相关论文: Two Results on Outer-String Graphs

An \emph{obstacle representation} of a graph $G$ is a straight-line drawing of $G$ in the plane together with a collection of connected subsets of the plane, called \emph{obstacles}, that block all non-edges of $G$ while not blocking any of…

Planar graphs can be represented as intersection graphs of different types of geometric objects in the plane, e.g., circles (Koebe, 1936), line segments (Chalopin \& Gon{\c{c}}alves, 2009), \textsc{L}-shapes (Gon{\c{c}}alves et al, 2018).…

An obstacle representation of a graph $G$ is a set of points in the plane representing the vertices of $G$, together with a set of polygonal obstacles such that two vertices of $G$ are connected by an edge in $G$ if and only if the line…

If a graph $G$ can be represented by means of paths on a grid, such that each vertex of $G$ corresponds to one path on the grid and two vertices of $G$ are adjacent if and only if the corresponding paths share a grid edge, then this graph…

An \emph{outer-RAC drawing} of a graph is a straight-line drawing where all vertices are incident to the outer cell and all edge crossings occur at a right angle. If additionally, all crossing edges are either horizontal or vertical, we…

Let $G=(V,E)$ be an undirected graph without loops and multiple edges. A subset $C\subseteq V$ is called \emph{identifying} if for every vertex $x\in V$ the intersection of $C$ and the closed neighbourhood of $x$ is nonempty, and these…

An \emph{s-graph} is a graph with two kinds of edges: \emph{subdivisible} edges and \emph{real} edges. A \emph{realisation} of an s-graph $B$ is any graph obtained by subdividing subdivisible edges of $B$ into paths of arbitrary length (at…

In this paper, we show that any $B_2$-VPG graph (i.e., an intersection graph of orthogonal curves with at most 2 bends) can be decomposed into $O(\log n)$ outerstring graphs or $O(\log^3 n)$ permutation graphs. This leads to better…

Given an ordered partition $\Pi =\{P_1,P_2, ...,P_t\}$ of the vertex set $V$ of a connected graph $G=(V,E)$, the \emph{partition representation} of a vertex $v\in V$ with respect to the partition $\Pi$ is the vector…

We investigate the two problems of computing the union join graph as well as computing the subset graph for acyclic hypergraphs and their subclasses. In the union join graph $G$ of an acyclic hypergraph $H$, each vertex of $G$ represents a…

An orientation $D$ of a graph $G=(V,E)$ is a digraph obtained from $G$ by replacing each edge by exactly one of the two possible arcs with the same end vertices. For each $v \in V(G)$, the indegree of $v$ in $D$, denoted by $d^-_D(v)$, is…

Deciding whether a given graph has a square root is a classical problem that has been studied extensively both from graph theoretic and from algorithmic perspectives. The problem is NP-complete in general, and consequently substantial…

The crossing number of a graph is the least number of crossings over all drawings of the graph in the plane. Computing the crossing number of a given graph is NP-hard, but fixed-parameter tractable (FPT) with respect to the natural…

A graph $G=(V,E)$ is a \emph{word-representable graph} if there exists a word $W$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $W$ if and only if $(x,y)\in E$ for each $x\neq y$. In this paper we give an effective…

A graph $G$ with vertex set $V(G)$ and edge set $E(G)$ is said to be word-representable if there exists a word $w$ over the alphabet $V(G)$ such that, for any two distinct letters $x,y \in V(G)$, the letters $x$ and $y$ alternate in $w$ if…

A \emph{$k$-radius sequence} for a graph $G$ is a sequence of vertices of $G$ (typically with repetitions) such that for every edge $uv$ of $G$ vertices $u$ and $v$ appear at least once within distance $k$ in the sequence. The length of a…

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…

A graph is outerplanar if it has a planar drawing for which all vertices belong to the outer face of the drawing. Let $H$ be a graph. The outerplanar Tur\'an number of $H$, denoted by $ex_\mathcal{OP}(n,H)$, is the maximum number of edges…

A string graph is an intersection graph of curves in the plane. A $k$-string graph is a graph with a string representation in which every pair of curves intersects in at most $k$ points. We introduce the class of $(=k)$-string graphs as a…

We consider the problem of drawing an outerplanar graph with $n$ vertices with at most one bend per edge if the outer face is already drawn as a simple polygon. We prove that it can be decided in $O(nm)$ time if such a drawing exists, where…