Why CS205ES has CS105ES as a prerequisite

The graph ADT

Exactly like every ADT in this course, a graph's interface stays the same regardless of whether it is stored as a matrix or a list underneath: add a vertex, add an edge, list a vertex's neighbors, check if two vertices are adjacent, and traverse. Code written against this interface — Lesson 3's BFS and DFS, for instance — never needs to change if the underlying representation switches from Lesson 2's matrix to its list, or back.

Three applications, three traversals in disguise

A road-mapping app finding the shortest route between two points is running a graph traversal — a variant of BFS when every road takes the same time, something fancier when road times differ, but the same underlying idea of exploring outward from a start vertex. A social network suggesting 'people you may know' is running BFS a fixed number of levels deep from your profile, exactly the same algorithm Lesson 3 built, just capped at depth two or three.

A course prerequisite chain — you cannot take CS205ES before CS105ES, exactly the rule stated in this very course's own record — is a directed graph where an edge from X to Y means X must be completed before Y. Such a graph must have no cycles, because a cycle would mean a course requires itself, directly or indirectly, which makes no sense. Finding a valid order to take every course in is a graph algorithm called topological sorting — a natural extension of the DFS you already know, deferred to further study, but worth naming here because you can now see exactly what problem it solves.