A refined algorithm for maximum independent set in degree-4 graphs

The maximum independent set problem is one of the most important problems in worstcase analysis for exact algorithms. Improvement on this problem in low-degree graphs can be used to get improvement on the problem in general graphs. In this paper, we show that the maximum independent set problem in an n-vertex graph with degree bounded by 4 can be solved in O∗(1.1376n) time and polynomial space, improving all previous exact algorithms for this problem. As most fast exact algorithms, this algorithm is a branch-and-search algorithm and analyzed by using the measure and conquer method. To effectively analyze the running time bound, we use the idea of ‘shift’ to save some decreasing on the measure from some good branches to some bad branches. After treating cycles of length 3 and 4 in the graph, we check carefully what will happen after branching on a degree-4 vertex (without any local structure), and then we can get the claimed improvement.