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The decision variant of the vertex cover problem is NP-complete, which means it is unlikely that there is an efficient algorithm to solve it exactly for arbitrary graphs. NP-completeness can be proven by reduction from 3-satisfiability or, as Karp did, by reduction from the clique problem .
A w-vertex-cover is a multiset of vertices ("multiset" means that each vertex may appear several times), in which each edge e is adjacent to at least w e vertices. Egerváry's theorem says: In any edge-weighted bipartite graph, the maximum w-weight of a matching equals the smallest number of vertices in a w-vertex-cover.
The vertex cover problem parameterized by the feedback vertex number of the input graph has a polynomial kernelization: [9] There is a polynomial-time algorithm that, given a graph whose feedback vertex number is , outputs a graph ′ on () vertices such that a minimum vertex cover in ′ can be transformed into a minimum vertex cover for in ...
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 vertex cover is a special case of a fractional vertex cover in which all weights are either 0 or 1. The size of a fractional vertex-cover is the sum of fractions of all vertices. The fractional vertex-cover number of a hypergraph H is the smallest size of a fractional vertex-cover in H. It is often denoted by τ*(H).
Minimal vertex covers have been studied in statistical mechanics in connection with the hard-sphere lattice gas model, a mathematical abstraction of fluid-solid state transitions. [ 2 ] Every maximal independent set is a dominating set , a set of vertices such that every vertex in the graph either belongs to the set or is adjacent to the set.
Vertex cover is another example for which iterative compression can be employed. In the vertex cover problem, a graph G = ( V , E ) and a natural number k are taken as inputs and the algorithm must decide whether there exists a set X of k vertices such that every edge is incident to a vertex in X .
A clique cover of a graph G may be seen as a graph coloring of the complement graph of G, the graph on the same vertex set that has edges between non-adjacent vertices of G. Like clique covers, graph colorings are partitions of the set of vertices, but into subsets with no adjacencies (independent sets) rather than cliques.