Graphs, degree, walks, connectedness, bipartite matching, coloring, spanning trees, tree traversal -- every one of these already has a name and a definition from Discrete Mathematics I and II. This chapter doesn't teach any of it for the first time; it puts the whole toolkit back on the table, in exactly the framing the rest of this course will lean on, so Chapter 1 can start building the formal, notation-heavy version of a graph without first stopping to ask "wait, what's a path again?"
Two prerequisite courses fed this vocabulary in at different points. Discrete Mathematics I introduced graphs themselves -- vertices, edges, degree, the handshaking theorem, walks and paths and cycles, connectedness, and, separately, trees and tree traversal. Discrete Mathematics II picked the thread back up for bipartite graphs, matchings, and graph coloring, and used graphs a second time as objects to count -- how many spanning trees a graph has, Cayley's formula for how many labeled trees exist on n vertices. Both courses' habits of mind are worth having cold: this course is about to ask harder structural and algorithmic questions of every one of these ideas -- Menger's theorem about connectivity, Dijkstra's and Floyd-Warshall's algorithms about weighted paths, Kuratowski's theorem about planarity -- and none of that lands if "degree," "connected," or "spanning tree" still needs a second of thought to recall.
Seven sections, one per major idea, each with its own quick refresher, a live widget to make the idea visible again, and a practice set to confirm it's solid before Chapter 1 raises the level of formality. By the end of this chapter, every piece of that toolkit is back in hand, ready to be sharpened.