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I am looking for an algorithm that finds minimal subset of vertices such that by removing this subset (and edges connecting these vertices) from graph all other vertices become unconnected (i.e. the graph won't have any edges).
I have a basic knowledge of graph theory so please excuse any incorrectness.
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A graph is disconnected if at least two vertices of the graph are not connected by a path. If a graph G is disconnected, then every maximal connected subgraph of G is called a connected component of the graph G.
Removing a Vertex in the Graph: To remove a vertex from the graph, we need to check if that vertex exists in the graph or not and if that vertex exists then we need to shift the rows to the left and the columns upwards of the adjacency matrix so that the row and column values of the given vertex gets replaced by the ...
An undirected graph G is therefore disconnected if there exist two vertices in G such that no path in G has these vertices as endpoints. A graph with just one vertex is connected. An edgeless graph with two or more vertices is disconnected.
1 Answer. Easiest explanation - Prim's algorithm iterates from one node to another, so it can not be applied for disconnected graph.
IIUC, this is the classic Minimum Vertex Cover problem, which is, unfortunately, NP Complete.
Fortunately, the most intuitive and greedy possible algorithm is as good as it gets in this case.
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The greedy algorithm is a 2-approximation for vertex cover, which in theory, under the Unique Games Conjecture, is as good as it gets. In practice, solving a formulation of vertex cover as an integer program will most likely yield much better results. The program is
min sum_{v in V} x(v)
s.t.
forall {u, v} in E, x(u) + x(v) >= 1
forall v in V, x(v) in {0, 1}.
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