Abstract
The Yutsis property of a simple, connected, and undirected graph is the property of partitioning its vertex set into two induced trees. Although the first impression is that such a property is quite particular, it is more general than Hamiltonicity on planar graphs since a planar graph satisfies the Yutsis property if and only if its dual is Hamiltonian. Despite the fact that recognizing Yutsis graphs is NP-complete even on planar graphs, it is still possible to consider two even more challenging problems: (i) the recognition of k-Yutsis graphs, which are graphs that have their vertex sets partitioned into k induced trees, for a fixed \(k\ge 2\); (ii) to find the minimum number of vertex-disjoint induced trees that cover all vertices of a graph G, which is called the tree cover number of G. The studies on Yutsis graphs emerge from the quantum theory of angular momenta since it appears as a graphical representation of general recoupling coefficients, and the studies on the tree cover number are motivated by its equality with the maximum positive semidefinite nullity on multigraphs with treewidth at most two.
Given the interest in the tree cover number on graphs with bounded treewidth, we investigate the parameterized complexity of the tree cover number computation. We prove that the tree cover number can be determined in \(2^{\mathcal {O}(tw\log tw)}\cdot n^{\mathcal {O}(1)}\), where tw is the treewidth of the input graph, but it cannot be solved in \(2^{o(tw\log tw)}\cdot n^{\mathcal {O}(1)}\) time unless ETH fails. Similarly, we conclude that recognizing k-Yutsis graphs can be done in \(k^{\mathcal {O}({tw})}\cdot n^{\mathcal {O}(1)}\) time, but it cannot be done in \((k-\epsilon )^{tw}\cdot n^{\mathcal {O}(1)}\) time assuming SETH. We also show that the problem of determining the tree cover number of a graph G is polynomial-time solvable on graphs with bounded clique-width, but it is W[1]-hard considering clique-width parameterization while recognizing k-Yutsis graphs can be done in FPT time. Furthermore, contrasting with the polynomial-time recognition of k-Yutsis chordal graphs, for split graphs G having a partition \(V(G)=(S,K)\) where S is an independent set and K is a clique, we prove that determining the tree cover number of G is NP-hard even when S has only vertices of degree 2 or 4, but it is polynomial-time solvable when each vertex of S has either odd degree or degree two in G. We also provide some characterizations for chordal k-Yutsis subclasses.
Supported by CNPq and FAPERJ.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aldred, R.E., Van Dyck, D., Brinkmann, G., Fack, V., McKay, B.D.: Graph structural properties of non-Yutsis graphs allowing fast recognition. Discret. Appl. Math., 157(2), 377–386 (2009)
Barioli, F., Fallat, S., Mitchell, L., Narayan, S.: Minimum semidefinite rank of outerplanar graphs and the tree cover number. Electron. J. Linear Algebra 22, 10–21 (2011)
Lawrence Christian Biedenharn and James D Louck. The Racah-Wigner algebra in quantum theory. Addison-Wesley (1981)
Bozeman, C., Catral, M., Cook, B., González, O., Reinhart, C.: On the tree cover number of a graph. Involve, a J. Math. 10(5), 767–779 (2017)
Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst. 33(2), 125–150 (2000)
Courcelle, B.: The monadic second-order logic of graphs. I. recognizable sets of finite graphs. Inform. Comput. 85(1), 12–75 (1990)
Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discret. Appl. Math. 101(1–3), 77–114 (2000)
Cygan, M., et al.: Parameterized Algorithms. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21275-3
Duarte, G., Oliveira, M.D.O., Souza, U.S.: Co-degeneracy and co-treewidth: using the complement to solve dense instances. In: 46th International Symposium on Mathematical Foundations of Computer Science, MFCS, volume 202 of LIPIcs, pp. 42:1–42:17 (2021)
Durate, G.L., et al.: Computing the largest bond and the maximum connected cut of a graph. Algorithmica 83(5), 1421–1458 (2021). https://doi.org/10.1007/s00453-020-00789-1
Duarte, G.L., Lokshtanov, D., Pedrosa, L.L., Schouery, R., Souza, U.S.: Computing the largest bond of a graph. In 14th International Symposium on Parameterized and Exact Computation, IPEC, volume 148 of LIPIcs, pp. 12:1–12:15 (2019)
Duarte, G.L., Souza, U.S.: On the minimum cycle cover problem on graphs with bounded co-degeneracy. In: 48th International Workshop on Graph-Theoretic Concepts in Computer Science, WG, vol. 13453 of LNCS, pp. 187–200 (2022). https://doi.org/10.1007/978-3-031-15914-5_14
Dyck, D.V., Fack, V.: On the reduction of Yutsis graphs. Discret. Math. 307(11), 1506–1515 (2007)
Ekstrand, J., Erickson, C., Hay, D., Hogben, L., Roat, J.: Note on positive semidefinite maximum nullity and positive semidefinite zero forcing number of partial 2-trees. Electron. J. Linear Algebra 23, 79–87 (2012)
Eto, H., Hanaka, T., Kobayashi, Y., Kobayashi, Y.: Parameterized algorithms for maximum cut with connectivity constraints. In: 14th International Symposium on Parameterized and Exact Computation, IPEC, vol. 148 of LIPIcs, pp. 13:1–13:15 (2019)
Fomin, F., Golovach, P., Lokshtanov, D., Saurabh, S.: Intractability of clique-width parameterizations. SIAM J. Comput. 39(5), 1941–1956 (2010)
Garey, M.R., Johnson, D.S., Tarjan, R.E.: The planar hamiltonian circuit problem is NP-complete. SIAM J. Comput. 5(4), 704–714 (1976)
Godsil, C., Royle, G.F.: Algebraic Graph Theory. Graduate texts in mathematics. Springer (2001)
Golumbic, M.C., Rotics, U.: On the clique—width of perfect graph classes. In: Widmayer, P., Neyer, G., Eidenbenz, S. (eds.) WG 1999. LNCS, vol. 1665, pp. 135–147. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-46784-X_14
Khani, M.R., Salavatipour, M.R.: Improved approximation algorithms for the min-max tree cover and bounded tree cover problems. Algorithmica 69(2), 443–460 (2013). https://doi.org/10.1007/s00453-012-9740-5
Lokshtanov, D., Marx, D., Saurabh, S.: Known algorithms on graphs of bounded treewidth are probably optimal. ACM Trans. Algorithms, 14(2) (2018)
Lokshtanov, D., Marx, D., Saurabh, S.: Slightly superexponential parameterized problems. SIAM J. Comput. 47(3), 675–702 (2018)
Oum, S.I., Seymour, P.: Approximating clique-width and branch-width. J. Comb. Theory, B, 96(4), 514–528 (2006)
Oxley, J.G.: Matroid theory, vol. 3. Oxford University Press, USA (2006)
Picouleau, C.: Complexity of the hamiltonian cycle in regular graph problem. Theoret. Comput. Sci. 131(2), 463–473 (1994)
Welsh, D.J.A.: Euler and bipartite matroids. J. Combinatorial Theory 6(4), 375–377 (1969)
Yutsis, A.P., Vanagas, V.V., Levinson, I.B.: Mathematical apparatus of the theory of angular momentum. Israel program for scientific translations (1962)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Cunha, L., Duarte, G., Protti, F., Nogueira, L., Souza, U. (2024). Induced Tree Covering and the Generalized Yutsis Property. In: Soto, J.A., Wiese, A. (eds) LATIN 2024: Theoretical Informatics. LATIN 2024. Lecture Notes in Computer Science, vol 14579. Springer, Cham. https://doi.org/10.1007/978-3-031-55601-2_10
Download citation
DOI: https://doi.org/10.1007/978-3-031-55601-2_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-55600-5
Online ISBN: 978-3-031-55601-2
eBook Packages: Computer ScienceComputer Science (R0)