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.