In graph theory, an edge cover of a graph is a set of edges such that every vertex of the graph is incident to at least one edge of the set.
In computer science, the minimum edge cover problem is the problem of finding an edge cover of minimum size. It is an optimization problem that belongs to the class of covering problems and can be solved in polynomial time.
Formally, an edge cover of a graph G is a set of edges C such that each vertex in G is incident with at least one edge in C. The set C is said to cover the vertices of G. The following figure shows examples of edge coverings in two graphs (the set C is marked with red).
A minimum edge covering is an edge covering of smallest possible size. The edge covering number is the size of a minimum edge covering. The following figure shows examples of minimum edge coverings (again, the set C is marked with red).
Note that the figure on the right is not only an edge cover but also a matching. In particular, it is a perfect matching: a matching M in which each vertex is incident with exactly one edge in M. A perfect matching (if it exists) is always a minimum edge covering.
The set of all edges is an edge cover, assuming that there are no degree-0 vertices.
The complete bipartite graph Km,n has edge covering number max(m, n).
A smallest edge cover can be found in polynomial time by finding a maximum matching and extending it greedily so that all vertices are covered. In the following figure, a maximum matching is marked with red; the extra edges that were added to cover unmatched nodes are marked with blue. (The figure on the right shows a graph in which a maximum matching is a perfect matching; hence it already covers all vertices and no extra edges were needed.)
On the other hand, the related problem of finding a smallest vertex cover is an NP-hard problem.
Looking at the image it already becomes obvious why, for a given minimal edge Cover and maximum matching , the following is true: .
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
This course covers numerous powerful algorithmic techniques (greedy, local search, linear programming, multiplicative weight update, ...). The concepts are studied in clean and simple settings so as t
A first graduate course in algorithms, this course assumes minimal background, but moves rapidly. The objective is to learn the main techniques of algorithm analysis and design, while building a reper
Maximum cardinality matching is a fundamental problem in graph theory. We are given a graph G, and the goal is to find a matching containing as many edges as possible; that is, a maximum cardinality subset of the edges such that each vertex is adjacent to at most one edge of the subset. As each edge will cover exactly two vertices, this problem is equivalent to the task of finding a matching that covers as many vertices as possible.
In graph theory, an edge cover of a graph is a set of edges such that every vertex of the graph is incident to at least one edge of the set. In computer science, the minimum edge cover problem is the problem of finding an edge cover of minimum size. It is an optimization problem that belongs to the class of covering problems and can be solved in polynomial time. Formally, an edge cover of a graph G is a set of edges C such that each vertex in G is incident with at least one edge in C.
In graph theory, a vertex cover (sometimes node cover) of a graph is a set of vertices that includes at least one endpoint of every edge of the graph. In computer science, the problem of finding a minimum vertex cover is a classical optimization problem. It is NP-hard, so it cannot be solved by a polynomial-time algorithm if P ≠ NP. Moreover, it is hard to approximate – it cannot be approximated up to a factor smaller than 2 if the unique games conjecture is true. On the other hand, it has several simple 2-factor approximations.