International Workshop on Combinatorial Algorithms

IWOCA 2014: Combinatorial Algorithms pp 164-175

# Reconfiguration of Vertex Covers in a Graph

• Takehiro Ito
• Hiroyuki Nooka
• Xiao Zhou
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8986)

## Abstract

Suppose that we are given two vertex covers $$C_{0}$$ and $$C_{t}$$ of a graph G, together with an integer threshold $$k\ge \max \{\left| C_0 \right| , \left| C_t \right| \}$$. Then, the vertex cover reconfiguration problem is to determine whether there exists a sequence of vertex covers of G which transforms $$C_{0}$$ into $$C_{t}$$ such that each vertex cover in the sequence is of cardinality at most $$k$$ and is obtained from the previous one by either adding or deleting exactly one vertex. This problem is PSPACE-complete even for planar graphs. In this paper, we first give a linear-time algorithm to solve the problem for even-hole-free graphs, which include several well-known graphs, such as trees, interval graphs and chordal graphs. We then give an upper bound on $$k$$ for which any pair of vertex covers in a graph G has a desired sequence. Our upper bound is best possible in some sense.

## Notes

### Acknowledgment

We are grateful to Ryuhei Uehara for fruitful discussions. This work is partially supported by JSPS KAKENHI 25106504 and 25330003.

## References

1. 1.
Bonamy, M., Johnson, M., Lignos, I., Patel, V., Paulusma, D.: Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs. J. Comb. Optim. 27, 132–143 (2014)
2. 2.
Bonsma, P., Cereceda, L.: Finding paths between graph colourings: PSPACE-completeness and superpolynomial distances. Theoret. Comput. Sci. 410, 5215–5226 (2009)
3. 3.
Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. Society for Industrial and Applied Mathematics, Philadelphia (1999)
4. 4.
Conforti, M., Cornuéjols, G., Kapoor, A., Vušković, K.: Even-hole-free graphs part i: decomposition theorem. J. Graph Theory 39, 6–49 (2002)
5. 5.
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)
6. 6.
Gopalan, P., Kolaitis, P.G., Maneva, E.N., Papadimitriou, C.H.: The connectivity of Boolean satisfiability: computational and structural dichotomies. SIAM J. Comput. 38, 2330–2355 (2009)
7. 7.
Hearn, R.A., Demaine, E.D.: PSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation. Theoret. Comput. Sci. 343, 72–96 (2005)
8. 8.
Ito, T., Demaine, E.D., Harvey, N.J.A., Papadimitriou, C.H., Sideri, M., Uehara, R., Uno, Y.: On the complexity of reconfiguration problems. Theoret. Comput. Sci. 412, 1054–1065 (2011)
9. 9.
Kamiński, M., Medvedev, P., Milanič, M.: Complexity of independent set reconfigurability problems. Theoret. Comput. Sci. 439, 9–15 (2012)
10. 10.
Mouawad, A.E., Nishimura, N., Raman, V.: Vertex cover reconfiguration and beyond. In: Ahn, H.-K., Shin, C.-S. (eds.) ISAAC 2014. LNCS, vol. 8889, pp. 452–463. Springer, Heidelberg (2014)
11. 11.
Mouawad, A.E., Nishimura, N., Raman, V., Simjour, N., Suzuki, A.: On the parameterized complexity of reconfiguration problems. In: Gutin, G., Szeider, S. (eds.) IPEC 2013. LNCS, vol. 8246, pp. 281–294. Springer, Heidelberg (2013)
12. 12.
Pirzada, S., Dharwadker, A.: Applications of graph theory. J. Korean Soc. Ind. Appl. Math. 11, 19–38 (2007)Google Scholar