Related papers: When can Graph Hyperbolicity be computed in Linear…
Boxicity of a graph $G(V,$ $E)$, denoted by $box(G)$, is the minimum integer $k$ such that $G$ can be represented as the intersection graph of axis parallel boxes in $\mathbb{R}^k$. The problem of computing boxicity is inapproximable even…
Diameter -- the task of computing the length of a longest shortest path -- is a fundamental graph problem. Assuming the Strong Exponential Time Hypothesis, there is no $O(n^{1.99})$-time algorithm even in sparse graphs [Roditty and…
We give an algorithm that for an input n-vertex graph G and integer k>0, in time 2^[O(k)]n either outputs that the treewidth of G is larger than k, or gives a tree decomposition of G of width at most 5k+4. This is the first algorithm…
Given a clique-width $k$-expression of a graph $G$, we provide $2^{O(k)}\cdot n$ time algorithms for connectivity constraints on locally checkable properties such as Node-Weighted Steiner Tree, Connected Dominating Set, or Connected Vertex…
Betweenness centrality measure assesses the importance of nodes in a graph and has been used in a variety of contexts. Betweenness centrality has also been extended to temporal graphs. Temporal graphs have edges that bear labels according…
The maximum modularity of a graph is a parameter widely used to describe the level of clustering or community structure in a network. Determining the maximum modularity of a graph is known to be NP-complete in general, and in practice a…
We study the problem of finding a temporal hybridization network for a set of phylogenetic trees that minimizes the number of reticulations. First, we introduce an FPT algorithm for this problem on an arbitrary set of $m$ binary trees with…
Connectivity (or equivalently, unweighted maximum flow) is an important measure in graph theory and combinatorial optimization. Given a graph $G$ with vertices $s$ and $t$, the connectivity $\lambda(s,t)$ from $s$ to $t$ is defined to be…
Generalised hypertree width ($ghw$) is a hypergraph parameter that is central to the tractability of many prominent problems with natural hypergraph structure. Computing $ghw$ of a hypergraph is notoriously hard. The decision version of the…
We consider the problem of augmenting an n-vertex graph embedded in a metric space, by inserting one additional edge in order to minimize the diameter of the resulting graph. We present exact algorithms for the cases when (i) the input…
The hypergraph unreliability problem asks for the probability that a hypergraph gets disconnected when every hyperedge fails independently with a given probability. For graphs, the unreliability problem has been studied over many decades,…
Treedepth is a central parameter to algorithmic graph theory. The current state-of-the-art in computing and approximating treedepth consists of a $2^{O(k^2)} n$-time exact algorithm and a polynomial-time $O(\text{OPT} \log^{3/2}…
We show that the VC-dimension of a graph can be computed in time $n^{\log d+1} d^{O(d)}$, where $d$ is the degeneracy of the input graph. The core idea of our algorithm is a data structure to efficiently query the number of vertices that…
Let $G = (V,E)$ be a connected directed graph on $n$ vertices. Assign values from the set $\{1,2,\dots,n\}$ to the vertices of $G$ and update the values according to the following rule: uniformly at random choose a vertex and update its…
Unbreakable decomposition, introduced by Cygan et al. (SICOMP'19) and Cygan et al. (TALG'20), has proven to be one of the most powerful tools for parameterized graph cut problems in recent years. Unfortunately, all known constructions…
We study algorithmic and structural aspects of connectivity in hypergraphs. Given a hypergraph $H=(V,E)$ with $n = |V|$, $m = |E|$ and $p = \sum_{e \in E} |e|$ the best known algorithm to compute a global minimum cut in $H$ runs in time…
The study of structural graph width parameters like tree-width, clique-width and rank-width has been ongoing during the last five decades, and their algorithmic use has also been increasing [Cygan et al., 2015]. New width parameters…
An L(2,1)-labeling of a graph $G$ is an assignment $f$ from the vertex set $V(G)$ to the set of nonnegative integers such that $|f(x)-f(y)|\ge 2$ if $x$ and $y$ are adjacent and $|f(x)-f(y)|\ge 1$ if $x$ and $y$ are at distance 2, for all…
We consider the problem of computing the diameter of a unicycle graph (i.e., a graph with a unique cycle). We present an O(n) time algorithm for the problem, where n is the number of vertices of the graph. This improves the previous best…
Stable gonality is a multigraph parameter that measures the complexity of a graph. It is defined using maps to trees. Those maps, in some sense, divide the edges equally over the edges of the tree; stable gonality asks for the map with the…