Related papers: Index notation of grid graphs
Let $G$ be a finite group. Define a graph on the set $G^{\#} = G \setminus \{ 1 \}$ by declaring distinct elements $x,y\in G^{\#}$ to be adjacent if and only if $\langle x,y\rangle$ is cyclic. Denote this graph by $\Delta(G)$. The graph…
Any graph which is not vertex transitive has a proper induced subgraph which is unique due to its structure or the way of its connection to the rest of the graph. We have called such subgraph as an anchor. Using an anchor which, in fact, is…
We classify the connected-homogeneous digraphs with more than one end. We further show that if their underlying undirected graph is not connected-homogeneous, they are highly-arc-transitive.
Signed graphs are graphs with signed edges. They are commonly used to represent positive and negative relationships in social networks. While balance theory and clusterizable graphs deal with signed graphs to represent social interactions,…
In this article, we considered a fractal image as a fractal curve, that is, as a walk on a grid in Euclidean space $\R^d$. We placed integers on the generating vectors of a grid, such that opposite directions have opposite numbers. This…
Let B be a finite collection of geometric (not necessarily convex) bodies in the plane. Clearly, this class of geometric objects naturally generalizes the class of disks, lines, ellipsoids, and even convex polygons. We consider geometric…
Chord diagrams, under the name of Gauss diagrams, are used in low-dimensional topology as an important tool for studying curves or knots. Those Gauss diagrams that correspond to curves or knots are called realizable. The theme of our paper…
A graph is called matching covered if for its every edge there is a maximum matching containing it. It is shown that minimal matching covered graphs contain a perfect matching.
A consistent path system in a graph $G$ is an intersection-closed collection of paths, with exactly one path between any two vertices in $G$. We call $G$ metrizable if every consistent path system in it is the system of geodesic paths…
Network models with latent geometry have been used successfully in many applications in network science and other disciplines, yet it is usually impossible to tell if a given real network is geometric, meaning if it is a typical element in…
The directions of an infinite graph $G$ are a tangle-like description of its ends: they are choice functions that choose compatibly for all finite vertex sets $X\subseteq V(G)$ a component of $G-X$. Although every direction is induced by a…
Understanding and interacting with everyday physical scenes requires rich knowledge about the structure of the world, represented either implicitly in a value or policy function, or explicitly in a transition model. Here we introduce a new…
A signed graph is a graph whose edges are labelled positive or negative. The sign of a circle (cycle, circuit) is the product of the signs of its edges. Most of the essential properties of a signed graph depend on the signs of its circles.…
Let $\mathcal{A}$ be a set of positive numbers. A graph $G$ is called an $\mathcal{A}$-embeddable graph in $\mathbb{R}^d$ if the vertices of $G$ can be positioned in $\mathbb{R}^d$ so that the distance between endpoints of any edge is an…
We propose a new framework for the recognition of online handwritten graphics. Three main features of the framework are its ability to treat symbol and structural level information in an integrated way, its flexibility with respect to…
A graph is periodic if it can be obtained by joining identical pieces in a cyclic fashion. It is shown that the limit crossing number of a periodic graph is computable. This answers a question of Benny Pinontoan and Bruce Richter (2004).
Graphs are widely used as a popular representation of the network structure of connected data. Graph data can be found in a broad spectrum of application domains such as social systems, ecosystems, biological networks, knowledge graphs, and…
These lecture notes are a personal introduction to signed graphs, concentrating on the aspects that have been most persistently interesting to me. They are just a few corners of signed graph theory; I am leaving out a great deal. The…
A hypergraph is a $T_0$-hypergraph if for every two different vertices of the hypergraph there exists an edge containing one of the vertices and not containing the other. A general method for the enumeration of certain classes of…
A visibility representation is a classical drawing style of planar graphs. It displays the vertices of a graph as horizontal vertex-segments, and each edge is represented by a vertical edge-segment touching the segments of its end vertices;…