Algorithms and Complexity
Several researchers have studied the complexity of exponential time algorithms restricted to cubic graphs. For instance, by applying dynamic programming to a path decomposition of the graph, Fomin and Høie showed how to find their maximum independent sets in time O(2n/6 + o(n)). The travelling salesman problem can be solved in cubic graphs in time O(1.251n).
Several important graph optimization problems are APX hard, meaning that, although they have approximation algorithms whose approximation ratio is bounded by a constant, they do not have polynomial time approximation schemes whose approximation ratio tends to 1 unless P=NP. These include the problems of finding a minimum vertex cover, maximum independent set, minimum dominating set, and maximum cut. The crossing number (the minimum number of edges which cross in any graph drawing) of a cubic graph is also NP-hard for cubic graphs but may be approximated.
Read more about this topic: Cubic Graph
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