Related papers: Light Edge Fault Tolerant Graph Spanners
Recently Elkin and Solomon gave a construction of spanners for doubling metrics that has constant maximum degree, hop-diameter O(log n) and lightness O(log n) (i.e., weight O(log n)w(MST). This resolves a long standing conjecture proposed…
In this draft we prove an interesting structural property related to the problem of computing {\em all the best swap edges} of a {\em tree spanner} in unweighted graphs. Previous papers show that the maximum stretch factor of the tree where…
As Graph Neural Networks (GNNs) become more pervasive, it becomes paramount to build reliable tools for explaining their predictions. A core desideratum is that explanations are \textit{faithful}, \ie that they portray an accurate picture…
Message passing graph neural networks (GNNs) are a popular learning architectures for graph-structured data. However, one problem GNNs experience is oversquashing, where a GNN has difficulty sending information between distant nodes.…
Temporal graphs have edge sets that change over discrete time steps. Such graphs are temporally connected (TC) if all pairs of vertices can reach each other using paths that traverse the edges in a time-respecting way (temporal paths).…
In this paper, we study two examples of minimum weight random graphs with edge constraints. First we consider the complete graph on ${n}$ vertices equipped with uniformly heavy edge weights and use iteration methods to obtain deviation…
Let $G$ be a connected graph and $d(a,b)$ be the distance between the vertices $a$ and $b$. A subset $U =\{u_1,u_2,\cdots,u_k\}$ of the vertices is called a resolving set for $G$ if for every two distinct vertices $a,b \in V(G)$, there is a…
A seminal work of [Ahn-Guha-McGregor, PODS'12] showed that one can compute a cut sparsifier of an unweighted undirected graph by taking a near-linear number of linear measurements on the graph. Subsequent works also studied computing other…
Motivated by a result of [1] which states that if F is a subgraph of a convex complete graph K_n and F contains no boundary edge of K_n and |E(F)| \leq n-3, then K_n - F admits a triangulation, we determine necessary and sufficient…
In this paper we study the problem of how resilient networks are to node faults. Specifically, we investigate the question of how many faults a network can sustain so that it still contains a large (i.e. linear-sized) connected component…
An $f$-edge (or vertex) connectivity certificate is a sparse subgraph that maintains connectivity under the failure of at most $f$ edges (or vertices). It is well known that any $n$-vertex graph admits an $f$-edge (or vertex) connectivity…
In 2016, a breakthrough result of Chechik and Wulff-Nilsen [SODA '16] established that every $n$-node graph $G$ has a $(1+\varepsilon)(2k-1)$-spanner of lightness $O_{\varepsilon}(n^{1/k})$, and recent followup work by Le and Solomon [STOC…
Grid graphs, and, more generally, $k\times r$ grid graphs, form one of the most basic classes of geometric graphs. Over the past few decades, a large body of works studied the (in)tractability of various computational problems on grid…
Given a connected graph $G=(V,E)$ and a length function $\ell:E\to {\mathbb R}$ we let $d_{v,w}$ denote the shortest distance between vertex $v$ and vertex $w$. A $t$-spanner is a subset $E'\subseteq E$ such that if $d'_{v,w}$ denotes…
A set of vertices $S\subseteq V(G)$ is a basis or resolving set of a graph $G$ if for each $x,y\in V(G)$ there is a vertex $u\in S$ such that $d(x,u)\neq d(y,u)$. A basis $S$ is a fault-tolerant basis if $S\setminus \{x\}$ is a basis for…
With applications in distribution systems and communication networks, the minimum stretch spanning tree problem is to find a spanning tree T of a graph G such that the maximum distance in T between two adjacent vertices is minimized. The…
We present an $f$-fault tolerant distance oracle for an undirected weighted graph where each edge has an integral weight from $[1 \dots W]$. Given a set $F$ of $f$ edges, as well as a source node $s$ and a destination node $t$, our oracle…
In the problem of structured prediction with graph representation learning (GRL for short), the hypothesis returned by the algorithm maps the set of features in the \emph{receptive field} of the targeted vertex to its label. To understand…
A class of graphs admits an adjacency labeling scheme of size $b(n)$, if the vertices in each of its $n$-vertex graphs can be assigned binary strings (called labels) of length $b(n)$ so that the adjacency of two vertices can be determined…
One way to define the Matching Cut problem is: Given a graph $G$, is there an edge-cut $M$ of $G$ such that $M$ is an independent set in the line graph of $G$? We propose the more general Conflict-Free Cut problem: Together with the graph…