LOGIC & PUZZLES / NETWORKS
Draw a problem
as a graph.
A good diagram is not a picture of the world; it is a model of the relationships you need.
THE MAIN IDEA
Vertices name things. Edges name their connections.
Graph theory represents a network as vertices—also called nodes—and edges connecting them. A map problem can become a graph when intersections are vertices and roads are edges. A friendship puzzle can use people as vertices and known relationships as edges. A task schedule can connect prerequisites to tasks. Once the relationship is clear, the drawing can reveal properties that a paragraph hides, such as disconnected regions, bottlenecks, cycles, and alternative routes. Edges can be directed when order matters or undirected when a relationship is mutual. They can carry weights for distance, cost, travel time, or capacity, and the meaning of those numbers should be written down. Two lines crossing on paper do not necessarily connect at an intersection; the model needs an explicit vertex if the crossing matters. Similarly, a road shown as one undirected edge may not represent a one-way street. Choosing what to omit is part of modeling. A city map includes far more detail than a routing calculation needs; a graph preserves only the features relevant to the question. A path is a sequence of connected vertices, while a cycle returns to its start. A tree is a connected graph with no cycles; a spanning tree connects every vertex using a minimal set of edges when all edges have equal cost. The degree of a vertex counts incident edges and can expose clues in route puzzles. An Euler trail, for example, asks whether you can use each edge exactly once; odd-degree vertices determine important constraints. A shortest-path question has a different goal: minimize the total edge weight, not necessarily the number of turns or road segments. A graph diagram is an aid to reasoning, not a guarantee that the problem data are accurate. Real travel needs current closures, access, traffic, and safety information; mathematical shortest routes should not be taken as practical navigation advice. Start by writing what each vertex and edge represents, then check the diagram against each fact. That discipline prevents decorative drawings from quietly changing the puzzle. Once you have a faithful graph, simple techniques—marking visited nodes, sorting edge weights, or considering degree—can replace repeated mental tracing.
TRY THIS
Model a familiar route map.
Draw four places as dots and connect only the paths that really exist between them. Mark any one-way direction and write approximate walking times on edges. Compare the fewest segments with the lowest total-time path; they need not be the same.