Graph Traversal: BFS and DFS Short Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Graph Traversal: BFS and DFS short notes for GATE DA: 3 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Quick Recap: Reachability

Core Principles

Asymmetry: In directed graphs, $v \to u$ does not imply $u \to v$ .

Reverse Graph ( $G^{R}$ ): Flipping edges turns "reachable from" into "can reach".

SCC Intersection: $SCC (v) = Reach (v in G) \cap Reach (v in G^{R})$ .

Algorithm Equivalence: BFS and DFS discover the exact same reachable set from a source.

Path vs Existence: Reachability only guarantees a path exists. Only BFS guarantees the shortest path in unweighted graphs.

Scope: One traversal = single-source reachability. $V$ traversals = transitive closure.

Quick Recap: DFS Edge Classification

Core Principles

Parenthesis Theorem: Time intervals $[d, f]$ are either strictly nested or completely disjoint.

Tree Edge: $d [u] < d [v] < f [v] < f [u]$ (and $v$ discovered via $u$ ).

Back Edge: $d [v] < d [u] < f [u] < f [v]$ (indicates a cycle).

Forward Edge: $d [u] < d [v] < f [v] < f [u]$ (but $v$ not discovered via $u$ ).

Cross Edge: $d [v] < f [v] < d [u] < f [u]$ (disjoint intervals).

Undirected Rule: DFS on undirected graphs yields only Tree and Back edges.

Quick Recap: Visual Tracing Checklist

Visual Tracing Checklist

Workspace: Always use a trace table and explicitly draw the Queue (BFS) or Stack (DFS).

Degree Check: Count the degrees of all vertices before starting to ensure no edges are missed.

Ordering: Pre-sort adjacency lists if the question specifies an ordering constraint.

BFS Signature: Expands level-by-level. All distance

k

vertices are visited before distance

k + 1

DFS Signature: Dives deep. Explores a single path as far as possible before backtracking.

Strictness: Follow the exact neighbor order specified; any deviation ruins the trace.

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Graph Traversal: BFS and DFS Short Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Graph Traversal: BFS and DFS short notes for GATE DA: 3 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

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