Related papers: Faster algorithms for graph homomorphism via tract…

A simple-triangle graph is the intersection graph of triangles that are defined by a point on a horizontal line and an interval on another horizontal line. The time complexity of the recognition problem for simple-triangle graphs was a…

We give a fixed-parameter tractable algorithm that, given a parameter $k$ and two graphs $G_1,G_2$, either concludes that one of these graphs has treewidth at least $k$, or determines whether $G_1$ and $G_2$ are isomorphic. The running time…

We study the Hamilton cycle problem with input a random graph G=G(n,p) in two settings. In the first one, G is given to us in the form of randomly ordered adjacency lists while in the second one we are given the adjacency matrix of G. In…

We study Hamiltonicity and pancyclicity in the graph obtained as the union of a deterministic $n$-vertex graph $H$ with $\delta(H)\geq\alpha n$ and a random $d$-regular graph $G$, for $d\in\{1,2\}$. When $G$ is a random $2$-regular graph,…

Consider the family of graphs without $ k $ node-disjoint odd cycles, where $ k $ is a constant. Determining the complexity of the stable set problem for such graphs $ G $ is a long-standing problem. We give a polynomial-time algorithm for…

In this paper, we prove that, given a clique-width $k$-expression of an $n$-vertex graph, \textsc{Hamiltonian Cycle} can be solved in time $n^{\mathcal{O}(k)}$. This improves the naive algorithm that runs in time $n^{\mathcal{O}(k^2)}$ by…

Let us be given two graphs $\Gamma_1$, $\Gamma_2$ of $n$ vertices. Are they isomorphic? If they are, the set of isomorphisms from $\Gamma_1$ to $\Gamma_2$ can be identified with a coset $H\cdot\pi$ inside the symmetric group on $n$…

A graph $G = (V,E)$ is $\textit{monopolar}$ if its vertex set admits a partition $V = (C \uplus{} I)$ where $G[C]$ is a $\textit{cluster graph}$ and $I$ is an $\textit{independent set}$ in $G$; this is a \textit{monopolar partition} of $G$.…

In the Hedge Cut problem, the edges of a graph are partitioned into groups called hedges, and the question is what is the minimum number of hedges to delete to disconnect the graph. Ghaffari, Karger, and Panigrahi [SODA 2017] showed that…

For graphs $G$ and $H$, a mapping $f: V(G)\dom V(H)$ is a homomorphism of $G$ to $H$ if $uv\in E(G)$ implies $f(u)f(v)\in E(H).$ If, moreover, each vertex $u \in V(G)$ is associated with costs $c_i(u), i \in V(H)$, then the cost of the…

Dirac's theorem (1952) is a classical result of graph theory, stating that an $n$-vertex graph ($n \geq 3$) is Hamiltonian if every vertex has degree at least $n/2$. Both the value $n/2$ and the requirement for every vertex to have high…

For graphs $G$ and $H$, an $H$-coloring of $G$ is an edge-preserving mapping from $V(G)$ to $V(H)$. In the $H$-Coloring problem the graph $H$ is fixed and we ask whether an instance graph $G$ admits an $H$-coloring. A generalization of this…

For a connected graph $G=(V,E)$, a matching $M\subseteq E$ is a matching cut of $G$ if $G-M$ is disconnected. It is known that for an integer $d$, the corresponding decision problem Matching Cut is polynomial-time solvable for graphs of…

We generalize the structure theorem of Robertson and Seymour for graphs excluding a fixed graph $H$ as a minor to graphs excluding $H$ as a topological subgraph. We prove that for a fixed $H$, every graph excluding $H$ as a topological…

Given an $n$-vertex $m$-edge graph $G$ with non negative edge-weights, the girth of $G$ is the weight of a shortest cycle in $G$. For any graph $G$ with polynomially bounded integer weights, we present a deterministic algorithm that…

We consider a generalization of finding a homomorphism from an input digraph $G$ to a fixed digraph $H$, HOM($H$). In this setting, we are given an input digraph $G$ together with a list function from $G$ to $2^H$. The goal is to find a…

A graph H is a vertex-minor of a graph G if it can be reached from G by the successive application of local complementations and vertex deletions. Vertex-minors have been the subject of intense study in graph theory over the last decades…

We improve the running time of the general algorithmic technique known as Baker's approach (1994) on H-minor-free graphs from O(n^{f(|H|)}) to O(f(|H|) n^{O(1)}). The numerous applications include e.g. a 2-approximation for coloring and…

We study the problem of determining whether an $n$-node graph $G$ has an even hole, i.e., an induced simple cycle consisting of an even number of nodes. Conforti, Cornu\'ejols, Kapoor, and Vu\v{s}kovi\'c gave the first polynomial-time…

Counting homomorphisms of a constant sized pattern graph $H$ in an input graph $G$ is a fundamental computational problem. There is a rich history of studying the complexity of this problem, under various constraints on the input $G$ and…