Related papers: 1-Fan-Bundle-Planar Drawings of Graphs
Graph Neural Networks (GNNs) extend basic Neural Networks (NNs) by additionally making use of graph structure based on the relational inductive bias (edge bias), rather than treating the nodes as collections of independent and identically…
Topological drawings are representations of graphs in the plane, where vertices are represented by points, and edges by simple curves connecting the points. A drawing is simple if two edges intersect at most in a single point, either at a…
Recommender systems play a crucial role in addressing the issue of information overload by delivering personalized recommendations to users. In recent years, there has been a growing interest in leveraging graph neural networks (GNNs) for…
We consider the task of drawing a graph on multiple horizontal layers, where each node is assigned a layer, and each edge connects nodes of different layers. Known algorithms determine the orders of nodes on each layer to minimize crossings…
Edge intelligence has arisen as a promising computing paradigm for supporting miscellaneous smart applications that rely on machine learning techniques. While the community has extensively investigated multi-tier edge deployment for…
Let $B$ be a set of Eulerian subgraphs of a graph $G$. We say $B$ forms a $k$-basis if it is a minimum set that generates the cycle space of $G$, and any edge of $G$ lies in at most $k$ members of $B$. The basis number of a graph $G$,…
1-planar graphs are graphs that can be drawn in the plane such that any edge intersects with at most one other edge. Ackerman showed that the edges of a 1-planar graph can be partitioned into a planar graph and a forest, and claims that the…
Many real life networks present an average path length logarithmic with the number of nodes and a degree distribution which follows a power law. Often these networks have also a modular and self-similar structure and, in some cases -…
Vertex splitting replaces a vertex by two copies and partitions its incident edges amongst the copies. This problem has been studied as a graph editing operation to achieve desired properties with as few splits as possible, most often…
The problem of maximising the number of cliques among $n$-vertex graphs from various graph classes has received considerable attention. We investigate this problem for the class of $1$-planar graphs where we determine precisely the maximum…
We propose a fixed-parameter tractable algorithm for the \textsc{Max-Cut} problem on embedded 1-planar graphs parameterized by the crossing number $k$ of the given embedding. A graph is called 1-planar if it can be drawn in the plane with…
A 1-planar graph refers to a graph that can be drawn on the plane such that each edge has at most one crossing. In this paper, focusing on the spectral Tur\'{a}n-type problems of $1$-planar graphs, we determine completely the unique…
We improve the best known upper bound on the number of edges in a unit-distance graph on $n$ vertices for each $n\in\{16,\ldots,30\}$. When $n\leq 21$, our bounds match the best known lower bounds, and we fully enumerate the densest…
A drawing of a graph is greedy if for each ordered pair of vertices u and v, there is a path from u to v such that the Euclidean distance to v decreases monotonically at every vertex of the path. The existence of greedy drawings has been…
Why do many modern neural-network-based graph generative models fail to reproduce typical real-world network characteristics, such as high triangle density? In this work we study the limitations of edge independent random graph models, in…
Graph neural networks (GNNs) are effective machine learning models for many graph-related applications. Despite their empirical success, many research efforts focus on the theoretical limitations of GNNs, i.e., the GNNs expressive power.…
This paper presents an empirical study of the relationship between the density of small-medium sized random graphs and their planarity. It is well known that, when the number of vertices tends to infinite, there is a sharp transition…
In extension problems of partial graph drawings one is given an incomplete drawing of an input graph $G$ and is asked to complete the drawing while maintaining certain properties. A prominent area where such problems arise is that of…
It is well-known that every maximal planar graph has a matching of size at least $\tfrac{n+8}{3}$ if $n\geq 14$. In this paper, we investigate similar matching-bounds for maximal \emph{1-planar} graphs, i.e., graphs that can be drawn such…
The $(k,r)$-fan is the graph consisting of $k$ copies of the complete graph $K_r$ which intersect in a single vertex, and is denoted by $F_{k,r}$. Erd\H{o}s, F\"uredi, Gould and Gunderson [J. Combin. Theory Ser. B 64 (1995) 89--100]…