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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.
In mathematics, an edge cycle cover (sometimes called simply cycle cover [1]) of a graph is a family of cycles which are subgraphs of G and contain all edges of G. If the cycles of the cover have no vertices in common, the cover is called vertex-disjoint or sometimes simply disjoint cycle cover. In this case, the set of the cycles constitutes a ...
In mathematics, a vertex cycle cover (commonly called simply cycle cover) of a graph G is a set of cycles which are subgraphs of G and contain all vertices of G . If the cycles of the cover have no vertices in common, the cover is called vertex-disjoint or sometimes simply disjoint cycle cover. This is sometimes known as exact vertex cycle cover.
Clique cover. In graph theory, a clique cover or partition into cliques of a given undirected graph is a collection of cliques that cover the whole graph. A minimum clique cover is a clique cover that uses as few cliques as possible. The minimum k for which a clique cover exists is called the clique cover number of the given graph.
Edge cover. 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 ...
Bipartite double cover. In graph theory, the bipartite double cover of an undirected graph G is a bipartite, covering graph of G, with twice as many vertices as G. It can be constructed as the tensor product of graphs, G × K2. It is also called the Kronecker double cover, canonical double cover or simply the bipartite double of G .
A vertex-cover (aka hitting set or transversal) in H is set T ⊆ V such that, for all hyperedges e ∈ E, it holds that T ∩ e ≠ ∅ . The vertex-cover number (aka transversal number) of a hypergraph H is the smallest size of a vertex cover in H. It is often denoted by τ(H). [ 1]: 466. For example, if H is this 3-uniform hypergraph:
The covering map is indicated by the vertex colors. In graph theory, a planar cover of a finite graph G is a finite covering graph of G that is itself a planar graph. Every graph that can be embedded into the projective plane has a planar cover; an unsolved conjecture of Seiya Negami states that these are the only graphs with planar covers. [ 1]