Improving Clarity in Nearly Planar Graphs

Graphs are a way to show relationships between different things. They consist of points, called nodes, and lines connecting those points, called edges. People often use graphs to visualize connections and information. Yet sometimes, these graphs can become messy and hard to read, especially when they are nearly planar. Nearly planar graphs have only a few crossings in their representations, but making them clear and easy to understand remains a challenge.

The goal is to create better Drawings of these graphs so that viewers can see the connections without being distracted by clutter. One way to improve graph layouts is by identifying edges that cause confusion in a drawing. By changing how these edges are weighted, or valued, we can create layouts that are easier to read, using specific techniques to position the nodes and edges.

#Understanding Graph Drawings

When constructing a graph, the aim is to produce a drawing that avoids clutter. Clutter is often caused by edges crossing each other. It is known that when there are fewer crossings in a graph, people can perform tasks better. Hence, a perfect drawing ideally has no crossing edges, which are called planar drawings.

However, not all graphs are planar. Many graphs can still be sparse or have a clear structure, which is why they are called nearly planar. The challenge lies in making these nearly planar graphs easier to read by highlighting their planar parts.

#The Challenge of Nearly Planar Graphs

Although there are many types of graphs, the methods used to draw them are often not good enough for nearly planar ones. Previous attempts to define what nearly planar means have led to complex problems. Some algorithms that try to make these drawings work better do exist, but much work still needs to be done to create practical solutions that produce clearer drawings.

This article focuses on improving the layouts of nearly planar graphs by using a specific approach that adjusts the weights of edges to reduce confusion and enhance clarity in the drawings.

#Our Approach

We propose a new way to identify edges that cause clutter and adjust their weights. By doing this, edges that create confusion can have less influence on the drawing process. Our approach uses a method called spring-based drawing, which is a way to visualize graphs by treating the edges as springs. This means that nodes will repel or attract each other based on how they are connected with edges.

#Identifying Problematic Edges

To make our approach work, we need to identify those edges that lead to clutter in the drawing. If two nodes connected by an edge can be linked through multiple shorter paths, the edge in question can likely be short and not congested. On the other hand, if the only connections available are longer, the edge could contribute to drawing confusion.

To find these paths, we use a method that simplifies the process while still providing good results. This way, we can focus on the edges that create clutter and deal with them appropriately.

#Using Isolation Forest Technique

Once the problematic edges are identified, we apply a technique called Isolation Forest. This method finds edges that behave differently compared to others, which means they may cause confusion in the layout. By isolating these edges, we can weigh them differently, making them less impactful when drawing the graph.

Normalizing these weights ensures that we fairly compare how different edges affect the drawing. This allows us to prioritize edges that are critical to maintaining the graph's overall structure.

#Experiments and Results

To test our method, we conducted several experiments using different types of graphs. We created grids, triangulations, and deep triangulations, all modified to include edges that disrupt their original structure.

#Grids and Triangulations

Starting with a grid, we introduced a number of edges between random pairs of nodes, resulting in what we called an augmented grid. We noticed that the initial graphs looked cluttered and hard to understand. By applying our Edge-weighting approach, we ran the experiments again, and the new layouts showed clear improvements. Edges that previously crossed less often appeared longer and less cluttered, revealing the underlying grid structure effectively.

For triangulations, we followed a similar process. By adjusting the weights of edges, we were able to see clearer drawings that made the triangular relationships more understandable.

#Deep Triangulations

Deep triangulations presented a greater challenge because they were more complex. However, even in these cases, our approach provided visual improvements. The clarity in these layouts suggested that our method could help reveal the underlying structure of complicated graphs.

#Qualitative Analysis

Looking at the drawings, significant improvements were noticeable when using our edge-weighting technique. The clutter that confused viewers was noticeably reduced. By analyzing how edges were positioned, it became apparent that our method allowed for a clearer depiction of the relationships between nodes.

#Quantitative Analysis

To quantify our findings, we evaluated the results based on the number of edge crossings, angular resolution, and overall layout distortion. The method successfully led to fewer crossings in the drawings when compared to traditional techniques. This shows that our adjustments had a positive impact not just visually, but also in technical terms.

In every case, the quality metrics demonstrated that our approach significantly improved the layout quality for augmented grids and triangulations. While results were not as positive for Rome graphs, this was expected because they do not contain dense planar structures.

#Conclusion and Future Work

In summary, we have proposed a method to detect and adjust problematic edges in nearly planar graphs to improve their layouts. The results indicate that our approach works well for a range of graph types, making them clearer and reducing clutter for viewers.

To enhance our findings further, future work will include deeper testing of more complex graphs and possibly exploring other techniques that can yield higher clarity and understanding. We aim to refine our methods and broaden their application, ensuring even challenging graphs can be visualized more effectively.

By advancing our grasp on nearly planar graphs and their layouts, we contribute to the ongoing effort to make data visualization more accessible and meaningful for everyone.