Related papers: Efficient recognition algorithms for $(1,2)$-, $(2…
The Path Contraction and Cycle Contraction problems take as input an undirected graph $G$ with $n$ vertices, $m$ edges and an integer $k$ and determine whether one can obtain a path or a cycle, respectively, by performing at most $k$ edge…
Connectivity related concepts are of fundamental interest in graph theory. The area has received extensive attention over four decades, but many problems remain unsolved, especially for directed graphs. A directed graph is 2-edge-connected…
We give an algorithm that decides whether the bipartite crossing number of a given graph is at most $k$. The running time of the algorithm is upper bounded by $2^{O(k)} + n^{O(1)}$, where $n$ is the number of vertices of the input graph,…
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 study finding and listing $k$-cliques in a graph, for constant $k\geq 3$, a fundamental problem of both theoretical and practical importance. Our main contribution is a new output-sensitive algorithm for listing $k$-cliques in graphs,…
A graph $G = (V,E)$ is a double-threshold graph if there exist a vertex-weight function $w \colon V \to \mathbb{R}$ and two real numbers $\mathtt{lb}, \mathtt{ub} \in \mathbb{R}$ such that $uv \in E$ if and only if $\mathtt{lb} \le…
We consider the (exact, minimum) $k$-cut problem: given a graph and an integer $k$, delete a minimum-weight set of edges so that the remaining graph has at least $k$ connected components. This problem is a natural generalization of the…
Given a graph $G = (V, E)$ and an integer $k$, the Minimum Membership Dominating Set problem asks to compute a set $S \subseteq V$ such that for each $v \in V$, $1 \leq |N[v] \cap S| \leq k$. The problem is known to be NP-complete even on…
A connected graph has a $(k,\ell)$-cover if each of its edges is contained in at least $\ell$ cliques of order $k$. Motivated by recent advances in extremal combinatorics and the literature on edge modification problems, we study the…
We study the time complexity of induced subgraph isomorphism problems where the pattern graph is fixed. The earliest known example of an improvement over trivial algorithms is by Itai and Rodeh (1978) who sped up triangle detection in…
In the EDGE CLIQUE COVER (ECC) problem, given a graph G and an integer k, we ask whether the edges of G can be covered with k complete subgraphs of G or, equivalently, whether G admits an intersection model on k-element universe. Gramm et…
Many NP-hard problems, such as Dominating Set, are FPT parameterized by clique-width. For graphs of clique-width $k$ given with a $k$-expression, Dominating Set can be solved in $4^k n^{O(1)}$ time. However, no FPT algorithm is known for…
We present linear time {\it in-place} algorithms for several basic and fundamental graph problems including the well-known graph search methods (like depth-first search, breadth-first search, maximum cardinality search), connectivity…
The $(n-\ell)$-deck of an $n$-vertex graph is the multiset of subgraphs obtained from it by deleting $\ell$ vertices. A family of $n$-vertex graphs is $\ell$-recognizable if every graph having the same $(n-\ell)$-deck as a graph in the…
2-joins are edge cutsets that naturally appear in the decomposition of several classes of graphs closed under taking induced subgraphs, such as balanced bipartite graphs, even-hole-free graphs, perfect graphs and claw-free graphs. Their…
A graph is $k$-clique-extendible if there is an ordering of the vertices such that whenever two $k$-sized overlapping cliques $A$ and $B$ have $k-1$ common vertices, and these common vertices appear between the two vertices $a,b\in…
A maximal matching $M$ that consists of independent edges is a subgraph of a simple and undirected graph $G$ for which $G-M$ forms an independent set. A graph $G$ is called equimatchable if all maximal matchings have the same number of…
We introduce the class of interval $H$-graphs, which is the generalization of interval graphs, particularly interval bigraphs. For a fixed graph $H$ with vertices $a_1,a_2,\dots,a_k$, we say that an input graph $G$ with given partition…
Various classes of induced subgraphs are involved in the deepest results of graph theory and graph algorithms. A prominent example concerns the {\em perfection} of $G$ that the chromatic number of each induced subgraph $H$ of $G$ equals the…
We give algorithms with running time $2^{O({\sqrt{k}\log{k}})} \cdot n^{O(1)}$ for the following problems. Given an $n$-vertex unit disk graph $G$ and an integer $k$, decide whether $G$ contains (1) a path on exactly/at least $k$ vertices,…