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The Planar Graph Product Structure Theorem of Dujmovi\'c et al. [J. ACM '20] says that every planar graph $G$ is contained in $H\boxtimes P\boxtimes K_3$ for some planar graph $H$ with treewidth at most 3 and some path $P$. This result has…

Let G=(V,E)(|V|=n and |E|=m) be an undirected graph with positive edge weights. Let P_{G}(s, t) be a shortest s-t path in G. Let l be the number of edges in P_{G}(s, t). The \emph{Edge Replacement Path} problem is to compute a shortest s-t…

We consider an inhomogeneous version of the Barak-Erd\H{o}s graph, i.e. a directed Er\H{o}s-R\'enyi random graph on $\{1,\ldots,n\}$ with no loop. Given $f$ a Riemann-integrable non-negative function on $[0,1]^2$ and $\gamma > 0$, we define…

A tree t-spanner of an unweighted graph G is a spanning tree T such that for every two vertices their distance in T is at most t times their distance in G. Given an unweighted graph G and a positive integer t as input, the tree t-spanner…

The dynamic shortest paths problem on planar graphs asks us to preprocess a planar graph $G$ such that we may support insertions and deletions of edges in $G$ as well as distance queries between any two nodes $u,v$ subject to the constraint…

The diameter of an undirected or a directed graph is defined to be the maximum shortest path distance over all pairs of vertices in the graph. Given an undirected graph $G$, we examine the problem of assigning directions to each edge of $G$…

We present two new and efficient algorithms for computing all-pairs shortest paths. The algorithms operate on directed graphs with real (possibly negative) weights. They make use of directed path consistency along a vertex ordering d. Both…

For the first time proposed: a method for representing the projections of a graph in computer memory and a description based on it of a quick search for shortest paths in unweighted dynamic graphs. The spatial complexity of the projection…

Let $G$ be an edge-weighted directed graph with $n$ vertices embedded on an orientable surface of genus $g$. We describe a simple deterministic lexicographic perturbation scheme that guarantees uniqueness of minimum-cost flows and shortest…

Given an undirected graph $G=(V,E)$ with positive edge lengths and two vertices $s$ and $t$, the next-to-shortest path problem is to find an $st$-path which length is minimum amongst all $st$-paths strictly longer than the shortest path…

Given an undirected graph $G=(V,E)$, vertices $s,t\in V$, and an integer $k$, Tracking Shortest Paths requires deciding whether there exists a set of $k$ vertices $T\subseteq V$ such that for any two distinct shortest paths between $s$ and…

The approximate single-source shortest-path problem is as follows: given a graph with nonnegative edge weights and a designated source vertex $s$, return estimates of the distances from~$s$ to each other vertex such that the estimate falls…

Let $G$ be an $n$-node simple directed planar graph with nonnegative edge weights. We study the fundamental problems of computing (1) a global cut of $G$ with minimum weight and (2) a~cycle of $G$ with minimum weight. The best previously…

In this paper we describe an algorithm that embeds a graph metric $(V,d_G)$ on an undirected weighted graph $G=(V,E)$ into a distribution of tree metrics $(T,D_T)$ such that for every pair $u,v\in V$, $d_G(u,v)\leq d_T(u,v)$ and…

Calculating the diameter of an undirected graph requires quadratic running time under the Strong Exponential Time Hypothesis and this barrier works even against any approximation better than 3/2. For planar graphs with positive edge…

In a geometric graph, $G$, the \emph{stretch factor} between two vertices, $u$ and $w$, is the ratio between the Euclidean length of the shortest path from $u$ to $w$ in $G$ and the Euclidean distance between $u$ and $w$. The \emph{average…

We study shortest-path routing in large weighted, undirected graphs, where expanding search frontiers raise time and memory costs for exact solvers. We propose \emph{SPHERE}, a query-aware partitioning heuristic that adaptively splits the…

The girth of a graph is the length of its shortest cycle. We give an algorithm that computes in O(n(log n)^3) time and O(n) space the (weighted) girth of an n-vertex planar digraph with arbitrary real edge weights. This is an improvement of…

We consider the parametric shortest paths problem in a linearly interpolated graph. Given two positively-weighted directed graphs $G_0=(V,E,\omega_0)$ and $G_1=(V,E,\omega_1),$ the linearly interpolated graph is the family of graphs…

The replacement paths problem for directed graphs is to find for given nodes s and t and every edge e on the shortest path between them, the shortest path between s and t which avoids e. For unweighted directed graphs on n vertices, the…