Related papers: Minimum Weight Connectivity Augmentation for Plana…
Augmented graphs play a vital role in regularizing Graph Neural Networks (GNNs), which leverage information exchange along edges in graphs, in the form of message passing, for learning. Due to their effectiveness, simple edge and node…
An $(\alpha,\beta)$-spanner of a weighted graph $G=(V,E)$, is a subgraph $H$ such that for every $u,v\in V$, $d_G(u,v) \le d_H(u,v)\le\alpha\cdot d_G(u,v)+\beta$. The main parameters of interest for spanners are their size (number of edges)…
Dynamic connectivity is a well-studied problem, but so far the most compelling progress has been confined to the edge-update model: maintain an understanding of connectivity in an undirected graph, subject to edge insertions and deletions.…
It is known that the vertex connectivity of a planar graph can be computed in linear time. We extend this result to the class of locally maximal 1-plane graphs: graphs that have an embedding with at most one crossing per edge such that the…
A searcher is tasked with exploring a graph with edge lengths and vertex weights, starting from a designated vertex. Initially, only the starting vertex is considered explored. At each step, the searcher adds an edge to the solution,…
In this paper we consider the problem of reconstructing a hidden weighted hypergraph of constant rank using additive queries. We prove the following: Let $G$ be a weighted hidden hypergraph of constant rank with n vertices and $m$…
We seek to augment a geometric network in the Euclidean plane with shortcuts to minimize its continuous diameter, i.e., the largest network distance between any two points on the augmented network. Unlike in the discrete setting where a…
An edge subset \( S \subseteq E(G) \) is called a 3-restricted edge-cut if $G-S$ is disconnected and each component of \( G - S \) contains at least three vertices. The 3-restricted edge-connectivity of a graph \( G \), denoted by \(…
A {\em simple drawing} $D(G)$ of a graph $G$ is one where each pair of edges share at most one point: either a common endpoint or a proper crossing. An edge $e$ in the complement of $G$ can be {\em inserted} into $D(G)$ if there exists a…
During the last 10 years it has become popular to study dynamic graph problems in a emergency planning or sensitivity setting: Instead of considering the general fully dynamic problem, we only have to process a single batch update of size…
We consider the problem on finding the edge weights that maximize the algebraic connectivity of a graph (smallest positive eigenvalue of the Laplacian), subject to the condition that the total effective resistance is kept constant. We…
We present a framework for dynamically maintaining $k$-edge-connectivity of an undirected simple graph $G$ under edge insertions and deletions, where $k$ is a fixed constant. After an edge insertion, the algorithm identifies and removes a…
The $g$-extra edge-connectivity is an important measure for the reliability of interconnection networks. Recently, Yang et al. [Appl. Math. Comput. 320 (2018) 464--473] determined the $3$-extra edge-connectivity of balanced hypercubes…
Traditionally, the quality of orthogonal planar drawings is quantified by either the total number of bends, or the maximum number of bends per edge. However, this neglects that in typical applications, edges have varying importance.…
A graph has \emph{diameter} D if every pair of vertices are connected by a path of at most D edges. The Diameter-D Augmentation problem asks how to add the a number of edges to a graph in order to make the resulting graph have diameter D.…
We consider approximations for computing minimum weighted cuts in directed graphs. We consider both rooted and global minimum cuts, and both edge-cuts and vertex-cuts. For these problems we give randomized Monte Carlo algorithms that…
Let $G=(V,E)$ be a strongly connected graph with $|V|\geq 3$. For $T\subseteq V$, the strongly connected graph $G$ is $2$-T-connected if $G$ is $2$-edge-connected and for each vertex $w$ in $T$, $w$ is not a strong articulation point. This…
We study the problem of guaranteeing the connectivity of a given graph by protecting or strengthening edges. Herein, a protected edge is assumed to be robust and will not fail, which features a non-uniform failure model. We introduce the…
There is a graph reduction system so that every optimal 1-planar graph can be reduced to an irreducible extended wheel graph, provided the reductions are applied such that the given graph class is preserved. A graph is optimal 1-planar if…
The edge-connectivity matrix of a weighted graph is the matrix whose off-diagonal $v$-$w$ entry is the weight of a minimum edge cut separating vertices $v$ and $w$. Its computation is a classical topic of combinatorial optimization since at…