Final answer:
The time efficiency of DFS and BFS using an adjacency matrix is O(V^2), as both require checking every vertex against all others, resulting in V*V operations in the worst case.
Step-by-step explanation:
The time efficiency of Depth-First Search (DFS) and Breadth-First Search (BFS) when using an adjacency matrix is generally determined by the number of vertices (V) and the number of edges (E) in the graph. However, the efficiency also depends on the data structure used to represent the graph. An adjacency matrix is a two-dimensional array where the presence of an edge is indicated by a non-zero value at the intersection of rows and columns corresponding to the vertices.
For DFS, the time complexity is O(V^2). This is because in the worst case, you would have to visit every vertex and examine all adjacent vertices through the matrix, which requires looking at all possible edges, even if they do not exist, i.e., a complete traversal through the matrix for each vertex. As there are V vertices and V potential edges per vertex in a matrix, the complexity becomes V*V or V^2.
Similarly, for BFS, the time complexity is also O(V^2) when using an adjacency matrix. While BFS traverses the graph level by level, it still requires checking each vertex in every level against all others to determine connections, leading to the same V*V operations in the worst case as for DFS. Therefore, the statement that DFS is O(V) and BFS is O(V) when using an adjacency matrix is incorrect. Both DFS and BFS have a time complexity of O(V^2) for adjacency matrix representations of graphs.