Vertex Rules: The Foundation of Logical Deduction

The Intersection of Truth

Slitherlink is fundamentally about connections. Lines connect dots, but they meet at vertices. A vertex is simply the point where grid lines intersect—a corner of a cell. Every vertex in the grid (except the border ones) is a meeting point for 4 potential edges. Understanding the rules governing these intersections is the key to advanced deduction.

The Fundamental Loop Rule

The most important rule in Slitherlink is that the line forms a single, continuous loop. This leads to the Vertex Rule:
Every vertex must have exactly 0 or 2 lines connected to it.

A vertex cannot have 1 line (that would be a dead end).
A vertex cannot have 3 lines (that would be a branch).
A vertex cannot have 4 lines (that would be a crossing, which is disallowed).

  ✓ Degree 0       ✓ Degree 2       ✗ Degree 1       ✗ Degree 3
                                     (dead end)        (branch)

  ×   ×             ═══·═══          ═══·   ×          ═══·═══
      ·                 ║                ×                  ║
  ×   ×             ×   ║            ×   ×              ×   ║

Forced Moves at Vertices

This "0 or 2" rule creates powerful forcing moves:

• The "2 Lines" Rule: If a vertex already has 2 lines connected to it, all other potential edges at that vertex MUST be marked with an X. The loop simply passes through; it cannot branch off. This is vital for cleaning up the board after you draw a segment.
• The "Forced Continuation" Rule: If a vertex has 2 edges marked with X (blocked), the remaining 2 edges MUST be lines (if the vertex is part of the loop path). Or, if a line enters a vertex and 2 other paths are blocked, it MUST exit through the only remaining open path.

  2 lines → X the rest       1 line + 2 Xs → forced exit

  ×   ×                       ×   ×
      ·                           ·
  ═══·═══                     ═══·
      ×                           ║ (forced!)

Dead End Prevention

This is a subtle but powerful elimination technique. If drawing a line segment would leave a neighboring vertex with only one possible connection (because the others are blocked by edges or Xs), then that original line segment is impossible. You can mark it with an X.

Essentially: "If I go here, I get stuck at the next intersection. Therefore, I cannot go here."

Vertex Scanning Technique

When you are stuck, stop looking at the numbers and start looking at the dots (vertices). Scan the grid intersection by intersection. Look for:

• Vertices with 3 Xs (the 4th must be X).
• Vertices with 1 Line and 2 Xs (the 4th must be a Line).
• Vertices with 2 Lines (mark the rest X).

  3 Xs → 4th is X      1 Line + 2 Xs → Line     2 Lines → X rest

  ×   ×                 ×   ×                     ═══·═══
      ·                     ·                         ×
  ×   × (→ all X)       ═══· (→ must exit ║)     ×   ×

This "vertex scanning" is often the only way to progress in Hard puzzles where the number clues are sparse.

Why It Matters

Numbers tell you about edges. Vertices tell you about flow. Combining these two perspectives—edge constraints from numbers, flow constraints from vertices—is what separates a guesser from a true solver.