Recognizing Planar Laman Graphs

Laman graphs are the minimally rigid graphs in the plane. We present two algorithms for recognizing planar Laman graphs. A simple algorithm with running time O(n^(3/2)) and a more complicated algorithm with running time O(n log^3 n) based on involved planar network flow algorithms. Both improve upon the previously fastest algorithm for general graphs by Gabow and Westermann [Algorithmica, 7(5-6):465 - 497, 1992] with running time O(n sqrt{n log n}). To solve this problem we introduce two algorithms (with the running times stated above) that check whether for a directed planar graph G, disjoint sets S, T subseteq V(G), and a fixed k the following connectivity condition holds: for each vertex s in S there are k directed paths from s to T pairwise having only vertex s in common. This variant of connectivity seems interesting on its own

Faster Algorithms for Rooted Connectivity in Directed Graphs

We consider the fundamental problems of determining the rooted and global edge and vertex connectivities (and computing the corresponding cuts) in directed graphs. For rooted (and hence also global) edge connectivity with small integer capacities we give a new randomized Monte Carlo algorithm that runs in time $\tilde{O}(n^2)$. For rooted edge connectivity this is the first algorithm to improve on the $\Omega(n^3)$ time bound in the dense-graph high-connectivity regime. Our result relies on a simple combination of sampling coupled with sparsification that appears new, and could lead to further tradeoffs for directed graph connectivity problems. We extend the edge connectivity ideas to rooted and global vertex connectivity in directed graphs. We obtain a $(1 + \epsilon)$-approximation for rooted vertex connectivity in $\tilde{O}(nW/\epsilon)$ time where $W$ is the total vertex weight (assuming integral vertex weights); in particular this yields an $\tilde{O}(n^2/\epsilon)$ time randomized algorithm for unweighted graphs. This translates to a $\tilde{O}(\kappa nW)$ time exact algorithm where $\kappa$ is the rooted connectivity. We build on this to obtain similar bounds for global vertex connectivity. Our results complement the known results for these problems in the low connectivity regime due to work of Gabow [9] for edge connectivity from 1991, and the very recent work of Nanongkai et al. [24] and Forster et al. [7] for vertex connectivity

Finding Densest kk-Connected Subgraphs

Dense subgraph discovery is an important graph-mining primitive with a variety of real-world applications. One of the most well-studied optimization problems for dense subgraph discovery is the densest subgraph problem, where given an edge-weighted undirected graph $G=(V,E,w)$, we are asked to find $S\subseteq V$ that maximizes the density $d(S)$, i.e., half the weighted average degree of the induced subgraph $G[S]$. This problem can be solved exactly in polynomial time and well-approximately in almost linear time. However, a densest subgraph has a structural drawback, namely, the subgraph may not be robust to vertex/edge failure. Indeed, a densest subgraph may not be well-connected, which implies that the subgraph may be disconnected by removing only a few vertices/edges within it. In this paper, we provide an algorithmic framework to find a dense subgraph that is well-connected in terms of vertex/edge connectivity. Specifically, we introduce the following problems: given a graph $G=(V,E,w)$ and a positive integer/real $k$, we are asked to find $S\subseteq V$ that maximizes the density $d(S)$ under the constraint that $G[S]$ is $k$-vertex/edge-connected. For both problems, we propose polynomial-time (bicriteria and ordinary) approximation algorithms, using classic Mader's theorem in graph theory and its extensions