Q7 — Waltz Line Labeling: Is the Drawing Consistent?

The Question

The paper's junction dictionary (each junction type with its legal labelings; in-plane rotations are valid too):

(+ convex, − concave, arrows = occluding edges.)

The drawing to label — junctions 1–6, with two edges running off the page at 2 and 6:

        3 ------------ 4 ------------- 5
        |              |               |
        |              |               |
        |      1 (junction below 4)    |
      2 ················1·············· 6
   ~~↖ /                  \ ↗~~
(dangling)                  (dangling)

Edges: 3–4, 4–5 (top bar) · 3–2, 5–6 (verticals) · 2–1, 1–6 (diagonals) · 1–4 (vertical) · plus dangling edges leaving the page at 2 (up-left) and 6 (up-right).

Enter the junction labels for 1, 2, 3, 4 — or NIL.

Official answer: NIL — the drawing has no consistent labeling.

The Topic in 60 Seconds

Waltz labeling = constraint satisfaction over edges: each junction must realize one of its legal dictionary patterns, and an edge shared by two junctions must carry the same label (and arrow direction) at both ends. If any junction runs out of legal patterns, the whole drawing is inconsistent → NIL.

Deep-dive: Waltz Line Labeling, Arc Consistency.

Step 1 — Classify every junction

Count edges and angles:

Step 2 — The T-junction speaks first (it's the most constrained)

In every T pattern, the bar arrows point away from the T. So:

• Edge 3–4: arrow pointing into junction 3
• Edge 4–5: arrow pointing into junction 5
• Stem 1–4: still free — arrow into 4 (T1), arrow away (T2), + (T3), or − (T4).

Step 3 — Boundary edges must be occluding arrows

An edge that runs off the page cannot be + or −: those join two visible faces and must terminate at junctions on both ends. A dangling edge is an occluding (arrow) edge — the object continues behind whatever the drawing cuts off.

Now look at the W dictionary for junction 2's dangling edge:

• W2 and W3 put − / + on the diagonals — impossible for a boundary edge.
• Only W1 puts an arrow on a diagonal, pointing into the junction.

Same at junction 6. So both W junctions are forced to W1:

• At 2: dangling edge = arrow into 2 ⟹ edge 2–1 = arrow into 2, stem edge 2–3 = +
• At 6: dangling edge = arrow into 6 ⟹ edge 1–6 = arrow into 6, stem edge 5–6 = +

Step 4 — Check the L junctions (they survive)

• Junction 3 (L): edge 3–4 arrives as an arrow into 3; edge 2–3 is + → pattern L5 (arrow-in + +) ✓ legal.
• Junction 5 (L): edge 4–5 arrives as an arrow into 5; edge 5–6 is + → L5/L6 ✓ legal.

Step 5 — Junction 1 kills everything → NIL

Junction 1 is a Y. Its legal patterns are only:

• Y1: arrows pointing into 1 on two edges, − on the third
• Y2: all three edges −
• Y3: all three edges +

But from Step 3:

• Edge 2–1 carries its arrowhead at the 2-end — from junction 1 it is an arrow pointing away
• Edge 1–6 carries its arrowhead at the 6-end — again pointing away from 1

No Y pattern allows arrows pointing away on two edges (Y1 wants them pointing in; Y2/Y3 want − / +). Junction 1 has no legal labeling — the domain empties — the drawing is inconsistent.

$\boxed{\text{Answer: NIL}}$

Step through the propagation — watch each junction's domain shrink until junction 1 starves:

Tricks & Tips for This Question Family

Answer: 7.1 NIL