Edge cover

Definition

An edge cover in a graph is a set of edges such that every vertex in the graph is incident to at least one edge from the set. This means that each vertex is 'covered' by one or more edges, ensuring that no vertex is left uncovered. Edge covers are useful in various applications, including network design and resource allocation, as they help ensure connectivity within the graph.

5 Must Know Facts For Your Next Test

1. An edge cover may not be unique; there can be multiple edge covers for a given graph.
2. The size of the smallest edge cover is called the edge covering number of the graph.
3. An edge cover can always be found by selecting all edges incident to a maximal matching in the graph.
4. If a graph has a perfect matching, then the edges in this matching form an edge cover.
5. Edge covers play an essential role in bipartite graphs, where they can help solve problems related to job assignments and resource distribution.

Review Questions

• How does an edge cover differ from a vertex cover in a graph?
  • An edge cover consists of a set of edges that ensures every vertex in the graph is incident to at least one edge from the set, while a vertex cover is formed by selecting vertices such that every edge in the graph has at least one endpoint among those vertices. This means that an edge cover focuses on covering all vertices through edges, whereas a vertex cover aims to cover all edges through selected vertices.
• Discuss the significance of finding the smallest edge cover in practical applications.
  • Finding the smallest edge cover is important because it helps optimize resources and connectivity within various applications, such as network design or transportation systems. A smaller edge cover minimizes the number of connections needed to ensure that every point (vertex) is connected, reducing costs and improving efficiency. This optimization can lead to better performance in network routing, resource allocation, and other logistical operations.
• Evaluate how the presence of a perfect matching in a bipartite graph influences the existence of an edge cover.
  • In bipartite graphs, if there exists a perfect matching, it implies that all vertices can be paired with exactly one other vertex through edges, leading to an efficient way to create an edge cover. The edges included in this perfect matching directly serve as an edge cover since every vertex will be incident to exactly one of these edges. This relationship highlights how matchings can provide effective solutions for ensuring coverage and connectivity within bipartite structures.

Related terms

vertex cover: A vertex cover is a set of vertices such that every edge in the graph is incident to at least one vertex from the set. It ensures that all edges are 'covered' by selecting specific vertices.

matching: A matching in a graph is a set of edges without common vertices, where no two edges share an endpoint. Matchings are often used to find optimal pairings in bipartite graphs.

independent set: An independent set is a set of vertices in a graph, none of which are adjacent. This means there are no edges connecting any pair of vertices within the set.