Directed Draft Graph Visualization

• Directed draft graphs are weighted, structured directed graphs explicitly designed under geometric constraints to render maximum-weight subgraphs within fixed dimensions.
• They employ an enhanced force-directed layout with custom repulsive, attractive, and edge–vertex forces to ensure clarity while respecting rigid area restrictions.
• Stress-based vertex and edge removal combined with weight transfer preserve critical structural details, making these graphs vital for visualizing student errors and other flow-based data.

A directed draft graph refers to a weighted, structured directed graph that is explicitly constructed or processed under geometric and physical constraints, particularly those arising in the visualization of calculation graphs, error flow diagrams, and educational analytics. The concept, as formalized by "Drawing Graphs within Restricted Area" (Aulbach et al., 2014), encompasses graph drawing methodologies that select and render maximum-weight subgraphs of a given directed graph such that the output is visually legible, respects fixed non-uniform vertex sizes, and fits within a prescribed rectangular area. Directed draft graphs are especially relevant in practice for visualizing step-by-step student errors in symbolic manipulations (e.g., addition of fractions), but the technical methodology applies broadly to weighted, label-driven, and application-specific directed graphs constrained by area and readability.

1. Maximum-Weight Subgraph Selection under Area Constraints

The central problem is to select a subgraph $G' = (V', E')$ of a directed graph $G = (V, E)$ such that:

• The total weight $$W(G') = \sum_{v \in V'} w(v) + \sum_{e \in E'} w(e)$$ is maximized, where $w(\cdot)$ denotes vertex and edge weights reflecting the frequency of occurrence (e.g., number of students making a specific calculation step or error).
• The subgraph can be drawn within a fixed rectangular area of width $W$ and height $H$, given that each vertex must be displayed at a fixed size (with width $w_v$ and height $h_v$ for each $v \in V$).
• The drawing remains readable and as faithful as possible to the original structure.

This dual optimization—maximizing the retained "heavy" structure while satisfying physical drawing constraints—is NP-hard, with the hardness demonstrated via reduction from Subset Sum (unit heights case).

2. Force-Directed Layout with Enhanced Physical Forces

The layout algorithm for directed draft graphs is an augmented force-directed system incorporating several custom forces:

• Repulsive Force (Fruchterman–Reingold):

$$F_{\mathrm{rep}}(u, v) = \frac{l_{\mathrm{unit}}^2}{d(u,v)} \cdot \hat{d}(u,v)$$

where $l_{\mathrm{unit}}$ is the desired edge length and $\hat{d}(u,v)$ is the unit vector from $u$ to $v$.

• Attractive Force (for edges):

$$F_{\mathrm{att}}(u, v) = \frac{d(u,v)^2}{l_{\mathrm{unit}}} \cdot \hat{d}(v,u)$$

applied for each $(u,v) \in E$.

• Edge–Vertex Repulsion:

Vertices are pushed away from any non-incident edge crossing their fixed-size rectangle, with repulsion calibrated to the distance between intersection points (method inspired by Bertault).

• Frame Compression:

The enclosing frame is gradually shrunk; vertices pushed outside receive a counterforce, and equilibrium updates are iteratively solved to ensure all vertices remain and no overlaps occur.

These forces iteratively reposition vertices while respecting rectangle sizes and update the drawing toward a feasible area-constrained configuration.

3. Stress-Based Vertex and Edge Removal

When area constraints are violated (e.g., too many vertices for given $W \times H$), the algorithm employs "stress" and "importance" metrics for strategic removal:

• Vertex Stress:

The aggregate "pressure" $p(v)$ exerted on a vertex $v$ is computed in eight octants; for each octant $i$, $l_i$ is summed, and pressure $p_i$ is derived via maximum pairwise minima across opposite directions. The overall stress is:

$s(v) = \frac{p(v)}{w(v) \cdot (\deg(v) + c)}$

Vertices with the highest $s(v)$ are prioritized for removal.

• Importance Based on Vertex Size:

Especially in directed calculation graphs, importance is defined as $i(v) = w(v) / h_v$. Tall vertices are penalized unless frequent.

• Edge Stress:

Edges may also be removed when their average length is high or crossings are excessive:

$$s(e) = \frac{\sum_{e' \in E'(e)} w(e') \cdot |E'(e)|}{w(e)}$$

where $E'(e)$ is the set of edges crossing $e$.

The removal routine always aims to preserve maximal total weight and the most informative portions of the original graph.

4. Layered Drawing of Directed Calculation Graphs (Modified Sugiyama Framework)

For calculation graphs, readability from left to right is crucial. The process specifically adapts hierarchical drawing techniques (Sugiyama's framework):

• Cycle Breaking: Cycles are reduced by flipping a minimal feedback set of edges, taking weights into account.
• Layer Assignment: Layers are assigned by heuristics not strictly based on drawing area height, but instead to optimize retention of important vertices (e.g., $n_\text{layers} \approx \lceil |V| / k \rceil$).
• Crossing Minimization: Edge crossings are minimized via adjacent exchanges in the vertex ordering.
• Edge Routing: Edges are drawn as smooth quadratic or cubic Bezier curves rather than bent orthogonal lines, with control points adjusted by forces to maximize separation and avoid acute bends.

This approach ensures that dense subgraphs, frequent in error flow diagrams, remain comprehensible and that paths correspond to plausible calculation trajectories.

5. Weight Transfer and Information Preservation

Whenever a vertex is removed for space, a "weight transfer" routine is applied:

• For incoming edge $(u, v)$ and outgoing edge $(v, w)$, an edge $(u, w)$ is inserted (or incremented by $\min\{w(u,v), w(v,w)\}$) to partially preserve reachability and aggregate frequencies.
• This systematic weight transfer enables partial retention of calculation information at intermediate steps, enhancing interpretability even in heavily pruned graphs.

6. Theoretical Significance and Practical Impact

The presented combination of area-constrained force-directed layout, stress-based pruning, hierarchical adaption, and smooth edge rendering yields a theoretically sharp solution to an NP-hard problem (maximum-weight subgraph selection under hard geometric constraints). Practically, directed draft graphs as constructed:

• Visualize dominant flow patterns and common errors in student calculation exercises.
• Allow educators to quickly locus frequent mistakes, optimize curriculum, or conduct pedagogical analysis.
• Are broadly applicable to any context demanding maximum-weight, structurally faithful visualization of dense directed graphs within strict layout bounds.

7. Representative Formulas and Decision Criteria

A canonical formula governing vertex removal:

$s(v) = \frac{p(v)}{w(v) \cdot (\deg(v) + c)}$

is used as the vertex stress metric. Importance is typically:

$i(v) = \frac{w(v)}{h_v}$

Edge stress leverages crossing information as above.

8. Broader Implications

Directed draft graphs, as conceived in this framework, encode general principles for information-preserving visualization under physical constraints. The methodology applies beyond educational graphs—for example, to flowcharts, circuit diagrams, or any domain requiring simultaneous maximization of structural content and visual feasibility within a restricted area. The systematic mapping of "stress," importance, and weight transfer establishes a template for scalable, readable, and interpretable directed graph visualization under real-world constraints.