New Branching Rules: Improvements on Independent Set and Vertex Cover in Sparse Graphs

We present an O(1.0919)-time algorithm for finding a maximum independent set in an n-vertex graph with degree bounded by 3, which improves the previously known algorithm of running time O(1.0977) by Bourgeois, Escoffier and Paschos [IWPEC 2008]. We also present an O(1.1923)-time algorithm to decide if a graph with degree bounded by 3 has a vertex cover of size k, which improves the previously known algorithm of running time O(1.1939) by Chen, Kanj and Xia [ISAAC 2003]. Two new branching techniques, branching on a bottle and branching on a 4-cycle, are introduced, which help us to design simple and fast algorithms for the maximum independent set and minimum vertex cover problems and avoid tedious branching rules.