Question

A simple undirected graph is complete if it contains an edge between every pair of distinct vertices. What does a depth-first search tree of a complete graph look like?

Short answer

A DFS tree of a complete graph has a star-like structure, with the root node connected directly to all other nodes.

Step-by-step solution

Understand the Graph

A complete graph is one in which there is an edge between every pair of distinct vertices. In simpler terms, each vertex is directly connected to every other vertex in the graph.

Concept of Depth-First Search (DFS)

Depth-First Search (DFS) is a graph traversal technique where you start at a root node (any node), explore as far as possible along a branch before backtracking. This means visiting a node, then visiting one of its adjacent nodes, and so on until you reach a node with no unvisited adjacent nodes.

Initial Step of DFS in Complete Graph

In a complete graph, starting from any chosen root node, the DFS will select one of the adjacent nodes to visit next (since all nodes are adjacent).

Recursive Nature of DFS

Since every node is adjacent in a complete graph, from the newly visited node, DFS will again pick an adjacent node to visit. This process continues, with DFS visiting as far along a branch as it can go (which, in a complete graph, usually means visiting all nodes).

Structure of DFS Tree in Complete Graph

In a DFS tree of a complete graph, the root node will be connected to each of the other nodes, forming the tree branches. Each node after the root will have just the initial edge connecting back to the root, forming a star-like structure. No additional branches are needed since all nodes are directly connected to the root node.

Key concepts

complete graph

A complete graph is a type of graph where there is an edge between every pair of distinct vertices. Imagine you have a group of friends, and each friend is friends with every other friend in the group. That is like a complete graph!
In mathematical terms, if the graph has \( n \) vertices, each vertex has an edge to the remaining \( n-1 \) vertices. This makes a complete graph incredibly interconnected, ensuring there are no isolated points.
Complete graphs are also denoted as \( K_n \), where \( n \) represents the number of vertices. For example, \( K_4 \) would be a complete graph with four vertices.

DFS tree

A Depth-First Search (DFS) tree is formed when using the DFS algorithm to traverse a graph.
When performing DFS on a graph, you start from a root node, visit an adjacent unvisited node, continue this process until no more unvisited nodes are reachable, then backtrack.
In the context of a complete graph, the DFS tree has a unique structure.
Starting from any node, the DFS process will first visit all other nodes directly connected to the root (which they all are in a complete graph). This means that right after visiting the root node, the DFS will pursue all other nodes as immediate children of the root, forming what looks like a star with all leaf nodes radiating out from the central root node.
In other words, if you start DFS at vertex \( v \), the DFS tree will look like a star with \( v \) at the center.

graph traversal

Graph traversal means visiting all the nodes in a graph in a specific manner. Two common types of graph traversal methods are Depth-First Search (DFS) and Breadth-First Search (BFS).

In DFS, you travel as far as possible along one path before backtracking to explore another path.
It uses a stack data structure, either implicitly through recursion or explicitly with a manual stack. Each time you push a node onto the stack and continue until there are no more nodes to visit along that path.
In BFS, you explore all neighbors at the present depth level before moving on to nodes at the next depth level. This uses a queue structure to keep track of the next node to visit.

Both traversal techniques are essential for solving different computational problems like searching, finding connected components, and even more complex tasks like topological sorting and finding cycles.