The Complexity of Graph Contractions
For a fixed pattern graph H, let H-CONTRACTIBILITY denote the problem of deciding whether a given input graph is contractible to H. We continue a line of research that was started in 1987 by Brouwer & Veldman, and we determine the computational complexity of H-CONTRACTIBILITY for certain classes of pattern graphs. In particular, we pin-point the complexity for all graphs H with five vertices.
Interestingly, in all 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.
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