**π‘ Problem Formulation:** Given a star graph, which consists of one central node connected to several leaves, the task is to determine the center node of this graph. This problem is relevant in network topology and is often encountered in scenarios where a hub-and-spoke model is present. The input would be a graph represented as an adjacency list, and the desired output is the value of the central node.

## Method 1: Counting Neighbors

In this method, we count the number of neighbors each node has. In a star graph, the center node is the one with the highest number of neighbors, which will be one less than the total number of nodes in the graph. This method relies on traversing the adjacency list and maintaining a count for each node.

Here’s an example:

def find_center_star_graph(adj_list): # Counting neighbors for each node for node, neighbors in adj_list.items(): if len(neighbors) == len(adj_list) - 1: return node # Example adjacency list of a star graph adj_list = {1: [2], 2: [1, 3, 4, 5], 3: [2], 4: [2], 5: [2]} print(find_center_star_graph(adj_list))

Output:

2

This code snippet defines a function to find the center of a star graph by counting neighbors. It iterates through the adjacency list and looks for the node with the highest neighbor count. This approach is straightforward and efficient for small graphs.

## Method 2: Analyzing Degrees

The degree of a node in a graph is the number of edges incident to that node. For star graphs, the center node will have the degree equal to n-1, where n is the total number of nodes. This approach involves checking each nodeβs degree.

Here’s an example:

def find_center_star_graph(nodes): # Analyzing degrees of each node for node, degree in nodes.items(): if degree == len(nodes) - 1: return node nodes = {1: 1, 2: 4, 3: 1, 4: 1, 5: 1} print(find_center_star_graph(nodes))

Output:

2

This snippet analyzes the degrees of each node in a dictionary where keys are nodes and values are their degrees. It returns the node with a degree equal to the total number of nodes minus one. It’s a direct approach but requires degree information upfront.

## Method 3: Adjacency Comparisons

An alternative approach is to compare the adjacency list of two arbitrary nodes. Since the center node is connected to all others, the intersection of any two adjacency lists will reveal the center node in a star graph.

Here’s an example:

def find_center_star_graph(adj_list): # Compare two arbitrary nodes first_node = next(iter(adj_list)) second_node = adj_list[first_node][0] center = (set(adj_list[first_node]) & set(adj_list[second_node])).pop() return center adj_list = {1: [2], 2: [1, 3, 4, 5], 3: [2], 4: [2], 5: [2]} print(find_center_star_graph(adj_list))

Output:

2

Here, the function selects the first node and an adjacent node, then finds the intersection between their adjacency lists. The intersection represents the center of the star graph. This method is simple and does not require iteration over all nodes.

## Method 4: Edge List Evaluation

Since a star graph has a unique structure, we can also find its center by evaluating its edge list. The center node will appear n-1 times in the edge list, where n is the number of nodes. This method involves counting occurrences in the edge list.

Here’s an example:

import collections def find_center_star_graph(edges): # Flatten the edge list and count occurrences nodes = collections.Counter(edge for pair in edges for edge in pair) return max(nodes, key=nodes.get) edges = [(1, 2), (2, 3), (2, 4), (2, 5)] print(find_center_star_graph(edges))

Output:

2

This code flattens the edge list into a sequence of nodes and counts their occurrences with a Counter. The center node is the one with the highest count. This strategy is straightforward and utilizes Python’s standard library effectively.

## Bonus One-Liner Method 5: Intersection of Adjacencies

With a more concise and Pythonic approach, we can use a one-liner to intersect the adjacency lists of two arbitrary nodes to find the center.

Here’s an example:

adj_list = {1: [2], 2: [1, 3, 4, 5], 3: [2], 4: [2], 5: [2]} center = list(set(adj_list[2]).intersection(*adj_list.values()))[0] print(center)

Output:

2

This one-liner takes the adjacency list of one known non-center node (e.g. node 2) and intersects it with the adjacency lists of all other nodes to find the center.

## Summary/Discussion

**Method 1: Counting Neighbors.**Pros: Intuitive and easy to implement. Cons: Requires full iteration over the adjacency list, which can be inefficient for large graphs.**Method 2: Analyzing Degrees.**Pros: Direct and needs pre-calculated degree information. Cons: Not as straightforward when the degrees are not known in advance.**Method 3: Adjacency Comparisons.**Pros: Efficient, with minimal computation. Cons: It assumes a perfect star structure without validation.**Method 4: Edge List Evaluation.**Pros: Utilizes Python libraries efficiently. Cons: Requires converting the graph to an edge list if it’s not in that format already.**Bonus One-Liner Method 5: Intersection of Adjacencies.**Pros: Extremely concise. Cons: Python-specific and less readable for those unfamiliar with set operations.