# Problem Description
This is a graph path validation problem where we need to determine if a given sequence of node values represents a valid path through the undirected graph. The graph contains specially marked "start" and "end" nodes, and the path must follow specific rules to be considered valid.

# Visual Facts
1. The image shows two example graphs:
   - Example 1: A simple graph with 3 nodes (1=START, 3=END)
   - Example 2: A larger graph with 6 nodes (2=START, 6=END)

2. Each graph contains:
   - A green "START" node
   - A red "END" node
   - Intermediate nodes (unmarked/white)
   - Undirected edges connecting nodes

3. Multiple example paths are provided for each graph:
   - Example 1 shows 4 paths (1 valid, 3 invalid)
   - Example 2 shows 6 paths (3 valid, 3 invalid)

4. Each path is marked as either "valid" or "invalid" with a reason for invalidity

# Visual Patterns
1. Path Validity Rules:
   - Must start from the START node (green)
   - Must end at the END node (red)
   - Can only move along connected edges
   - Must visit all the nodes in the graph

2. Invalid Path Patterns:
   - Paths not starting from START node are invalid
   - Paths not ending at END node are invalid
   - Paths missing any node in the graph are invalid
   - Paths must follow the graph's edge connections

3. Valid Path Patterns:
   - Multiple valid paths can exist between START and END
   - The same node can be visited multiple times in a valid path


# Solution
from typing import List

class GraphNode:
    """
    Represents a node in a graph. Each node contains an integer value, a list of connected nodes, and a type attribute.

    Attributes:
        value (int): The integer value of the node.
        type (str): The type of the node ('start', 'end', or '').
        connected (list): A list of connected nodes (GraphNode objects).
    """
    def __init__(self, value: int, type: str = '', connected: list = None):
        self.value = value
        self.type = type
        self.connected = connected if connected is not None else []

def solution(start: GraphNode, path: List[int]) -> bool:
    """
    Determines whether a given path through the graph is valid according to the specified rules.

    Args:
        start: The starting node of the path, represented as a GraphNode object
        path: A sequence of node values representing the traversal path through the graph. path[0] = start.value

    Returns:
        True if the path is valid according to the rules, False otherwise.
    """
    if not start or not path:
        return False

    # Create a mapping from node value to GraphNode for quick lookup
    node_map = {}
    stack = [start]
    while stack:
        node = stack.pop()
        if node.value in node_map:
            continue
        node_map[node.value] = node
        for connected_node in node.connected:
            stack.append(connected_node)

    # Check if all nodes are in the path
    if set(node_map.keys()) != set(path):
        return False

    # Check if the path starts with 'start' and ends with 'end'
    if node_map[path[0]].type != 'start' or node_map[path[-1]].type != 'end':
        return False

    # Check if each step in the path is valid (i.e., nodes are connected)
    for i in range(len(path) - 1):
        current_node = node_map[path[i]]
        next_node = node_map[path[i + 1]]
        if next_node not in current_node.connected:
            return False

    return True

# Test Script
from typing import List

class GraphNode:
    """
    Represents a node in a graph. Each node contains an integer value, a list of connected nodes, and a type attribute.
    
    Attributes:
        value (int): The integer value of the node.
        type (str): The type of the node, can be 'start', 'end', or ''.
        connected (list): A list of connected nodes (GraphNode objects).
    """
    def __init__(self, value: int, type: str = '', connected: list = None):
        self.value = value
        self.type = type
        self.connected = connected if connected is not None else []

def format_graph(start_node):
    """Helper function to format graph as dictionary for error messages"""
    visited = set()
    nodes = []
    edges = {}
    
    def dfs(node):
        if node in visited:
            return
        visited.add(node)
        nodes.append({
            'value': node.value,
            'type': node.type
        })
        edges[f'g{node.value}'] = [f'g{n.value}' for n in node.connected]
        for next_node in node.connected:
            dfs(next_node)
            
    dfs(start_node)
    return {
        'nodes': nodes,
        'edges': edges
    }

def test():
    test_cases = []
    
    # Test 1
    g1 = GraphNode(1, "start")
    g2 = GraphNode(2, "end") 
    g1.connected = [g2]
    g2.connected = [g1]
    test_cases.append((g1, [1,2], True))

    # Test 2
    g1 = GraphNode(1, "start")
    g2 = GraphNode(2, "end")
    g1.connected = [g2]
    g2.connected = [g1]
    test_cases.append((g1, [1,2,1], False))

    # Test 3
    g1 = GraphNode(1, "start")
    g2 = GraphNode(2, "")
    g3 = GraphNode(3, "end")
    g1.connected = [g2]
    g2.connected = [g1, g3]
    g3.connected = [g2]
    test_cases.append((g1, [1,2,3], True))

    # Test 4  
    g1 = GraphNode(1, "start")
    g2 = GraphNode(2, "")
    g3 = GraphNode(3, "")
    g4 = GraphNode(4, "end")
    g1.connected = [g2]
    g2.connected = [g1, g3]
    g3.connected = [g2, g4]
    g4.connected = [g3]
    test_cases.append((g1, [1,2,3,4], True))

    # Test 5
    g1 = GraphNode(1, "")
    g2 = GraphNode(2, "start")
    g3 = GraphNode(3, "end")
    g1.connected = [g2]
    g2.connected = [g1, g3]
    g3.connected = [g2]
    test_cases.append((g2, [2,1,2,3], True))

    # Test 6
    g1 = GraphNode(1, "start")
    g2 = GraphNode(2, "")
    g3 = GraphNode(3, "end")
    g1.connected = [g2, g3]
    g2.connected = [g1]
    g3.connected = [g1]
    test_cases.append((g1, [1,2,1,3], True))

    # Test 7
    g1 = GraphNode(1, "start")
    g2 = GraphNode(2, "")
    g3 = GraphNode(3, "end")
    g1.connected = [g2, g3]
    g2.connected = [g1]
    g3.connected = [g1]
    test_cases.append((g1, [1,2,3], False))

    # Test 8
    g1 = GraphNode(1, "")
    g2 = GraphNode(2, "start")
    g3 = GraphNode(3, "")
    g4 = GraphNode(4, "")
    g5 = GraphNode(5, "")
    g6 = GraphNode(6, "end")
    g1.connected = [g2]
    g2.connected = [g1, g3, g4]
    g3.connected = [g2]
    g4.connected = [g2, g5, g6]
    g5.connected = [g4]
    g6.connected = [g4]
    test_cases.append((g2, [2,3,2,1,2,4,5,4,6], True))

    # Test 9
    g1 = GraphNode(1, "")
    g2 = GraphNode(2, "start")
    g3 = GraphNode(3, "")
    g4 = GraphNode(4, "")
    g5 = GraphNode(5, "")
    g6 = GraphNode(6, "end")
    g1.connected = [g2]
    g2.connected = [g1, g3, g4]
    g3.connected = [g2]
    g4.connected = [g2, g5, g6]
    g5.connected = [g4]
    g6.connected = [g4]
    test_cases.append((g2, [2,1,2,4,5,4,6], False))

    # Test 10
    g1 = GraphNode(1, "")
    g2 = GraphNode(2, "start")
    g3 = GraphNode(3, "")
    g4 = GraphNode(4, "")
    g5 = GraphNode(5, "")
    g6 = GraphNode(6, "end")
    g1.connected = [g2]
    g2.connected = [g1, g3, g4]
    g3.connected = [g2]
    g4.connected = [g2, g5, g6]
    g5.connected = [g4]
    g6.connected = [g4]
    test_cases.append((g2, [2,1,2,3,2,4,5,4,6], True))

    for i, (start, path, expected_output) in enumerate(test_cases):
        try:
            output = solution(start, path)
        except Exception as e:
            error_message = (
                f"An exception occurred while running test case {i+1}:\n"
                f"`start`: Graph structure: {format_graph(start)}\n"
                f"`path`: {path}\n\n"
                f"Exception: '{str(e)}'"
            )
            assert False, error_message

        if output != expected_output:
            error_message = (
                f"Test case {i+1} failed:\n"
                f"`start`: Graph structure: {format_graph(start)}\n"
                f"`path`: {path}\n\n"
                f"Actual Output: {output}\n"
                f"Expected Output: {expected_output}"
            )
            assert False, error_message

test()