Find the minimum number of nodes to be removed to convert a graph into a tree.
number of nodes < 1e5
number of edges < 2e5
Any help, please...
Find the minimum number of nodes to be removed to convert a graph into a tree.
number of nodes < 1e5
number of edges < 2e5
Any help, please...
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This problem is NP complete.
I quote from the paper "Maximum induced trees in graphs" by Paul Erdös, Michael Saks and Vera T Sós:
"The problem “given a graph H and integer k does H have an independent set of size k?” is a well-known NP-complete problem. Given H and k, let n be the number of vertices of H and let G be the graph obtained by adjoining a path on n vertices to H, one endpoint of which is joined by an edge to every vertex in H. The problem of whether H has an independent set of size k is easily seen to be equivalent to whether G has an induced tree of size n + k."
Source: https://doi.org/10.1016/0095-8956(86)90028-6
Therefore I doubt that an algorithm exists that can solve this problem for such large graphs in reasonable time.