Optimal decremental connectivity in planar graphs

We show an algorithm for dynamic maintenance of connectivity information in an undirected planar graph subject to edge deletions. Our algorithm may answer connectivity queries of the form `Are vertices $u$ and $v$ connected with a path?' in constant time. The queries can be intermixed with any sequence of edge deletions, and the algorithm handles all updates in $O(n)$ time. This results improves over previously known $O(n \log n)$ time algorithm

Sigma Coloring and Edge Deletions

A vertex coloring c : V(G) → N of a non-trivial graph G is called a sigma coloring if σ(u) is not equal to σ(v) for any pair of adjacent vertices u and v. Here, σ(x) denotes the sum of the colors assigned to vertices adjacent to x. The sigma chromatic number of G, denoted by σ(G), is defined as the fewest number of colors needed to construct a sigma coloring of G. In this paper, we consider the sigma chromatic number of graphs obtained by deleting one or more of its edges. In particular, we study the difference σ(G)−σ(G−e) in general as well as in restricted scenarios; here, G−e is the graph obtained by deleting an edge e from G. Furthermore, we study the sigma chromatic number of graphs obtained via multiple edge deletions in complete graphs by considering the complements of paths and cycles

On the Kontsevich integral for knotted trivalent graphs

We construct an extension of the Kontsevich integral of knots to knotted trivalent graphs, which commutes with orientation switches, edge deletions, edge unzips, and connected sums. In 1997 Murakami and Ohtsuki [MO] first constructed such an extension, building on Drinfel'd's theory of associators. We construct a step by step definition, using elementary Kontsevich integral methods, to get a one-parameter family of corrections that all yield invariants well behaved under the graph operations above.Comment: Journal version, 47 page

Path-Contractions, Edge Deletions and Connectivity Preservation

We study several problems related to graph modification problems under connectivity constraints from the perspective of parameterized complexity: {\sc (Weighted) Biconnectivity Deletion}, where we are tasked with deleting~$k$ edges while preserving biconnectivity in an undirected graph, {\sc Vertex-deletion Preserving Strong Connectivity}, where we want to maintain strong connectivity of a digraph while deleting exactly~$k$ vertices, and {\sc Path-contraction Preserving Strong Connectivity}, in which the operation of path contraction on arcs is used instead. The parameterized tractability of this last problem was posed by Bang-Jensen and Yeo [DAM 2008] as an open question and we answer it here in the negative: both variants of preserving strong connectivity are $\sf W[1]$-hard. Preserving biconnectivity, on the other hand, turns out to be fixed parameter tractable and we provide a $2^{O(k\log k)} n^{O(1)}$-algorithm that solves {\sc Weighted Biconnectivity Deletion}. Further, we show that the unweighted case even admits a randomized polynomial kernel. All our results provide further interesting data points for the systematic study of connectivity-preservation constraints in the parameterized setting

Improved Algorithms for Decremental Single-Source Reachability on Directed Graphs

Recently we presented the first algorithm for maintaining the set of nodes reachable from a source node in a directed graph that is modified by edge deletions with $o(mn)$ total update time, where $m$ is the number of edges and $n$ is the number of nodes in the graph [Henzinger et al. STOC 2014]. The algorithm is a combination of several different algorithms, each for a different $m$ vs. $n$ trade-off. For the case of $m = \Theta(n^{1.5})$ the running time is $O(n^{2.47})$, just barely below $mn = \Theta(n^{2.5})$. In this paper we simplify the previous algorithm using new algorithmic ideas and achieve an improved running time of $\tilde O(\min(m^{7/6} n^{2/3}, m^{3/4} n^{5/4 + o(1)}, m^{2/3} n^{4/3+o(1)} + m^{3/7} n^{12/7+o(1)}))$. This gives, e.g., $O(n^{2.36})$ for the notorious case $m = \Theta(n^{1.5})$. We obtain the same upper bounds for the problem of maintaining the strongly connected components of a directed graph undergoing edge deletions. Our algorithms are correct with high probabililty against an oblivious adversary.Comment: This paper was presented at the International Colloquium on Automata, Languages and Programming (ICALP) 2015. A full version combining the findings of this paper and its predecessor [Henzinger et al. STOC 2014] is available at arXiv:1504.0795

Editing to a Graph of Given Degrees

We consider the Editing to a Graph of Given Degrees problem that asks for a graph G, non-negative integers d,k and a function \delta:V(G)->{1,...,d}, whether it is possible to obtain a graph G' from G such that the degree of v is \delta(v) for any vertex v by at most k vertex or edge deletions or edge additions. We construct an FPT-algorithm for Editing to a Graph of Given Degrees parameterized by d+k. We complement this result by showing that the problem has no polynomial kernel unless NP\subseteq coNP/poly

On dynamic breadth-first search in external-memory

We provide the first non-trivial result on dynamic breadth-first search (BFS) in external-memory: For general sparse undirected graphs of initially $n$ nodes and O(n) edges and monotone update sequences of either $\Theta(n)$ edge insertions or $\Theta(n)$ edge deletions, we prove an amortized high-probability bound of O(n/B^{2/3}+\sort(n)\cdot \log B) I/Os per update. In contrast, the currently best approach for static BFS on sparse undirected graphs requires \Omega(n/B^{1/2}+\sort(n)) I/Os. 1998 ACM Subject Classification: F.2.2. Key words and phrases: External Memory, Dynamic Graph Algorithms, BFS, Randomization