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We study the complexity of a class of promise graph homomorphism problems. For a fixed graph H, the H-colouring problem is to decide whether a given graph has a homomorphism to H. By a result of Hell and Ne\v{s}et\v{r}il, this problem is…
A storyplan visualizes a graph $G=(V,E)$ as a sequence of $\ell$ frames $\Gamma_1, \dots, \Gamma_\ell$, each of which is a drawing of the induced subgraph $G[V_i]$ of a vertex subset $V_i \subseteq V$. Moreover, each vertex $v \in V$ is…
The NP-hard Metric Dimension problem is to decide for a given graph G and a positive integer k whether there is a vertex subset of size at most k that separates all vertex pairs in G. Herein, a vertex v separates a pair {u,w} if the…
A $k$-bend path is a non-self-intersecting polyline in the plane made of at most $k+1$ axis-parallel line segments. B$_k$-VPG is the class of graphs which can be represented as intersection graphs of $k$-bend paths in the same plane. In…
A string graph is the intersection graph of curves in the plane. Kratochv\'il previously showed the existence of infinitely many obstacles: graphs that are not string graphs but for which any edge contraction or vertex deletion produces a…
Let $G$ be a graph that is topologically embedded in the plane and let $\mathcal{A}$ be an arrangement of pseudolines intersecting the drawing of $G$. An aligned drawing of $G$ and $\mathcal{A}$ is a planar polyline drawing $\Gamma$ of $G$…
In line with the recent development in topological graph theory, we are considering undirected graphs that are allowed to contain {\em multiple edges}, {\em loops}, and {\em semi-edges}. A graph is called {\em simple} if it contains no…
IC-planar graphs are those graphs that admit a drawing where no two crossed edges share an end-vertex and each edge is crossed at most once. They are a proper subfamily of the 1-planar graphs. Given an embedded IC-planar graph $G$ with $n$…
An (h,s,t)-representation of a graph G consists of a collection of subtrees of a tree T, where each subtree corresponds to a vertex of G such that (i) the maximum degree of T is at most h, (ii) every subtree has maximum degree at mots s,…
The K-way vertex cut problem} consists in, given a graph G, finding a subset of vertices of a given size, whose removal partitions G into the maximum number of connected components. This problem has many applications in several areas. It…
The regular number of a graph G denoted by reg(G) is the minimum number of subsets into which the edge set of G can be partitioned so that the subgraph induced by each subset is regular. In this work we answer to the problem posed as an…
A fixed-mobile bigraph G is a bipartite graph such that the vertices of one partition set are given with fixed positions in the plane and the mobile vertices of the other part, together with the edges, must be added to the drawing. We…
Many NP-hard graph problems become easy for some classes of graphs. For example, coloring is easy for bipartite graphs, but NP-hard in general. So we can ask question like when does a hard problem become easy? What is the minimum…
In this survey, we explore recent literature on finding the cores of higher graphs using geometric and topological means. We study graphs, hypergraphs, and simplicial complexes, all of which are models of higher graphs. We study the notion…
Stick graphs are defined as follows. Let A (respectively B) be a set of vertical (respectively horizontal) segments in the plane such that the bottom endpoints of the segments in A and the left endpoints of the segments in B lie on the same…
A geometric intersection graph is constructed over a set of geometric objects, where each vertex represents a distinct object and an edge connects two vertices if and only if the corresponding objects intersect. We examine the problem of…
A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components. The toughness of a graph is the largest $t$ for which the graph is $t$-tough. A graph is minimally $t$-tough if…
A graph G is (d_1,..,d_l)-colorable if the vertex set of G can be partitioned into subsets V_1,..,V_l such that the graph G[V_i] induced by the vertices of V_i has maximum degree at most d_i for all 1 <= i <= l. In this paper, we focus on…
In the area of beyond-planar graphs, i.e. graphs that can be drawn with some local restrictions on the edge crossings, the recognition problem is prominent next to the density question for the different graph classes. For 1-planar graphs,…
An $H$-graph is an intersection graph of connected subgraphs of a suitable subdivision of a fixed graph $H$. Many important classes of graphs, including interval graphs, circular-arc graphs, and chordal graphs, can be expressed as…