ISAAC 2011: Algorithms and Computation pp 90-99

# Hamiltonian Paths in the Square of a Tree

• Wojciech Rytter
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7074)

## Abstract

We introduce a new family of graphs for which the Hamiltonian path problem is non-trivial and yet has a linear time solution. The square of a graph G = (V,E), denoted as G 2, is a graph with the set of vertices V, in which two vertices are connected by an edge if there exists a path of length at most 2 connecting them in G. Harary & Schwenk (1971) proved that the square of a tree T contains a Hamiltonian cycle if and only if T is a caterpillar, i.e., it is a single path with several leafs connected to it. Our first main result is a simple graph-theoretic characterization of trees T for which T 2 contains a Hamiltonian path: T 2 has a Hamiltonian path if and only if T is a horsetail (the name is due to the characteristic shape of these trees, see Figure 1). Our next results are two efficient algorithms: linear time testing if T 2 contains a Hamiltonian path and finding such a path (if there is any), and linear time preprocessing after which we can check for any pair (u,v) of nodes of T in constant time if there is a Hamiltonian path from u to v in T 2.

## Keywords

tree square of a graph Hamiltonian path

## References

1. 1.
Abderrezzak, M.E.K., Flandrin, E., Ryjáček, Z.: Induced S(K$$_{\mbox{1, 3}}$$) and Hamiltonian cycles in the square of a graph. Discrete Mathematics 207(1-3), 263–269 (1999)
2. 2.
Diestel, R.: Graph Theory, 4th edn. Springer, Heidelberg (2010)
3. 3.
Fleischner, H.: The square of every two-connected graph is Hamiltonian. J. Combin. Theory (Series B) 16, 29–34 (1974)
4. 4.
Georgakopoulos, A.: A short proof of Fleischner’s theorem. Discrete Mathematics 309(23-24), 6632–6634 (2009)
5. 5.
Georgakopoulos, A.: Infinite Hamilton cycles in squares of locally finite graphs (2006) (preprint)Google Scholar
6. 6.
Harary, F., Schwenk, A.: Trees with Hamiltonian square. Mathematika 18, 138–140 (1971)
7. 7.
Hendry, G., Vogler, W.: The square of a S(K$$_{\mbox{1, 3}}$$)-free graph is vertex pancyclic. Journal of Graph Theory 9, 535–537 (1985)
8. 8.
Karaganis, J.J.: On the cube of a graph. Canad. Math. Bull. 11, 295–296 (1968)
9. 9.
Lin, Y.-L., Skiena, S.: Algorithms for square roots of graphs. SIAM J. Discrete Math. 8(1), 99–118 (1995)
10. 10.
Sekanina, M.: On an ordering of the set of vertices of a connected graph. Technical Report 412, Publ. Fac. Sci. Univ. Brno. (1960)Google Scholar
11. 11.
Thomassen, C.: Hamiltonian paths in squares of infinite locally finite blocks. Annals of Discr. Math. 3, 269–277 (1978)
12. 12.
Řiha, S.: A new proof of the theorem by Fleischner. J. Comb. Theory Ser. B 52, 117–123 (1991)