VectorPaint minimum spanning connections: cover five nodes before comparing four-edge totals

English translation of the Chinese original, translated on 2026-11-11. Original publication dates, attribution, code and version notes are retained.

A shortest A-to-E route need not be the least-cost network connecting every site. Use an original five-node undirected graph, checking coverage/connectivity and absence of cycles before total edge weight. Weights are fictional costs, not real distances or quotes. VectorPaint supplies a static diagram, with calculations and minimality checked manually rather than by an assumed graph algorithm.

List five vertices and eight undirected edges

Use A–E for 甲乙丙丁戊 and eight edges: AB=1, BC=2, CD=3, DE=4, AC=5, BD=6, CE=7, AE=9. Undirected AB/BA is one edge, without arrows. List edges externally before layout, placing each weight clearly on its own edge. Unmarked crossings create no sixth vertex. Geometric lengths encode no costs; use the table’s values only. Title it fictional, without inferring real roads or terrain.

Check coverage with the four chain edges

Candidate one selects AB, BC, CD, DE: all five vertices connect along a four-edge chain without a cycle. Trace A through B/C/D to E, not just the endpoints. A spanning tree here connects all five without cycles; AE alone omits three vertices and is not a candidate. Emphasise selected edges while retaining pale unselected source edges with a legend, or use patterns/numbers. Four edges alone is not sufficient; connectivity and acyclicity still require checks.

Establish minimality for this graph

1+2+3+4=10. A five-vertex spanning tree has four edges, and this graph’s four smallest weights are 1,2,3,4. Every four-edge selection therefore costs at least 10. Since these four also form a spanning tree, they attain that bound. State both lower bound and feasibility: simply choosing four cheapest edges does not work universally, because another graph may yield a cycle or omit vertices. Distinct weights do not license a visual uniqueness claim, and no software enumeration is claimed. Sum each selected edge once, not separate A-to-every-vertex routes, which would repeatedly charge shared chain edges.

Compare a feasible thirteen-cost tree

Candidate two uses AB, BC, CD, CE, costing 1+2+3+7=13. It still connects every vertex without a cycle: E joins through C and D remains reachable. It is feasible but more expensive, not structurally invalid. Hold the edge table/layout fixed, including CE=7. Separate feasibility from optimality; higher cost does not mean disconnected. Restore selections by edge identities rather than colour memory. One connected example alone cannot establish minimality without a bound or another reliable comparison.

Separate shortest-route and spanning objectives

For A-to-E alone, direct AE costs 9, less than chain cost 10, but omits B/C/D and cannot replace the spanning tree. Conversely, the tree’s A-to-E path of 10 does not make 10 the original graph’s shortest-route answer. State each objective explicitly. The earlier shortest-route exercise is a comparison, while covering every node is the new requirement. Use only listed nonnegative edges, without invented shortcuts. Real capacity, reliability and geographic constraints are absent from this small graph.

Test a cheap cyclic selection

The incorrect AB, BC, AC selection costs 8 but cycles within ABC and omits D/E. Adding DE gives four edges and cost 12, yet leaves two components and the cycle. Record cost, edge count, coverage and connectivity separately: four edges do not pass. Repair edge selection rather than moving omitted labels inside a circle and pretending they connect. Check both failure reasons manually without automatic cycle detection. Preserve the valid 10 and 13 versions as separate evidence.

Inspect output against the edge table

Save and inspect PNG all eight weights, four selected edges and five labels, retaining any no-node-at-crossing legend. Move misattached labels rather than altering costs. If pale unselected edges disappear small, include the edge table so the original is not mistaken for a four-edge graph. Distinguish selection in monochrome and hide unwanted checking arrows/numbers. Open other formats separately; inspect endpoints/occlusion for gaps. An image export is no network-connectivity test. Keep source table and editable drawing retrievable.

Review coverage, cycles and total weight

For omitted vertices revisit the full set; for cheap invalid selections inspect cycles/connectivity; for 13 called minimum revisit the bound of 10; for endpoint-objective confusion revisit titles. After repair trace reachability as well as adding weights. Retain edge table, 10/13 trees, 9-route comparison, 8/12 controls and lower-bound argument. Princeton’s minimum-spanning-tree text supplies terminology/objective only; no program is run. This is no automatic planning, quote or optimum under every real constraint. Actual systems require separate data and professional validation.