Undirected graph

Definition

An undirected graph is a collection of vertices connected by edges where the edges have no direction, meaning the connection between two vertices is bidirectional. This lack of direction allows for simpler representations of relationships, making undirected graphs useful in modeling many real-world scenarios like social networks, where connections are mutual. The concept of undirected graphs is foundational in graph theory and plays a crucial role in understanding algorithms that deal with paths and connectivity.

5 Must Know Facts For Your Next Test

1. In an undirected graph, the edges do not have arrows; they simply connect two vertices without indicating any specific order.
2. Undirected graphs can be used to represent symmetrical relationships, such as friendship in social networks, where both individuals mutually know each other.
3. The degree of a vertex in an undirected graph is the number of edges connected to it, and it is important for understanding the structure of the graph.
4. Algorithms like Depth-First Search (DFS) and Breadth-First Search (BFS) are commonly used on undirected graphs to explore their structure and find paths.
5. In an undirected graph, cycles can exist, which are paths that start and end at the same vertex without retracing any edges.

Review Questions

• How does the absence of direction in an undirected graph affect its representation and the relationships it models?
  • The absence of direction in an undirected graph allows for a bidirectional representation of relationships between vertices. This means that if vertex A is connected to vertex B, there is an implicit connection back from B to A. This characteristic makes undirected graphs ideal for modeling scenarios where interactions are mutual, such as friendships in social networks or collaboration between teams. The simplicity of this representation helps simplify various algorithms used to analyze such graphs.
• Discuss how the concept of degree in an undirected graph contributes to understanding its overall structure and connectivity.
  • In an undirected graph, the degree of a vertex indicates how many edges are incident to it. Understanding the degree distribution among vertices helps reveal important structural characteristics of the graph, such as identifying highly connected nodes or potential bottlenecks. For example, in a social network represented as an undirected graph, vertices with a high degree may represent influential individuals with numerous connections. Analyzing these degrees allows for deeper insights into the connectivity and cohesion within the entire network.
• Evaluate the implications of using undirected graphs for pathfinding algorithms compared to directed graphs.
  • When evaluating pathfinding algorithms on undirected graphs versus directed graphs, one key difference is the nature of traversable routes. In undirected graphs, every edge is bi-directional, allowing algorithms like Dijkstra's or A* to explore routes freely without concern for directionality. This simplifies the logic needed for certain operations since every connection is valid both ways. However, directed graphs may be more appropriate for certain scenarios where relationships are inherently one-way, such as web page links or traffic flow. Understanding these implications helps choose the right graph type depending on the specific problem being addressed.

Related terms

Vertex: A fundamental unit of a graph representing a point where edges meet, often symbolized as a dot or circle.

Edge: The connection between two vertices in a graph, which can represent relationships or pathways.

Connected Graph: A type of graph where there is a path between every pair of vertices, ensuring that all vertices are reachable from one another.