One-Line Drawing: Graph Generation Algorithms and Difficulty Scaling

One-Stroke: Treat Levels as Graphs for Automatic Difficulty Grading

The Essence in One Sentence

On the surface, it's a geometric challenge where players experience "finishing in one stroke"; actually, the core mechanism leverages Euler path theory from graph theory to guarantee level solvability; this article focuses on explaining how to automatically generate and grade level difficulty using quantitative metrics (branching count, loop count, visual crossings).

Traffic Reason: Dual attributes of education + casual; viewers are willing to watch "how to finish in one stroke" and also willing to try it themselves.

1) Graph Generation Algorithm: Guarantee Solvability Before Fun

One-stroke is essentially graph theory:

• Point = Node
• Line = Edge
• "Finish in one stroke" = Traverse every edge once (Euler path/circuit)

The simplest condition for solvability (extremely useful):

• Number of odd-degree nodes = 0 → Has an Euler circuit (can return to start from anywhere)
• Number of odd-degree nodes = 2 → Has an Euler path (start/end points are fixed)

The generation process can be highly engineering-oriented:

1. First generate a graph that meets the conditions
2. Then perform "visual mapping" (draw the graph as a geometric shape/animal/letter)

2) Difficulty Grading Logic: Don't Rely on Intuition, Rely on Metrics

Three difficulty metrics that can be directly quantified:

• Branching Count: The higher the maximum degree of a node, the easier it is to make a mistake.
• Loop Count: The more loops, the easier it is to get stuck prematurely.
• Visual Crossings: The more line segments cross, the more likely players are to misread the path (even if they aren't connected on the graph).

Suggested Grading:

• Beginner: Maximum degree ≤ 3, few loops
• Intermediate: Presence of 4-degree nodes or multiple loops
• Hardcore: Many loops + high visual crossing density

3) Making One-Stroke Feel More Like a Game: Add "Cost of Mistakes," But Not "Frustration"

Lightweight design can boost retention:

• Allow undoing one step (reduces anxiety)
• Provide "highlight traversable edges" (lowers the barrier to reading the map)
• On failure, do not clear the board; return to the nearest branching point instead.

Boundary Conditions

• The number of odd-degree nodes must be 0 or 2; otherwise, a solvable one-stroke path cannot be generated.
• If the number of odd-degree nodes is neither 0 nor 2, the graph is deemed unsolvable.
• The generation process must first construct a graph satisfying Euler conditions, then perform visual mapping.
• After failure, the player must return to the nearest branching point, rather than clearing the current progress directly.
• Players can execute an undo operation to handle situations where they take the wrong path.

Design Trade-offs

• Benefits: Allowing one-step undo reduces player anxiety; providing "highlight traversable edges" lowers the barrier to reading the map; not clearing the board on failure and retreating to the nearest branching point reduces frustration.
• Cost: The aforementioned lightweight designs (undo, highlight, retreat) may increase system logical complexity or resource consumption.
• Alternatives: If the aforementioned lightweight designs are not adopted, the system would rely entirely on player trial-and-error, potentially leading to a steep learning curve or decreased retention rates.

One-Sentence Summary: One-stroke content production can be fully automated: Euler guarantees solvability, metrics determine difficulty, and visuals handle dissemination.