Add Edges to Make Degrees of All Nodes Even

Problem Description

You are given an undirected graph consisting of n nodes and a 2D array edges where each edge connects two nodes. You can add at most two additional edges to ensure that every node in the graph has an even degree. Return true if it's possible to achieve this, otherwise return false.

Key Insights

• A node has an even degree if it is connected to an even number of edges.
• The sum of the degrees of all nodes in a graph is twice the number of edges, which means the total degree is even.
• The parity (even or odd nature) of a node's degree can be determined by the number of edges connected to it.
• To make all degrees even, we need to count how many nodes have odd degrees.
• If there are:
  • 0 odd degrees: All nodes are already even.
  • 2 odd degrees: We can connect these two nodes.
  • 4 odd degrees: We can connect them in pairs (two edges).
  • More than 4 odd degrees: We cannot make all degrees even with just two edges.

Space and Time Complexity

Time Complexity: O(n + e) where n is the number of nodes and e is the number of edges (for traversing the list of edges). Space Complexity: O(n) for storing the degree count of each node.

Solution

To solve the problem, we will:

1. Count the degree of each node using an array where the index corresponds to the node number.
2. Determine how many nodes have odd degrees.
3. Apply the insights from above to decide if we can make all degrees even by adding at most two edges.

Code Solutions