# Degree-Constrained Subgraph Reconfiguration is in P

## Abstract

The degree-constrained subgraph problem asks for a subgraph of a given graph such that the degree of each vertex is within some specified bounds. We study the following reconfiguration variant of this problem: Given two solutions to a degree-constrained subgraph instance, can we transform one solution into the other by adding and removing individual edges, such that each intermediate subgraph satisfies the degree constraints and contains at least a certain minimum number of edges? This problem is a generalization of the matching reconfiguration problem, which is known to be in P. We show that even in the more general setting the reconfiguration problem is in P.

## Notes

### Acknowledgements

We would like to thank the anonymous referees for their constructive comments and valuable remarks on this paper.

## References

- 1.Bonsma, P.S.: The complexity of rerouting shortest paths. Theor. Comput. Sci.
**510**, 1–12 (2013)MathSciNetCrossRefMATHGoogle Scholar - 2.Cereceda, L., van den Heuvel, J., Johnson, M.: Finding paths between 3-colorings. J. Graph Theory
**67**(1), 69–82 (2011)MathSciNetCrossRefMATHGoogle Scholar - 3.Gabow, H.N.: An efficient reduction technique for degree-constrained subgraph and bidirected network flow problems. In: Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, STOC 1983, pp. 448–456. ACM, New York (1983)Google Scholar
- 4.Ito, T., Demaine, E.D., Harvey, N.J.A., Papadimitriou, C.H., Sideri, M., Uehara, R., Uno, Y.: On the complexity of reconfiguration problems. Theor. Comput. Sci.
**412**(1214), 1054–1065 (2011)MathSciNetCrossRefMATHGoogle Scholar - 5.Shiloach, Y.: Another look at the degree constrained subgraph problem. Inf. Process. Lett.
**12**(2), 89–92 (1981)MathSciNetCrossRefMATHGoogle Scholar - 6.van den Heuvel, J.: The complexity of change. In: Blackburn, S.R., Gerke, S., Wildon, M.: (eds.) Surveys in Combinatorics 2013. London Mathematical Society Lectures Note Series, vol. 409 (2013)Google Scholar
- 7.Wrochna, M.: Homomorphism reconfiguration via homotopy. In: 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, 4–7 March 2015, Garching, Germany, pp. 730–742 (2015)Google Scholar