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Examples of vertex covers Examples of minimum vertex covers. Formally, a vertex cover ′ of an undirected graph = (,) is a subset of such that ′ ′, that is to say it is a set of vertices ′ where every edge has at least one endpoint in the vertex cover ′.
A vertex cover in a graph is a set of vertices that includes at least one endpoint of every edge, and a vertex cover is minimum if no other vertex cover has fewer vertices. [1] A matching in a graph is a set of edges no two of which share an endpoint, and a matching is maximum if no other matching has more edges. [2]
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.
Via this result, the minimum vertex cover, maximum independent set, and maximum vertex biclique problems may be solved in polynomial time for bipartite graphs. Hall's marriage theorem provides a characterization of bipartite graphs which have a perfect matching and the Tutte theorem provides a characterization for arbitrary graphs.
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. A ...
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.
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