Related papers: A $2\ell k$ Kernel for $\ell$-Component Order Conn…

We prove that for any integer $k\geq 2$ and $\varepsilon>0$, there is an integer $\ell_0\geq 1$ such that any $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $(1/2+\varepsilon)n$ has a fractional decomposition into…

A popular model to measure the stability of a network is k-core - the maximal induced subgraph in which every vertex has at least k neighbors. Many studies maximize the number of vertices in k-core to improve the stability of a network. In…

Let $S$ be a nonempty set of vertices of a connected graph $G$. A collection $T_1,..., T_\ell$ of trees in $G$ is said to be internally disjoint trees connecting $S$ if $E(T_i)\cap E(T_j)= \emptyset$ and $V(T_i)\cap V(T_j)=S$ for any pair…

A matching cut of a graph is a partition of its vertex set in two such that no vertex has more than one neighbor across the cut. The Matching Cut problem asks if a graph has a matching cut. This problem, and its generalization d-cut, has…

Our work concerns algorithms for an unweighted variant of Maximum Flow. In the All-Pairs Connectivity (APC) problem, we are given a graph $G$ on $n$ vertices and $m$ edges, and are tasked with computing the maximum number of edge-disjoint…

Let $G$ be a graph on $n$ vertices. We call a subset $A$ of the vertex set $V(G)$ \emph{$k$-small} if, for every vertex $v \in A$, $\deg(v) \le n - |A| + k$. A subset $B \subseteq V(G)$ is called \emph{$k$-large} if, for every vertex $u \in…

A graph H is a square root of a graph G if G can be obtained from H by the addition of edges between any two vertices in H that are of distance 2 from each other. The Square Root problem is that of deciding whether a given graph admits a…

For $\alpha\colon\mathbb{N}\rightarrow\mathbb{R}$, an $\alpha$-approximate bi-kernel is a polynomial-time algorithm that takes as input an instance $(I, k)$ of a problem $Q$ and outputs an instance $(I',k')$ of a problem $Q'$ of size…

We develop an FPT algorithm and a bi-kernel for the Weighted Edge Clique Partition (WECP) problem, where a graph with $n$ vertices and integer edge weights is given together with an integer $k$, and the aim is to find $k$ cliques, such that…

Given a graph $G=(V, E)$ and a positive integer $k$, in Maximum $k$-Order Bounded Component Set (Max-$k$-OBCS), it is required to find a vertex set $S \subseteq V$ of maximum size such that each component in the induced graph $G[S]$ has at…

We show a kernel of at most $13k$ vertices for the Planar Feedback Vertex Set problem restricted to planar graphs, i.e., a polynomial-time algorithm that transforms an input instance $(G,k)$ to an equivalent instance with at most $13k$…

For a connected graph, a {\em minimum vertex separator} is a minimum set of vertices whose removal creates at least two connected components. The vertex connectivity of the graph refers to the size of the minimum vertex separator and a…

The $P_2$-packing problem asks for whether a graph contains $k$ vertex-disjoint paths each of length two. We continue the study of its kernelization algorithms, and develop a $5k$-vertex kernel.

Given an undirected graph $G$ with $m$ edges, $n$ vertices, and non-negative edge weights, and given an integer $k\geq 2$, we show that a $(2k-1)$-approximate distance oracle for $G$ of size $O(kn^{1 + 1/k})$ and with $O(\log k)$ query time…

For $S\subseteq V(G)$ and $|S|\geq 2$, $\lambda(S)$ is the maximum number of edge-disjoint trees connecting $S$ in $G$. For an integer $k$ with $2\leq k\leq n$, the \emph{generalized $k$-edge-connectivity} $\lambda_k(G)$ of $G$ is then…

For two positive integers $k$ and $\ell$, a $(k \times \ell)$-spindle is the union of $k$ pairwise internally vertex-disjoint directed paths with $\ell$ arcs between two vertices $u$ and $v$. We are interested in the (parameterized)…

Consider the following problem: given a graph with edge costs and a subset Q of vertices, find a minimum-cost subgraph in which there are two edge-disjoint paths connecting every pair of vertices in Q. The problem is a failure-resilient…

We give the first analysis of the computational complexity of {\it coalition structure generation over graphs}. Given an undirected graph $G=(N,E)$ and a valuation function $v:2^N\rightarrow\RR$ over the subsets of nodes, the problem is to…

A subgraph $T$ of a digraph $D$ is an {\em out-branching} if $T$ is an oriented spanning tree with only one vertex of in-degree zero (called the {\em root}). The vertices of $T$ of out-degree zero are {\em leaves}. In the {\sc Directed…

We present a data structure that, given a graph $G$ of $n$ vertices and $m$ edges, and a suitable pair of nested $r$-divisions of $G$, preprocesses $G$ in $O(m+n)$ time and handles any series of edge-deletions in $O(m)$ total time while…