Related papers: Star-Based Separators for Intersection Graphs of $…

Let $F$ be a set of $n$ objects in the plane and let $G(F)$ be its intersection graph. A balanced clique-based separator of $G(F)$ is a set $S$ consisting of cliques whose removal partitions $G(F)$ into components of size at most $\delta…

Let $d$ be a (well-behaved) shortest-path metric defined on a path-connected subset of $\mathbb{R}^2$ and let $\mathcal{D}=\{D_1,\ldots,D_n\}$ be a set of geodesic disks with respect to the metric $d$. We prove that…

We prove a geometric version of the graph separator theorem for the unit disk intersection graph: for any set of $n$ unit disks in the plane there exists a line $\ell$ such that $\ell$ intersects at most $O(\sqrt{(m+n)\log{n}})$ disks and…

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…

For undirected graphs $G=(V,E)$ and $G_0=(V_0,E_0)$, say that $G$ is a region intersection graph over $G_0$ if there is a family of connected subsets $\{ R_u \subseteq V_0 : u \in V \}$ of $G_0$ such that $\{u,v\} \in E \iff R_u \cap R_v…

The planar separator theorem by Lipton and Tarjan [FOCS '77, SIAM Journal on Applied Mathematics '79] states that any planar graph with $n$ vertices has a balanced separator of size $O(\sqrt{n})$ that can be found in linear time. This…

In SoCG 2022, Conroy and T\'oth presented several constructions of sparse, low-hop spanners in geometric intersection graphs, including an $O(n\log n)$-size 3-hop spanner for $n$ disks (or fat convex objects) in the plane, and an $O(n\log^2…

We consider the existence and construction of \textit{biclique covers} of graphs, consisting of coverings of their edge sets by complete bipartite graphs. The \textit{size} of such a cover is the sum of the sizes of the bicliques.…

A graph separator is a subset of vertices of a graph whose removal divides the graph into small components. Computing small graph separators for various classes of graphs is an important computational task. In this paper, we present 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…

Alon, Seymour, and Thomas generalized Lipton and Tarjan's planar separator theorem and showed that a $K_h$-minor free graph with $n$ vertices has a separator of size at most $h^{3/2}\sqrt n$. They gave an algorithm that, given a graph $G$…

The biclique partition number of a graph \(G\), denoted \( \operatorname{bp}(G)\), is the minimum number of biclique subgraphs needed to partition the edge set of $G$. Lyu and Hicks \cite{lyu2023finding} posed the open problem of whether \(…

We provide a simple proof of the existence of a planar separator by showing that it is an easy consequence of the circle packing theorem. We also reprove other results on separators, including: (A) There is a simple cycle separator if the…

We establish that a simple polynomial-time algorithm that we call reweighted spectral partitioning obtains small 2/3-balanced vertex-separators for a number of graph classes, including $O(\sqrt{n})$-sized separators for planar graphs,…

Graph spanners are sparse subgraphs that faithfully preserve the distances in the original graph up to small stretch. Spanner have been studied extensively as they have a wide range of applications ranging from distance oracles, labeling…

The biclique partition number of a graph \(G\), denoted \( \operatorname{bp}(G)\), is the minimum number of biclique subgraphs that partition the edge set of \(G\). The Graham-Pollak theorem states that the complete graph on \( n \)…

Many real-world networks, such as transportation or trade networks, are dynamic in the sense that the edge set may change over time, but these changes are known in advance. This behavior is captured by the temporal graphs model, which has…

In fault-tolerant distance labeling we wish to assign short labels to the vertices of a graph $G$ such that from the labels of any three vertices $u,v,f$ we can infer the $u$-to-$v$ distance in the graph $G\setminus \{f\}$. We show that any…

A minimal separator of a graph $G$ is a set $S \subseteq V(G)$ such that there exist vertices $a,b \in V(G) \setminus S$ with the property that $S$ separates $a$ from $b$ in $G$, but no proper subset of $S$ does. For an integer $k\ge 0$, we…

We study separating systems of the edges of a graph where each member of the separating system is a path. We conjecture that every $n$-vertex graph admits a separating path system of size $O(n)$ and prove this in certain interesting special…