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Example graph that has a vertex cover comprising 2 vertices (bottom), but none with fewer. 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.
The most prominent examples of covering problems are the set cover problem, which is equivalent to the hitting set problem, and its special cases, the vertex cover problem and the edge cover problem. Covering problems allow the covering primitives to overlap; the process of covering something with non-overlapping primitives is called decomposition.
A simple example of an approximation algorithm is one for the minimum vertex cover problem, where the goal is to choose the smallest set of vertices such that every edge in the input graph contains at least one chosen vertex. One way to find a vertex cover is to repeat the following process: find an uncovered edge, add both its endpoints to the ...
In the above example, each vertex of H has exactly 2 preimages in C. Hence C is a 2-fold cover or a double cover of H. For any graph G, it is possible to construct the bipartite double cover of G, which is a bipartite graph and a double cover of G. The bipartite double cover of G is the tensor product of graphs G × K 2:
The complement of a vertex cover in any graph is an independent set, so a minimum vertex cover is complementary to a maximum independent set; finding maximum independent sets is another NP-complete problem. The equivalence between matching and covering articulated in Kőnig's theorem allows minimum vertex covers and maximum independent sets to ...
Set cover problem; V. Vertex cover This page was last edited on 20 November 2020, at 03:35 (UTC). Text is available under the Creative Commons ...
In the fractional set cover problem, it is allowed to select fractions of sets, rather than entire sets. A fractional set cover is an assignment of a fraction (a number in [0,1]) to each set in , such that for each element x in the universe, the sum of fractions of sets that contain x is at least 1. The goal is to find a fractional set cover in ...
An optimal clique cover of the line graph () may be formed with one clique for each triangle in that has two or three degree-2 vertices, and one clique for each vertex that has degree at least two and is not a degree-two vertex of one of these triangles. The intersection number is the number of cliques of these two types.