# Directed graphs: from arrows to dependency plans This original twelve-question course develops a practical graph vocabulary for reading job dependencies, following their effects and checking a proposed order. It is intended for learners who can read short lists and count steps; no programming or prior graph theory is required. Every example supplies its own vertices, edges and interpretation. There is no empirical learner data or claim that these questions measure competence. The course file is [dependency-graphs.json](dependency-graphs.json), using the existing `recallweave-deck/1` format. Its four concepts each have three questions: directed links, reachability, dependency orders, and cycles/updates. The course's prerequisite links form an acyclic graph. Correct positions are balanced across the four options; neither option length nor alphabetic order identifies the answers consistently. ## Explore the prerequisite decisions Open the [offline dependency-planning companion](dependency-graphs-explorer.html) to edit a small graph, complete ready jobs, undo or reset progress, and distinguish cycle members from downstream blocked jobs. Its course examples use the arrow convention below. The companion also downloads this original course unchanged for the existing learner; its calculations and download work directly from the single HTML file. For an optional extension beyond these twelve questions, open the [offline critical-path timing companion](dependency-timing.html). Add bounded integer durations to see earliest/latest times, total slack and all critical branches under unlimited parallelism, then download the exact timing trace. Durations use abstract time units; this does not estimate calendars or staffing. ## Open and adapt it with the integrated studio 1. Open the repository's existing `author.html` directly in a browser, or serve the repository and visit `author/`. 2. Choose **Open a deck to edit** and select `dependency-graphs.json`. 3. Inspect the studio's preview, then choose **Replace draft** when you intend to replace the current editor draft. 4. Inspect or adapt the questions. **Check and preview** validates the draft and exposes its answer key. **Download deck (.json)** saves the checked course. The studio is the existing owner's application. This contribution supplies course content and does not alter its import, preview, replacement or export behavior. Learner-side local course selection is the separate [issue #7](https://github.com/Jacob-Met/RecallWeave/issues/7) contribution; this course does not add a second importer or claim that path is integrated. The original bundled cellular-energy lesson and other course contributions remain unchanged. ## Conventions that matter An arrow is directed. For the dependency-order questions, `X → Y` means that X must come before Y. Other applications can choose the reverse convention, so the arrow legend is part of the problem. Direct neighbors are one edge away; reachability can use several edges. Where the starting vertex must be excluded, the question says so explicitly. Shortest distance in this course counts edges equally. A weighted cost question would be different. Ready jobs have all graph prerequisites complete; this says nothing about available workers or shared resources. A cycle prevents a complete topological order, but a node blocked by that cycle need not itself lie on the cycle. These distinctions are mathematical properties of the supplied examples, not claims about a particular production scheduler. ## Worked answer map The JSON contains the complete learner feedback and transfer prompts. The table below makes each answer and its principal distractor checks easy to review. | Question | Correct result | Derivation and rejected alternatives | |---|---|---| | `dg-links-1` | `Parse → Render` | Parse is the prerequisite. Reversing the arrow reverses the requirement; adding a return arrow or a self-loop does not encode the single stated dependency. | | `dg-links-2` | 2 | Only B and E have edges ending at D. A and C are indirect predecessors through B. | | `dg-links-3` | `{B, C}` | These are the endpoints of A's outgoing edges. D requires two edges; A has no self-loop. | | `dg-reach-1` | `{B, C, D}` | B and C are one step from A; D is two steps away. E and F are disconnected from A, and there is no positive-length route back to A. | | `dg-reach-2` | 2 | `S → C → T` takes two edges. No direct edge gives one step. The route through A and B is longer. | | `dg-reach-3` | `{Clean, Chart, Summary}` | Follow both branches below Clean. The explicit rule excludes Raw and unrelated Glossary; omitting Summary misses a downstream consumer. | | `dg-order-1` | `B, A, C, D` | Both A and B precede C, which precedes D. Each other option violates at least one of those edges. | | `dg-order-2` | `{B}` | C still needs B, and D still needs C. One completed prerequisite does not satisfy all prerequisites. | | `dg-order-3` | 2 orders | C is last; A and B can occur in either order. The two possibilities exhaust all arrangements. | | `dg-update-1` | add `C → A` | The existing `A → B → C` path gives the directed return route. The other three additions leave at least one valid topological order. | | `dg-update-2` | `{A, B, C}` remain | Remove D, then E. A and B remain circular; their dependent C stays blocked despite not being a cycle member. | | `dg-update-3` | same valid-order set | `A → B → C` already forces A before C. The original and augmented graphs both allow exactly `A,B,C,D`, `A,B,D,C`, and `A,D,B,C`. | The last example preserves ordering only. A redundant edge could carry other meaning in a data-transfer graph; the course does not recommend deleting such edges from an application. ## References and original-content provenance The following primary teaching and reference materials were read on 2026-10-08 to check terminology and the general graph properties: - Paul E. Black, **directed graph**, NIST *Dictionary of Algorithms and Data Structures*, entry updated 2023-06-08: . - Paul E. Black, **topological sort**, NIST *Dictionary of Algorithms and Data Structures*, entry updated 2022-04-21: . - Erik Demaine, Jason Ku and Justin Solomon, MIT OpenCourseWare **6.006, Lecture 9: Breadth-First Search**, Spring 2020, especially pages 1–4: . - The same MIT course, **Recitation 10**, especially the topological-sort definition on page 3: . All scenarios, vertex sets, question wording, options, explanations and transfer prompts were newly authored for this course by the HAMON estate contribution `e137a7f86391` with AI assistance. No reference exercise, figure, example graph or passage is copied. Original course content is offered under **CC BY 4.0**; the linked sources retain their own terms. Reference links are background reading, not an endorsement of this course or a claim of measured learning effectiveness. ## Exact-content receiving Run the focused native check with Node 20 or later from the repository root: ```sh node --test tests/dependency-graphs.test.mjs ``` It admits the literal UTF-8 course through the existing validator, checks its bounded size and round trip, and derives the example answers from the graph declarations in the actual learner prompts. Exhaustive enumeration covers the small ordering examples. Two intentionally changed course copies remain structurally valid but fail the mathematical answer check: a reversed prerequisite and a blocked descendant mislabeled as removable. The existing selector, first-answer review, separate practice, study-note export and trace archive are also exercised with these twelve questions. That native consumer check is distinct from actual browser/studio receiving. Source and receiving records retain their exact pins; no learner-importer or browser acceptance is inferred from format validation alone.