Related papers: The graph tessellation cover number: extremal boun…
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 complete graph is the graph in which every two vertices are adjacent. For a graph $G=(V,E)$, the complete width of $G$ is the minimum $k$ such that there exist $k$ independent sets $\mathtt{N}_i\subseteq V$, $1\le i\le k$, such that the…
Bootstrap percolation in (random) graphs is a contagion dynamics among a set of vertices with certain threshold levels. The process is started by a set of initially infected vertices, and an initially uninfected vertex with threshold $k$…
In this paper we resolve the complexity of the isomorphism problem on all but finitely many of the graph classes characterized by two forbidden induced subgraphs. To this end we develop new techniques applicable for the structural and…
An ordered graph is a graph enhanced with a linear order on the vertex set. An ordered graph is a core if it does not have an order-preserving homomorphism to a proper subgraph. We say that $H$ is the core of $G$ if (i) $H$ is a core, (ii)…
Motivated by applications in graph drawing and information visualization, we examine the planar split thickness of a graph, that is, the smallest $k$ such that the graph is $k$-splittable into a planar graph. A $k$-split operation…
The positive zero forcing number of a graph is a graph parameter that arises from a non-traditional type of graph colouring, and is related to a more conventional version of zero forcing. We establish a relation between the zero forcing and…
We consider the problem of covering an input graph $H$ with graphs from a fixed covering class $G$. The classical covering number of $H$ with respect to $G$ is the minimum number of graphs from $G$ needed to cover the edges of $H$ without…
Covering all edges of a graph by a small number of vertices, this is the NP-complete Vertex Cover problem. It is among the most fundamental graph-algorithmic problems. Following a recent trend in studying temporal graphs (a sequence of…
We demonstrate that graphs embedded on surfaces are a powerful and practical tool to generate, characterize and simulate networks with a broad range of properties. Remarkably, the study of topologically embedded graphs is non-restrictive…
A graph covering projection, also referred to as a locally bijective homomorphism, is a mapping between the vertices and edges of two graphs that preserves incidences and is a local bijection. This concept originates in topological graph…
The paper considers the NP-hard graph vertex coloring problem, which differs from traditional problems in which it is required to color vertices with a given (or minimal) number of colors so that adjacent vertices have different colors. In…
A proper edge-coloring of a graph is an interval coloring if the labels on the edges incident to any vertex form an interval of consecutive integers. Interval thickness s(G) of a graph G is the smallest number of interval colorable graphs…
Given an n-vertex graph G, a drawing of G in the plane is a mapping of its vertices into points of the plane, and its edges into continuous curves, connecting the images of their endpoints. A crossing in such a drawing is a point where two…
We show that the following problems are NP-complete. 1. Can the vertex set of a graph be partitioned into two sets such that each set induces a perfect graph? 2. Is the difference between the chromatic number and clique number at most $1$…
The intersection graph of a collection of trapezoids with corner points lying on two parallel lines is called a trapezoid graph. Using binary indexed tree data structure, we improve algorithms for calculating the size and the number of…
We consider the selective graph coloring problem, which is a generalization of the classical graph coloring problem. Given a graph together with a partition of its vertex set into clusters, we want to choose exactly one vertex per cluster…
In this paper we show that the problem of identifying an edge $(i,j)$ in a graph $G$ such that there exists an optimal vertex cover $S$ of $G$ containing exactly one of the nodes $i$ and $j$ is NP-hard. Such an edge is called a weak edge.…
An embedding of a graph on a translation surface is said to be \emph{systolic} if each vertex of the graph corresponds to a singular point (or marked point) and each edge corresponds to a shortest saddle connection on the translation…
Temporal graphs naturally model graphs whose underlying topology changes over time. Recently, the problems TEMPORAL VERTEX COVER (or TVC) and SLIDING-WINDOW TEMPORAL VERTEX COVER(or $\Delta$-TVC for time-windows of a fixed-length $\Delta$)…