On the Independence Number of Graphs with Maximum Degree 3

Abstract

Let G be an undirected graph with maximum degree at most 3 such that G does not contain any of the three graphs shown in Figure [1] as a subgraph. We prove that the independence number of G is at least n(G)/3 + nt(G)/42, where n(G) is the number of vertices in G and nt(G) is the number of nontriangle vertices in G. This bound is tight as implied by the well-known tight lower bound of 5n(G)/14 on the independence number of triangle-free graphs of maximum degree at most 3. We then proceed to show some algorithmic applications of the aforementioned combinatorial result to the area of parameterized complexity. We present a linear-time kernelization algorithm for the independent set problem on graphs with maximum degree at most 3 that computes a kernel of size at most 140 k/47 < 3k, where k is the given parameter. This improves the known 3k upper bound on the kernel size for the problem, and implies a lower bound of 140k/93 on the kernel size for the vertex cover problem on graphs with maximum degree at most 3.KeywordsMaximum DegreeCommon NeighborKernel SizeReduction RuleIndependence NumberThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.