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Anson Chan
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The Vertex Rule in Slitherlink: Why Every Intersection Matters

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In Slitherlink, the vertex rule is the single most powerful constraint after the number clues themselves. Every intersection point (vertex) on the grid must have either zero or exactly two lines passing through it. Never one, never three, never four.

This sounds simple, but it drives the majority of advanced deductions.

Why 0 or 2?

The answer is a single closed loop. When the loop passes through a vertex, one line enters and one line exits — that's 2 lines. If the loop doesn't touch that vertex at all — that's 0 lines. There's no valid state where a vertex has exactly 1 line (that would be a dead end) or 3+ lines (that would be a branch, and loops don't branch).

How to Use It: Elimination

When a vertex already has 2 lines, all remaining edges at that vertex must be empty (marked ×):

·───·
│    
· × ·
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The top-left vertex has a horizontal line (right) and a vertical line (down). It's already at degree 2 — the remaining edge (going right from the lower-left vertex) cannot be part of the loop.

How to Use It: Forced Lines

When a vertex has 3 of its 4 edges marked ×, the fourth edge must also be ×:

· × ·
×    
· × ·
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If three edges are eliminated, the only way to reach degree 2 would require both remaining edges — but there's only one left. Since degree 1 is illegal, the last edge must also be ×. The vertex stays at degree 0.

Conversely, when a vertex has 2 edges marked × and 1 confirmed line, the remaining edge must be a line (to reach degree 2):

·───·
× ?  
·   ·
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The top-left vertex has one line going right and one × going down-left. The only remaining edge (going down) must be a line to give the vertex degree 2.

The Chain Reaction

The vertex rule creates cascading deductions. Mark one edge as × → a nearby vertex now has only 2 possible edges → one of those is forced → that forces the next vertex → and so on.

Example chain in a 3×3 section:

·───·   ·
│ 3   1  
·   · × ·
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The 3 in the corner gives us the top and left lines (corner 3 rule). The corner vertex now has degree 2 → the edge going right from top-right vertex gets ×. This propagates to the 1's constraints, potentially resolving it entirely.

Vertex Rule + Number Clues Together

The most powerful solving technique combines vertex constraints with number clues. Consider:

·   ·   ·
  2   3  
·   ·   ·
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If analysis of surrounding cells forces the shared vertex (center-top) to have 2 lines, and one of those lines is on the 2's side, you've constrained both cells simultaneously. The 3 needs 3 lines but its vertex is already partially committed.

At Grid Edges and Corners

Edge vertices have only 3 edges; corner vertices have only 2. This makes the vertex rule even more constraining:

  • Corner vertex with 1 line → the other edge must also be a line (degree 2)
  • Corner vertex with 1 × → if there's only 1 edge left, it must be × (can't have degree 1)
  • Edge vertex with 2 ×s and 1 line → the remaining edge is forced to be a line

This is exactly why corner number patterns (0, 1, 2, 3 in corners) are so powerful — they're vertex rule applications with maximum constraint.

Practice Strategy

  1. After each deduction, scan the affected vertices
  2. Check if any vertex now has 2 lines (close remaining edges)
  3. Check if any vertex has only 1 possible edge left (force or eliminate)
  4. Repeat until no more progress

The vertex rule alone can solve many Level 1-3 puzzles when combined with basic number patterns.

Practice on real puzzles: slitherlinks.com — 3000+ free puzzles, 10 difficulty levels.

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