The computational complexity of graph contractions II: Two tough polynomially solvable cases

For a fixed pattern graph H , let H -CONTRACTIBILITY denote the problem of deciding whether a given input graph is contractible to H . This article is part II of our study on the computational complexity of the H -CONTRACTIBILITY problem. In the first article we pinpointed the complexity for all pattern graphswith five vertices except for two pattern graphs H . Here, we present polynomial time algorithms for these two remaining pattern graphs. Interestingly, in all connected cases that are known to be polynomially solvable, the pattern graph H has a dominating vertex, whereas in all cases that are known to be NP-complete, the pattern graph H does not have a dominating vertex. © 2008Wiley Periodicals, Inc. NETWORKS, Vol. 52(1), 32–56 2008