{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-tree"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-tree-01","title":{"zh-Hant":"樹：判定一個 6 頂點圖是樹，求葉、唯一道路與加邊後的唯一環","en":"Trees: deciding that a 6-vertex graph is a tree, with its leaves, a unique path and the unique cycle after adding an edge"},"summary":{"zh-Hant":"設 G=(V,E)，V={1,2,3,4,5,6}，E={{1,2},{2,3},{2,4},{4,5},{4,6}}。(1) 依 MKO 定義證明 G 是樹（V 非空、連通、無簡單環）；(2) 列出葉；(3) 求從 3 到 6 的道路並證明它唯一；(4) 加入邊 {3,6}，寫出所形成的簡單環並證明它是唯一的簡單環；(5) 對照：三角形加一個孤立點雖有 |E|=|V|−1 卻不是樹；單點無邊圖是樹但沒有葉。","en":"Let G=(V,E) with V={1,2,3,4,5,6} and E={{1,2},{2,3},{2,4},{4,5},{4,6}}. (1) Prove from the MKO's definition that G is a tree (V nonempty, connected, no simple cycle); (2) list its leaves; (3) find the path from 3 to 6 and prove it is unique; (4) add the edge {3,6}, write the simple cycle it creates and prove it is the only simple cycle; (5) contrast: a triangle plus an isolated vertex has |E|=|V|−1 but is not a tree; a single vertex without edges is a tree with no leaf."},"mko_ids":["mko-tree"]}],"title":"Concept derivation guides"}