Which storage representation for graphs requires O(V^2) space?

Which storage representation for graphs requires O(V^2) space?

• A. Adjacency List
• B. Edge List
• C. Incidence Matrix
• D. Adjacency Matrix

Answer: (D)

Storage representations for graphs differ in how they allocate memory. The representation that requires space proportional to the square of the number of vertices uses a two-dimensional grid with V rows and V columns, where each cell indicates whether a pair of vertices is connected (and may store a weight). Since every pair of vertices has a slot, the total space is V times V, i.e., O(V^2). This holds for both directed and undirected graphs when using the full matrix. Other representations scale with the number of edges. An edge list stores one entry per edge, giving O(E). An incidence matrix uses a row per vertex and a column per edge, giving O(VE). An adjacency list stores the neighbors for each vertex, totaling O(V+E). Therefore, the matrix-based approach is the one that inherently uses O(V^2) space.

Storage representations for graphs differ in how they allocate memory. The representation that requires space proportional to the square of the number of vertices uses a two-dimensional grid with V rows and V columns, where each cell indicates whether a pair of vertices is connected (and may store a weight). Since every pair of vertices has a slot, the total space is V times V, i.e., O(V^2). This holds for both directed and undirected graphs when using the full matrix.

Other representations scale with the number of edges. An edge list stores one entry per edge, giving O(E). An incidence matrix uses a row per vertex and a column per edge, giving O(VE). An adjacency list stores the neighbors for each vertex, totaling O(V+E). Therefore, the matrix-based approach is the one that inherently uses O(V^2) space.