Sur le probléme des courbes gauches en Topologie

An annotated bibliography on 1-planarity

Graph Drawing Beyond Planarity is a rapidly growing research area that classifies and studies geometric representations of nonplanar graphs in terms of forbidden crossing configurations. The aim of this survey is to describe the main research directions in this area, the most prominent known results, and some of the most challenging open problems.

A Survey on Graph Drawing Beyond Planarity

Graph Drawing Beyond Planarity is a rapidly growing research area that classifies and studies geometric representations of non-planar graphs in terms of forbidden crossing configurations. Aim of this survey is to describe the main research directions in this area, the most prominent known results, and some of the most challenging open problems.

A Survey on Graph Drawing Beyond Planarity.

Objective: Aesthetics are important in algorithm design and graph evaluation. This paper presents two user studies that were conducted to investigate the impact of crossing angles on human graph comprehension. Method and results: These two studies together demonstrate our newly proposed two-step approach for testing graph aesthetics. The first study is a controlled experiment with purposely-generated graphs. Twenty-two subjects participated in the study and were asked to determine the length of a path which was crossed by a set of parallel edges at different angles. The result of an analysis of variance showed that larger crossing angles induced better task performance. The second study was a non-controlled experiment with general real world graphs. Thirty-seven subjects participated in the study and were asked to find the shortest path of two pre-selected nodes in a set of graph drawings. The results of simple regression tests confirmed the negative effect of small crossing angles. This study also showed that among our four proposed candidates, the minimum crossing angle on the path was the best measure for the aesthetic when path finding is important. Conclusion: Larger crossing angles make graphs easier to read. Implications: In situations where crossings cannot be completely removed (for example, graphs are non-planar, or a drawing convention is applied), or where effort needed to remove all crossings cannot be justified, the crossing angle should be maximized to reduce the negative impact of crossings to the minimum.

Larger crossing angles make graphs easier to read

There is strong empirical evidence that human perception of a graph drawing is negatively correlated with the number of edge crossings. However, recent experiments show that one can reduce the nega...

Every outer-1-plane graph has a right angle crossing drawing

We tackle the problem of constructing increasing-chord graphs spanning point sets. We prove that, for every point set P with n points, there exists an increasing-chord planar graph with O(n) Steiner points spanning P. Further, we prove that, for every convex point set P with n points, there exists an increasing-chord graph with O(n log n) edges (and with no Steiner points) spanning P.

Increasing-Chord Graphs On Point Sets

Circular right-angle crossing drawings in linear time

A graph $G$ is a non-separating planar graph if there is a drawing $D$ of $G$ on the plane such that (1) no two edges cross each other in $D$ and (2) for any cycle $C$ in $D$, any two vertices not in $C$ are on the same side of $C$ in $D$. 
Non-separating planar graphs are closed under taking minors and are a subclass of planar graphs and a superclass of outerplanar graphs. 
In this paper, we show that a graph is a non-separating planar graph if and only if it does not contain $K_1 \cup K_4$ or $K_1 \cup K_{2,3}$ or $K_{1,1,3}$ as a minor. 
Furthermore, we provide a structural characterisation of this class of graphs. More specifically, we show that any maximal non-separating planar graph is either an outerplanar graph or a subgraph of a wheel or it can be obtained by subdividing some of the side-edges of the 1-skeleton of a triangular prism (two disjoint triangles linked by a perfect matching). 
Lastly, to demonstrate an application of non-separating planar graphs, we use the characterisation of non-separating planar graphs to prove that there are maximal linkless graphs with $3n-3$ edges which provides an answer to a question asked by Horst Sachs about the number of edges of linkless graphs in 1983.

Hooman Reisi Dehkordi

Papers

Every outer-1-plane graph has a right angle crossing drawing

Increasing-Chord Graphs On Point Sets

Increasing-Chord Graphs On Point Sets

Circular right-angle crossing drawings in linear time

Non-separating Planar Graphs