Minimum Degree of a Connected Trio in a Graph

Minimum Degree of a Connected Trio in a Graph

1. What is this problem about?

The Minimum Degree of a Connected Trio in a Graph problem asks you to find a "trio" of nodes $(u, v, w)$ such that every pair of these nodes is connected by an edge. Once you find such a trio, you calculate its "degree," which is the number of edges connected to $u, v,$ and $w$ that are not part of the trio. The goal is to find the minimum such degree among all connected trios in the graph. If no trio exists, return -1.

2. Why is this asked in interviews?

This "Hard" problem is asked by companies like Amazon and GoDaddy to test a candidate's ability to perform efficient graph enumeration. The Minimum Degree of a Connected Trio in a Graph interview question evaluates how you handle $O(N^3)$ constraints and whether you can optimize the search by using an adjacency matrix for fast edge lookups. It assesses your understanding of graph topology and degree calculations.

3. Algorithmic pattern used

The algorithmic pattern is Enumeration on a Graph. We first calculate the degree of every node. Then, we iterate through all possible triples $(u, v, w)$. To check if they form a trio, we check if edges $(u, v), (v, w),$ and $(u, w)$ all exist. If they do, the degree of the trio is $(degree[u] + degree[v] + degree[w] - 6)$. The "-6" accounts for the six endpoints of the three edges within the trio (each node's degree includes two edges to the other two members). This "Graph, Enumeration interview pattern" is feasible for $N$ up to a few hundreds.

4. Example explanation

Imagine nodes 1, 2, 3 form a triangle.

• Node 1 is also connected to node 4. (Total degree of 1 is 3).
• Node 2 is also connected to node 5 and 6. (Total degree of 2 is 4).
• Node 3 is only in the triangle. (Total degree of 3 is 2). Trio Degree = $(3 + 4 + 2) - 6 = 3$. The external edges are (1,4), (2,5), and (2,6).

5. Common mistakes candidates make

In the Minimum Degree of a Connected Trio in a Graph coding problem, many candidates struggle with the degree formula, often forgetting to subtract 6. Another common mistake is using an adjacency list for the innermost check, which makes the complexity $O(N \times E)$, whereas an adjacency matrix allows for $O(1)$ edge verification. Some also fail to handle the case where no trio is found, forgetting to initialize the result to a large value and checking it at the end.

6. Interview preparation tip

When dealing with "triples" or "triangles" in a graph, an adjacency matrix is your best friend if $N$ is small enough (usually $N \leq 1000$). This "Enumeration interview pattern" is a brute-force approach that is often the expected solution when the structure being searched for is small and specific. Practice calculating degrees and managing adjacency structures effectively.