Size and independence in triangle-free graphs with maximum degree three
Journal of Graph Theory
11/30 (finding large independent sets in connected triangle-free 3-regular graphs)
Journal of Combinatorial Theory Series B
A new proof of the independence ratio of triangle-free cubic graphs
Discrete Mathematics
Some simplified NP-complete problems
STOC '74 Proceedings of the sixth annual ACM symposium on Theory of computing
Parametric Duality and Kernelization: Lower Bounds and Upper Bounds on Kernel Size
SIAM Journal on Computing
A simple and fast algorithm for maximum independent set in 3-degree graphs
WALCOM'10 Proceedings of the 4th international conference on Algorithms and Computation
Parameterized Complexity
Nonblocker in h-minor free graphs: kernelization meets discharging
IPEC'12 Proceedings of the 7th international conference on Parameterized and Exact Computation
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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