Induced Tree Covering and the Generalized Yutsis Property

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.