Related papers: Homothetic triangle representations of planar grap…
In this paper we show that every finite spatial graph is a connected sum of a planar graph, which is a forest, i.e. disjoint union of finite number of trees and a tangle. As a consequence we get that any finite spatial graph is a connected…
An outerplanar graph is a planar graph that has a planar drawing with all vertices on the unbounded face. The matching complex of a graph is the simplicial complex whose faces are subsets of disjoint edges of the graph. In this paper we…
A classification is given of all the countable homogeneous ordered bipartite graphs.
An \emph{obstacle representation} of a graph consists of a set of polygonal obstacles and a distinct point for each vertex such that two points see each other if and only if the corresponding vertices are adjacent. Obstacle representations…
A {\em string graph} is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the…
The 3-Decomposition Conjecture states that every connected cubic graph can be decomposed into a spanning tree, a 2-regular subgraph and a matching. We show that this conjecture holds for the class of connected plane cubic graphs.
A permutation graph is a graph that can be derived from a permutation, where the vertices correspond to letters of the permutation, and the edges represent inversions. We provide a construction to show that there are infinitely many…
We consider several classes of intersection graphs of line segments in the plane and prove new equality and separation results between those classes. In particular, we show that: (1) intersection graphs of grounded segments and intersection…
An invariant for cospectral graphs is a property shared by all cospectral graphs. In this paper, we establish three novel arithmetic invariants for cospectral graphs, revealing deep connections between spectral properties and combinatorial…
Planar graphs can be represented as intersection graphs of different types of geometric objects in the plane, e.g., circles (Koebe, 1936), line segments (Chalopin \& Gon{\c{c}}alves, 2009), \textsc{L}-shapes (Gon{\c{c}}alves et al, 2018).…
Let $G=(V,E)$ be a graph of order $n$ and let $1\leq k< n$ be an integer. The $k$-token graph of $G$ is the graph whose vertices are all the $k$-subsets of $V$, two of which are adjacent whenever their symmetric difference is a pair of…
While planar graphs are flat from a topological viewpoint, we observe that they are not from a geometric one. We prove that every planar graph can be embedded into a surface consisting of spheres, glued together in a tree-like fashion. As a…
Tutte showed that a graph $G$ is planar if and only if the conflict graph associated to every cycle of $G$ is bipartite. We define a (not necessarily unique) signed conflict graph associated to a maximally planar subgraph of a nonplanar…
We calculate the intersection ring of three-dimensional graph manifolds with rational coefficients and give an algebraic characterization of these rings when the manifold's underlying graph is a tree. We are able to use this…
In this paper we prove that a graph is a string graph (the intersection graph of curves in the plane) if and only if it admits a drawing in the plane with certain properties. This also allows us to define an algebraic obstruction, similar…
In this paper the concept of homothetic parallelogram is introduced. This concept is a generalization of Varignon's and Wittenbauer's parallelograms of an arbitrary quadrangle, whose diagonals are not parallel. A formula for the area and…
The planar Tur\'{a}n number of a given graph $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges over all planar graphs on $n$ vertices that do not contain a copy of $H$ as a subgraph. Let $H_k$ be a friendship graph,…
We introduce and study a new graph representation where vertices are embedded in three or more dimensions, and in which the edges are drawn on the projections onto the axis-parallel planes. We show that the complete graph on $n$ vertices…
Consider two horizontal lines in the plane. A pair of a point on the top line and an interval on the bottom line defines a triangle between two lines. The intersection graph of such triangles is called a simple-triangle graph. This paper…
We consider arrangements of axis-aligned rectangles in the plane. A geometric arrangement specifies the coordinates of all rectangles, while a combinatorial arrangement specifies only the respective intersection type in which each pair of…