Related papers: Halving Lines and Their Underlying Graphs

We investigate straight-line drawings of topological graphs that consist of a planar graph plus one edge, also called almost-planar graphs. We present a characterization of such graphs that admit a straight-line drawing. The…

An octilinear drawing of a planar graph is one in which each edge is drawn as a sequence of horizontal, vertical and diagonal at 45 degrees line-segments. For such drawings to be readable, special care is needed in order to keep the number…

In this paper, we study the signed domination numbers of graphs and present new sharp lower and upper bounds for this parameter. As an example, we present a lower bound on signed domination number of trees in terms of the order, leaves and…

This article emphasizes an extension of the study of metric and par- tition dimension to hypergraphs. We give a sharp lower bounds for the metric and partition dimension of hypergraphs in general and give exact values under specified…

We study the average number $\mathcal{A}(G)$ of colors in the non-equivalent colorings of a graph $G$. We show some general properties of this graph invariant and determine its value for some classes of graphs. We then conjecture several…

We present, to the best of the authors' knowledge, all known results for the (planar) crossing numbers of specific graphs and graph families. The results are separated into various categories; specifically, results for general graph…

We introduce and study the concept which we call the splitting of a graph and compare algebraic properties of the edge ideals of graphs and those of their splitting graphs.

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…

We exactly evaluate the entanglement of a six vertex and a nine vertex graph states which correspond to non ''two-colorable'' graphs. The upper bound of entanglement for five vertices ring graph state is improved to 2.9275, less than upper…

The study of nonplanar drawings of graphs with restricted crossing configurations is a well-established topic in graph drawing, often referred to as beyond-planar graph drawing. One of the most studied types of drawings in this area are the…

We determine the maximum number of maximal independent sets of arbitrary graphs in terms of their covering numbers and we completely characterize the extremal graphs. As an application, we give a similar result for K\"onig-Egerv\'ary graphs…

The edge-length ratio of a straight-line drawing of a graph is the ratio between the lengths of the longest and of the shortest edge in the drawing. The planar edge-length ratio of a planar graph is the minimum edge-length ratio of any…

We investigate the structure of connected graphs, not necessarily locally finite, with infinitely many ends. On the one hand we study end-transitive such graphs and on the other hand we study such graphs with the property that the…

We extend the concept of the law of a finite graph to graphings, which are, in general, infinite graphs whose vertices are equipped with the structure of a probability space. By doing this, we obtain a vast array of new unimodular measures.…

We give an overview of different approaches to measuring the similarity of, or the distance between, two graphs, highlighting connections between these approaches. We also discuss the complexity of computing the distances.

Expository article on the problem of determining the maximum number of equiangular lines with a fixed angle, and the associated problem of second eigenvalue multiplicity in graphs.

We study spaces of realisations of linkages (weighted graphs) whose underlying graph is a series parallel graph. In particular, we describe an algorithm for determining whether or not such spaces are connected.

We prove that any graph of multicurves satisfying certain natural properties is either hyperbolic, relatively hyperbolic, or thick. Further, this geometric characterization is determined by the set of subsurfaces that intersect every vertex…

We characterise the structure of those graphs of a given order which maximise the number of connected induced subgraphs for seven different graph classes, each with other prescribed parameters like minimum degree, independence number,…

Bipartite graphs model the relationships between two disjoint sets of entities in several applications and are naturally drawn as 2-layer graph drawings. In such drawings, the two sets of entities (vertices) are placed on two parallel lines…