Understanding Graph Properties and BFS Tree Height

This problem asks us to analyze the properties of a Breadth-First Search (BFS) tree derived from an undirected, unweighted graph with specific characteristics. We are given a graph $G(V, E)$ with 100 vertices. The distance $d(u, v)$ represents the length of the shortest path between vertices $u$ and $v$. We are told that the maximum shortest path distance between any two distinct vertices in the graph is 30. This value is known as the diameter of the graph, often denoted as $D$. So, $D = \max_{u, v \in V, u \neq v} d(u, v) = 30$. We need to determine a correct statement about the height of any BFS tree $T$ constructed for this graph $G$.

Key Concepts: BFS Tree and Height

• BFS Tree: A BFS tree is a subgraph formed by performing a Breadth-First Search starting from a source vertex $s$. It includes the source vertex and all vertices reachable from $s$, with edges connecting a vertex to its neighbors discovered first during the BFS. Importantly, the edges in a BFS tree always correspond to shortest paths from the source vertex $s$.
• Height of a BFS Tree: The height of a BFS tree $T$ rooted at $s$, denoted as $H(T)$, is the maximum distance from the root $s$ to any other vertex $v$ in the tree. Mathematically, $H(T) = \max_{v \in V} d(s, v)$.
• Graph Diameter: The diameter $D$ of a graph is the longest shortest path between any pair of vertices in the graph.

Relating Diameter and BFS Tree Height

Let $T$ be any BFS tree of the graph $G$, rooted at an arbitrary vertex $s$. Let the height of this tree be $H(T)$. We know the diameter of the graph is $D = 30$. By definition of the diameter, there must exist at least one pair of vertices, let's call them $x$ and $y$, such that the shortest distance between them is $d(x, y) = 30$.

Consider the path from $x$ to $y$. We can use the triangle inequality for shortest paths, which states that for any vertices $x, s, y$, the shortest path distance $d(x, y)$ is less than or equal to the sum of the shortest path distances $d(x, s)$ and $d(s, y)$.

So, we have:

Since $T$ is a BFS tree rooted at $s$, the distances $d(x, s)$ and $d(s, y)$ represent shortest path distances from $s$. The height of the BFS tree $H(T)$ is the maximum shortest distance from $s$ to any vertex. Therefore, the distance from $s$ to $x$ cannot be greater than the height, and similarly for $y$:

Substituting these into the inequality:

Dividing both sides by 2, we get:

This inequality shows that the height of any BFS tree $T$ for this graph must be greater than or equal to 15.

Analyzing the Options

• Option 1: The height of T is exactly 15. This is not necessarily true. While the height can be 15 (e.g., if the BFS is rooted near the middle of a diameter path), it could also be greater.
• Option 2: The height of T is exactly 30. This is also not necessarily true. The height depends on the choice of the root $s$. It could be 15 or some value between 15 and 30.
• Option 3: The height of T is at least 15. Our derivation showed that $H(T) \ge 15$, which means the height is at least 15. This statement holds true for all possible BFS trees.
• Option 4: The height of T is at least 30. This is not necessarily true. As shown in the thought process example of a path graph $P_{31}$ rooted at its center vertex, the height can be 15, which is less than 30.

Conclusion

Based on the relationship derived between the graph diameter and the height of any BFS tree ($H(T) \ge D/2$), the height of the BFS tree $T$ must be at least 15.