Related papers: Hard Problems That Quickly Become Very Easy

We study a general family of problems that form a common generalization of classic hitting (also referred to as covering or transversal) and packing problems. An instance of X-HitPack asks: Can removing k (deletable) vertices of a graph G…

An upper dominating set in a graph is a minimal (with respect to set inclusion) dominating set of maximum cardinality. The problem of finding an upper dominating set is generally NP-hard. We study the complexity of this problem in classes…

We consider the complexity of finding weighted homomorphisms from intersection graphs of curves (string graphs) with $n$ vertices to a fixed graph $H$. We provide a complete dichotomy for the problem: if $H$ has no two vertices sharing two…

A graph with degree sequence $\pi$ is a \emph{unigraph} if it is isomorphic to every graph that has degree sequence $\pi$. The class of unigraphs is not hereditary and in this paper we study the related hereditary class HCU, the hereditary…

For a graph $H$, the $H$-free Edge Deletion problem asks whether there exist at most $k$ edges whose deletion from the input graph $G$ results in a graph without any induced copy of $H$. We prove that $H$-free Edge Deletion is NP-complete…

A graph is distance-hereditary if for any pair of vertices, their distance in every connected induced subgraph containing both vertices is the same as their distance in the original graph. The Distance-Hereditary Vertex Deletion problem…

A class of graphs is nowhere dense if for every integer r there is a finite upper bound on the size of cliques that occur as (topological) r-minors. We observe that this tameness notion from algorithmic graph theory is essentially the…

Testing if a given graph $G$ contains the $k$-vertex path $P_k$ as a minor or as an induced minor is trivial for every fixed integer $k\geq 1$. However, the situation changes for the problem of checking if a graph can be modified into $P_k$…

We prove that it is NP-hard to decide whether a graph is the square of a 6-apex graph. This shows that the square root problem is not tractable for squares of sparse graphs (or even graphs from proper minor-closed classes).

For a graph $G$ and a hereditary property $\mathcal{P}$, let $\text{ex}(G,\mathcal{P})$ denote the maximum number of edges of a subgraph of $G$ that belongs to $\mathcal{P}$. We prove that for every non-trivial hereditary property…

Let $G$ be a graph on $n$ vertices with adjacency matrix $A$, and let $\mathbf{1}$ be the all-ones vector. We call $G$ controllable if the set of vectors $\mathbf{1}, A\mathbf{1}, \dots, A^{n-1}\mathbf{1}$ spans the whole space…

A graph is split if there is a partition of its vertex set into a clique and an independent set. The present paper is devoted to the splitness of some graphs related to finite simple groups, namely, prime graphs and solvable graphs, and…

A graph is $(H_1,H_2)$-free for a pair of graphs $H_1,H_2$ if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. In 2001, Kr\'al', Kratochv\'{\i}l, Tuza, and Woeginger initiated a study into the complexity of Colouring for…

In social networks the {\sc Strong Triadic Closure} is an assignment of the edges with strong or weak labels such that any two vertices that have a common neighbor with a strong edge are adjacent. The problem of maximizing the number of…

We consider combinatorial problems that can be solved in polynomial time for graphs of bounded treewidth but where the order of the polynomial that bounds the running time is expected to depend on the treewidth bound. First we review some…

For a graph $H$, the $H$-free Edge Deletion problem asks whether there exist at most $k$ edges whose deletion from the input graph $G$ results in a graph without any induced copy of $H$. $H$-free Edge Completion and $H$-free Edge Editing…

Given a hereditary graph property $\mathcal{P}$, consider distributions of random orderings of vertices of graphs $G\in\mathcal{P}$ that are preserved under isomorphisms and under taking induced subgraphs. We show that for many properties…

For a fixed set ${\cal H}$ of graphs, a graph $G$ is ${\cal H}$-subgraph-free if $G$ does not contain any $H \in {\cal H}$ as a (not necessarily induced) subgraph. A recently proposed framework gives a complete classification on ${\cal…

Temporal graphs are a popular modelling mechanism for dynamic complex systems that extend ordinary graphs with discrete time. Simply put, time progresses one unit per step and the availability of edges can change with time. We consider the…

We study the complexity of graph problems on graphs defined on groups, especially power graphs. We observe that an isomorphism invariant problem, such as Hamiltonian Path, Partition into Cliques, Feedback Vertex Set, Subgraph Isomorphism,…