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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 ...
An example of a bipartite graph, with a maximum matching (blue) and minimum vertex cover (red) both of size six. In the mathematical area of graph theory, Kőnig's theorem, proved by Dénes Kőnig (), describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs.
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:
A disjoint cycle cover of an undirected graph (if it exists) can be found in polynomial time by transforming the problem into a problem of finding a perfect matching in a larger graph. [1] [2] If the cycles of the cover have no edges in common, the cover is called edge-disjoint or simply disjoint cycle cover.
ARP4761, Guidelines for Conducting the Safety Assessment Process on Civil Aircraft, Systems, and Equipment is an Aerospace Recommended Practice from SAE International. [1]
The W2SAT problem includes as a special case the vertex cover problem, of finding a set of k vertices that together touch all the edges of a given undirected graph. For any given instance of the vertex cover problem, one can construct an equivalent W2SAT problem with a variable for each vertex of a graph.