Related papers: Conformal Hypergraphs: Duality and Implications fo…
In the past decades for more and more graph classes the Graph Isomorphism Problem was shown to be solvable in polynomial time. An interesting family of graph classes arises from intersection graphs of geometric objects. In this work we show…
A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of $H$-free graphs, that is, graphs that do not contain some graph $H$ as an induced subgraph, have…
For a graph $H$, the $H$-recolouring problem $\operatorname{Recol}(H)$ asks, for two given homomorphisms from a given graph $G$ to $H$, if one can get between them by a sequence of homomorphisms of $G$ to $H$ in which consecutive…
For any finite set $\mathcal{H} = \{H_1,\ldots,H_p\}$ of graphs, a graph is $\mathcal{H}$-subgraph-free if it does not contain any of $H_1,\ldots,H_p$ as a subgraph. In recent work, meta-classifications have been studied: these show that if…
A $3$-uniform hypergraph is a generalization of simple graphs where each hyperedge is a subset of vertices of size $3$. The degree of a vertex in a hypergraph is the number of hyperedges incident with it. The degree sequence of a hypergraph…
The notion of graph covers (also referred to as locally bijective homomorphisms) plays an important role in topological graph theory and has found its computer science applications in models of local computation. For a fixed target graph…
We consider the Minimum Coverage Kernel problem: given a set $B$ of $d$-dimensional boxes, find a subset of $B$ of minimum size covering the same region as $B$. This problem is $\mathsf{NP}$-hard, but as for many $\mathsf{NP}$-hard problems…
A graph $G$ is {\em matching-decyclable} if it has a matching $M$ such that $G-M$ is acyclic. Deciding whether $G$ is matching-decyclable is an NP-complete problem even if $G$ is 2-connected, planar, and subcubic. In this work we present…
A matching in a hypergraph $H$ is a set of pairwise vertex disjoint edges in $H$ and the matching number of $H$ is the maximum cardinality of a matching in $H$. A transversal in $H$ is a subset of vertices in $H$ that has a nonempty…
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…
In a graph, a perfect matching cut is an edge cut that is a perfect matching. Perfect Matching Cut (PMC) is the problem of deciding whether a given graph has a perfect matching cut, and is known to be NP-complete. We revisit the problem and…
A clique-coloring of a graph $G$ is a coloring of the vertices of $G$ so that no maximal clique of size at least two is monochromatic. The clique-hypergraph, $\mathcal{H}(G)$, of a graph $G$ has $V(G)$ as its set of vertices and the maximal…
The problem of finding the densest subgraph in a given graph has several applications in graph mining, particularly in areas like social network analysis, protein and gene analyses etc. Depending on the application, finding dense subgraphs…
A clique of a graph is a maximal set of vertices of size at least 2 that induces a complete graph. A $k$-clique-colouring of a graph is a colouring of the vertices with at most $k$ colours such that no clique is monochromatic. D\'efossez…
In the last decade, algorithmic frameworks based on a structural graph parameter called mim-width have been developed to solve generally NP-hard problems. However, it is known that the frameworks cannot be applied to the Clique problem, and…
The MEG (minimum equivalent graph) problem is, given a directed graph, to find a small subset of the edges that maintains all reachability relations between nodes. The problem is NP-hard. This paper gives a proof that, for graphs where each…
A graph $G$ realizes the degree sequence $S$ if the degrees of its vertices is $S$. Hakimi gave a necessary and sufficient condition to guarantee that there exists a connected multigraph realizing $S$. Taylor later proved that any connected…
${ NP}$-complete problem "Hamiltonian cycle"\ for graph $G=(V,E)$ is extended to the "Hamiltonian Complement of the Graph"\ problem of finding the minimal cardinality set $H$ containing additional edges so that graph $G=(V,E\cup H)$ is…
In this paper, we relate the problem of finding a maximum clique to the intersection number of the input graph (i.e. the minimum number of cliques needed to edge cover the graph). In particular, we consider the maximum clique problem for…
We investigate the problem of simultaneously dominating all spanning trees of a given graph. We prove that on 2-connected graphs, a subset of the vertices dominates all spanning trees of the graph if and only if it is a vertex cover. Using…