Apply the BF-algorithm of Section 11.7 to determine a...

Apply the BF-algorithm of Section $11.7$ to determine a BFS-tree for the following: (a) The graph of the regular dodecahedron (any root) (b) GraphBuster (any root) (c) A graph of order $n$ whose edges are arranged in a cycle (any root) (d) A complete graph $K_{n}$ (any root) (e) A complete bipartite graph $K_{m, n}$ (a left-vertex root, and a right-vertex root) In each case, determine the breadth-first numbers and the distance of each vertex from the root chosen.

Apply the BF-algorithm of Section $11.7$ to determine a BFS-tree for the following:
(a) The graph of the regular dodecahedron (any root)
(b) GraphBuster (any root)
(c) A graph of order $n$ whose edges are arranged in a cycle (any root)
(d) A complete graph $K_{n}$ (any root)
(e) A complete bipartite graph $K_{m, n}$ (a left-vertex root, and a right-vertex root)
In each case, determine the breadth-first numbers and the distance of each vertex from the root chosen.

Key Concepts

Breadth-First Search (BFS) Algorithm

The BFS algorithm is a method for exploring all the vertices of a graph systematically by levels. It starts from a designated root vertex and visits all its neighbors before moving on to the neighbors of those vertices. This method uses a queue data structure to keep track of the order in which vertices are to be examined and ensures that each vertex is visited in the order of its distance from the root.

BFS Tree Construction

A BFS tree is a spanning tree that results from running the BFS algorithm on a graph. This tree captures the structure of the graph by recording the predecessor of each vertex as the vertex from which it was first discovered. The edges of the BFS tree connect each vertex to the vertex from which it was reached, thereby preserving the minimal distance from the root for each vertex.

Level Numbering and Distance Labeling

BFS naturally partitions the graph's vertices into levels based on their distance from the starting root. The root is at level 0, its direct neighbors are at level 1, and so on. In this context, breadth-first numbers or timestamps can be assigned as each vertex is visited, and the number of levels from the root provides a direct measure of the shortest-path distance to any vertex in the graph.

Graph Structures and Their Properties in BFS

Different types of graphs, such as cyclic graphs, complete graphs, and bipartite graphs, possess unique characteristics that influence the structure of the BFS tree. For instance, complete graphs yield trees where all non-root vertices are direct neighbors of the root, while cycle graphs yield a sequential ordering. Similarly, the inherent symmetry and regularity present in certain graphs affect the order of discovery and the level structure of the resulting BFS tree.

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