameter tractable with respect to treewidth, and that computing the 2-page crossing number is fixed-parameter tractable with respect to the sum of the number of crossings and the treewidth of the input graph. We prove these results via Courcelle's theorem on the fixed-parameter tractability of properties expressible in monadic second order logic for graphs of bounded treewidth.Comment: Graph Drawing 201

Directed Acyclic Outerplanar Graphs Have Constant Stack Number

The stack number of a directed acyclic graph $G$ is the minimum $k$ for which there is a topological ordering of $G$ and a $k$-coloring of the edges such that no two edges of the same color cross, i.e., have alternating endpoints along the topological ordering. We prove that the stack number of directed acyclic outerplanar graphs is bounded by a constant, which gives a positive answer to a conjecture by Heath, Pemmaraju and Trenk [SIAM J. Computing, 1999]. As an immediate consequence, this shows that all upward outerplanar graphs have constant stack number, answering a question by Bhore et al. [GD 2021] and thereby making significant progress towards the problem for general upward planar graphs originating from Nowakowski and Parker [Order, 1989]. As our main tool we develop the novel technique of directed $H$-partitions, which might be of independent interest. We complement the bounded stack number for directed acyclic outerplanar graphs by constructing a family of directed acyclic 2-trees that have unbounded stack number, thereby refuting a conjecture by N\"ollenburg and Pupyrev [arXiv:2107.13658, 2021]

The Inverse Voronoi Problem in Graphs I: Hardness

International audienceWe introduce the inverse Voronoi diagram problem in graphs: given a graph G with positive edge-lengths and a collection of subsets of vertices of V (G), decide whether is a Voronoi diagram in G with respect to the shortest-path metric. We show that the problem is NP-hard, even for planar graphs where all the edges have unit length. We also study the parameterized complexity of the problem and show that the problem is W[1]-hard when parameterized by the number of Voronoi cells or by the pathwidth of the graph

Layout of Graphs with Bounded Tree-Width

A \emph{queue layout} of a graph consists of a total order of the vertices, and a partition of the edges into \emph{queues}, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its \emph{queue-number}. A \emph{three-dimensional (straight-line grid) drawing} of a graph represents the vertices by points in $\mathbb{Z}^3$ and the edges by non-crossing line-segments. This paper contributes three main results: (1) It is proved that the minimum volume of a certain type of three-dimensional drawing of a graph $G$ is closely related to the queue-number of $G$. In particular, if $G$ is an $n$-vertex member of a proper minor-closed family of graphs (such as a planar graph), then $G$ has a $O(1)\times O(1)\times O(n)$ drawing if and only if $G$ has O(1) queue-number. (2) It is proved that queue-number is bounded by tree-width, thus resolving an open problem due to Ganley and Heath (2001), and disproving a conjecture of Pemmaraju (1992). This result provides renewed hope for the positive resolution of a number of open problems in the theory of queue layouts. (3) It is proved that graphs of bounded tree-width have three-dimensional drawings with O(n) volume. This is the most general family of graphs known to admit three-dimensional drawings with O(n) volume. The proofs depend upon our results regarding \emph{track layouts} and \emph{tree-partitions} of graphs, which may be of independent interest.Comment: This is a revised version of a journal paper submitted in October 2002. This paper incorporates the following conference papers: (1) Dujmovic', Morin & Wood. Path-width and three-dimensional straight-line grid drawings of graphs (GD'02), LNCS 2528:42-53, Springer, 2002. (2) Wood. Queue layouts, tree-width, and three-dimensional graph drawing (FSTTCS'02), LNCS 2556:348--359, Springer, 2002. (3) Dujmovic' & Wood. Tree-partitions of $k$-trees with applications in graph layout (WG '03), LNCS 2880:205-217, 200

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum