Given a graph G = (V, E) on n vertices, the Maximum Υ-REGULAR Induced Subgraph (M-Υ-RIS) problems ask for a maximum sized subset of vertices R is contained in V such that the induced subgraph on R, G[R], is Υ-regular. We give an O(c~n) time algorithm for these problems for any fixed constant r, where c is a positive constant strictly less than 2, solving a well known open problem. These algorithms are then generalized to solve counting and enumeration version of these problems in the same time. An interesting consequence of the enumeration algorithm is, that it shows that the number of maximal Υ-regular induced subgraphs for a fixed constant r on any graph on n vertices is upper bounded by o(2~n). We then give combinatorial lower bounds on the number of maximal r-regular induced subgraphs possible on a graph on n vertices and also give matching algorithmic upper bounds. We use the techniques and results obtained in the paper to obtain an improved exact algorithm for a special case of Induced Subgraph Isomorphism that is Induced Υ-Regular Subgraph Isomorphism, where r is a constant. All the algorithms in the paper are simple but their analyses are not. Some of the upper bound proofs or algorithms require a new and different measure than the usual number of vertices or edges to measure the progress of the algorithm, and require solving an interesting system of polynomials.
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