Related papers: Parameterized Complexity of Simultaneous Planarity

Given a graph $G=(V,E)$, a set $\mathcal{F}$ of forbidden subgraphs, we study $\mathcal{F}$-Free Edge Deletion, where the goal is to remove minimum number of edges such that the resulting graph does not contain any $F\in \mathcal{F}$ as a…

We show that every $n$-vertex planar graph admits a simultaneous embedding with no mapping and with fixed edges with any $(n/2)$-vertex planar graph. In order to achieve this result, we prove that every $n$-vertex plane graph has an induced…

We discuss the problem of embedding graphs in the plane with restrictions on the vertex mapping. In particular, we introduce a technique for drawing planar graphs with a fixed vertex mapping that bounds the number of times edges bend. An…

Given an undirected graph $G=(V,E)$ and an integer $\ell$, the Eccentricity Shortest Path (ESP) asks to find a shortest path $P$ such that for every vertex $v\in V(G)$, there is a vertex $w\in P$ such that $d_G(v,w)\leq \ell$, where…

In this paper we study the complexity of the following problems: Given a colored graph X=(V,E,c), compute a minimum cardinality set S of vertices such that no nontrivial automorphism of X fixes all vertices in S. A closely related problem…

We introduce the Graph Sylvester Embedding (GSE), an unsupervised graph representation of local similarity, connectivity, and global structure. GSE uses the solution of the Sylvester equation to capture both network structure and…

In this paper we study the Spanning Tree Congestion problem, where we are given a graph $G=(V,E)$ and are asked to find a spanning tree $T$ of minimum maximum congestion. Here, the congestion of an edge $e\in T$ is the number of edges…

Consider a graph $G = (V,E)$ and a coloring $c$ of vertices with colors from $[\ell]$. A vertex $v$ is said to be happy with respect to $c$ if $c(v) = c(u)$ for all neighbors $u$ of $v$. Further, an edge $(u,v)$ is happy if $c(u) = c(v)$.…

We revisit two well-studied problems, Bounded Degree Vertex Deletion and Defective Coloring, where the input is a graph $G$ and a target degree $\Delta$ and we are asked either to edit or partition the graph so that the maximum degree…

In this work, we study how far one can deviate from optimal behavior when embedding a planar graph. For a planar graph $G$, we say that a plane subgraph $H\subseteq G$ is a \textit{plane-saturated subgraph} if adding any edge (possibly with…

Let $G$ be a graph such that each edge has its list of available colors, and assume that each list is a subset of the common set consisting of $k$ colors. Suppose that we are given two list edge-colorings $f_0$ and $f_r$ of $G$, and asked…

The C-Planarity problem asks for a drawing of a $\textit{clustered graph}$, i.e., a graph whose vertices belong to properly nested clusters, in which each cluster is represented by a simple closed region with no edge-edge crossings, no…

A graph on $n$ vertices is equitably $k$-colorable if it is $k$-colorable and every color is used either $\left\lfloor n/k \right\rfloor$ or $\left\lceil n/k \right\rceil$ times. Such a problem appears to be considerably harder than vertex…

The classical NP-complete problem Vertex Cover requires us to determine whether a graph contains at most $k$ vertices that cover all edges. In spite of its intractability, the problem can be solved in FPT time for parameter $k$ by various…

Let $F_G(P)$ be a functional defined on the set of all the probability distributions on the vertex set of a graph $G$. We say that $G$ is \emph{symmetric with respect to $F_G(P)$} if the uniform distribution on $V(G)$ maximizes $F_G(P)$.…

Let $G=(V,E)$ be a graph on $n$ vertices and $R$ be a set of pairs of vertices in $V$ called \emph{requests}. A \emph{multicut} is a subset $F$ of $E$ such that every request $xy$ of $R$ is cut by $F$, \i.e. every $xy$-path of $G$…

Given a graph $G(V,E)$, a vertex subset $S$ of $G$ is called an open packing in $G$ if no pair of distinct vertices in $S$ have a common neighbour in $G$. The size of a largest open packing in $G$ is called the open packing number,…

We study the parameterized complexity of separating a small set of vertices from a graph by a small vertex-separator. That is, given a graph $G$ and integers $k$, $t$, the task is to find a vertex set $X$ with $|X| \le k$ and $|N(X)| \le…

We revisit the Maximum Node-Disjoint Paths problem, the natural optimization version of Node-Disjoint Paths, where we are given a graph $G$, $k$ pairs of vertices $(s_i, t_i)$ and an integer $\ell$, and are asked whether there exist at…

In Two-Sets Cut-Uncut, we are given an undirected graph $G=(V,E)$ and two terminal sets $S$ and $T$. The task is to find a minimum cut $C$ in $G$ (if there is any) separating $S$ from $T$ under the following ``uncut'' condition. In the…