Related papers: Graph Layouts via Layered Separators

A graph $G$ is a non-separating planar graph if there is a drawing $D$ of $G$ on the plane such that (1) no two edges cross each other in $D$ and (2) for any cycle $C$ in $D$, any two vertices not in $C$ are on the same side of $C$ in $D$.…

A queue layout of a graph $G$ consists of a vertex ordering of $G$ and a partition of the edges into so-called queues such that no two edges in the same queue nest, i.e., have their endpoints ordered in an ABBA-pattern. Continuing the…

A mixed s-stack q-queue layout of a graph consists of a linear order of its vertices and of a partition of its edges into s stacks and q queues, such that no two edges in the same stack cross and no two edges in the same queue nest. In…

Given finitely many connected polygonal obstacles $O_1,\dots,O_k$ in the plane and a set $P$ of points in general position and not in any obstacle, the {\em visibility graph} of $P$ with obstacles $O_1,\dots,O_k$ is the (geometric) graph…

A vertex k-ranking is a labeling of the vertices of a graph with integers from 1 to k so any path connecting two vertices with the same label will pass through a vertex with a greater label. The rank number of a graph is defined to be the…

Given a graph $G = (V, E)$ and an integer $k$, we study $k$-Vertex Seperator (resp. $k$-Edge Separator), where the goal is to remove the minimum number of vertices (resp. edges) such that each connected component in the resulting graph has…

A vertex k-labeling of graph G is distinguishing if the only automorphism that preserves the labels of G is the identity map. The distinguishing number of G, D(G), is the smallest integer k for which G has a distinguishing k-labeling. In…

Graph separators are a ubiquitous tool in graph theory and computer science. However, in some applications, their usefulness is limited by the fact that the separator can be as large as $\Omega(\sqrt{n})$ in graphs with $n$ vertices. This…

An intersection graph of curves in the plane is called a string graph. Matousek almost completely settled a conjecture of the authors by showing that every string graph of m edges admits a vertex separator of size O(\sqrt{m}\log m). In the…

In spite of the extensive study of stack and queue layouts, many fundamental questions remain open concerning the complexity-theoretic frontiers for computing stack and queue layouts. A stack (resp. queue) layout places vertices along a…

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…

The page 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…

Dujmovi\'c et al. [\emph{J.~ACM}~'20] recently proved that every planar graph is isomorphic to a subgraph of the strong product of a bounded treewidth graph and a path. Analogous results were obtained for graphs of bounded Euler genus or…

An $h$-queue layout of a graph $G$ consists of a linear order of its vertices and a partition of its edges into $h$ queues, such that no two independent edges of the same queue nest. The minimum $h$ such that $G$ admits an $h$-queue layout…

A vertex colouring of a graph is \emph{nonrepetitive} if there is no path for which the first half of the path is assigned the same sequence of colours as the second half. The \emph{nonrepetitive chromatic number} of a graph $G$ is the…

The track number $\tau(G)$ of a graph $G$ is the minimum number of interval graphs whose union is $G$. We show that the track number of the line graph $L(G)$ of a triangle-free graph $G$ is at least $\lg \lg \chi(G) + 1$, where $\chi(G)$ is…

A bipartite graph $G = (X \cup Y, E)$ is a 2-layer $k$-planar graph if it admits a drawing on the plane such that the vertices in $X$ and $Y$ are placed on two parallel lines respectively, edges are drawn as straight-line segments, and…

A $k$-stack layout (or $k$-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 of a graph is the…

In a graph whose vertices are assigned integer ranks, a path is well-ranked if the endpoints have distinct ranks or some interior point has a higher rank than the endpoints. A ranking is an assignment of ranks such that all nontrivial paths…

Several types of linear layouts of graphs are obtained by leveraging known data structures; the most notable representatives are the stack and the queue layouts. In this content, given a data structure, one seeks to specify an order of the…