Spanning directed trees with many leaves

The {\sc Directed Maximum Leaf Out-Branching} problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number of leaves in out-branchings. We show that - every strongly connected $n$-vertex digraph $D$ with minimum in-degree at least 3 has an out-branching with at least $(n/4)^{1/3}-1$ leaves; - if a strongly connected digraph $D$ does not contain an out-branching with $k$ leaves, then the pathwidth of its underlying graph UG($D$) is $O(k\log k)$. Moreover, if the digraph is acyclic, the pathwidth is at most $4k$. The last result implies that it can be decided in time $2^{O(k\log^2 k)}\cdot n^{O(1)}$ whether a strongly connected digraph on $n$ vertices has an out-branching with at least $k$ leaves. On acyclic digraphs the running time of our algorithm is $2^{O(k\log k)}\cdot n^{O(1)}$

Spanning Trees with Many Leaves in Graphs without Diamonds and Blossoms

It is known that graphs on n vertices with minimum degree at least 3 have spanning trees with at least n/4+2 leaves and that this can be improved to (n+4)/3 for cubic graphs without the diamond K_4-e as a subgraph. We generalize the second result by proving that every graph with minimum degree at least 3, without diamonds and certain subgraphs called blossoms, has a spanning tree with at least (n+4)/3 leaves, and generalize this further by allowing vertices of lower degree. We show that it is necessary to exclude blossoms in order to obtain a bound of the form n/3+c. We use the new bound to obtain a simple FPT algorithm, which decides in O(m)+O^*(6.75^k) time whether a graph of size m has a spanning tree with at least k leaves. This improves the best known time complexity for MAX LEAF SPANNING TREE.Comment: 25 pages, 27 Figure

Reconfiguration of Spanning Trees with Many or Few Leaves

Let G be a graph and T?,T? be two spanning trees of G. We say that T? can be transformed into T? via an edge flip if there exist two edges e ? T? and f in T? such that T? = (T??e) ? f. Since spanning trees form a matroid, one can indeed transform a spanning tree into any other via a sequence of edge flips, as observed in [Takehiro Ito et al., 2011]. We investigate the problem of determining, given two spanning trees T?,T? with an additional property ?, if there exists an edge flip transformation from T? to T? keeping property ? all along. First we show that determining if there exists a transformation from T? to T? such that all the trees of the sequence have at most k (for any fixed k ? 3) leaves is PSPACE-complete. We then prove that determining if there exists a transformation from T? to T? such that all the trees of the sequence have at least k leaves (where k is part of the input) is PSPACE-complete even restricted to split, bipartite or planar graphs. We complete this result by showing that the problem becomes polynomial for cographs, interval graphs and when k = n-2

Spanning trees with many leaves: new extremal results and an improved FPT algorithm

We present two lower bounds for the maximum number of leaves in a spanning tree of a graph. For connected graphs without triangles, with minimum degree at least three, we show that a spanning tree with at least (n+4)/3 leaves exists, where n is the number of vertices of the graph. For connected graphs with minimum degree at least three, that contain D diamonds induced by vertices of degree three (a diamond is a K4 minus one edge), we show that a spanning tree exists with at least (2n-D+12)/7 leaves. The proofs use the fact that spanning trees with many leaves correspond to small connected dominating sets. Both of these bounds are best possible for their respective graph classes. For both bounds simple polynomial time algorithms are given that find spanning trees satisfying the bounds. \ud \ud The second bound is used to find a new fastest FPT algorithm for the Max-Leaf Spanning Tree problem. This problem asks whether a graph G on n vertices has a spanning tree with at least k leaves. The time complexity of our algorithm is f(k)g(n), where g(n) is a polynomial, and f(k) Î O(8.12k).\ud \ud \u

Parameterized Algorithms for Directed Maximum Leaf Problems

We prove that finding a rooted subtree with at least $k$ leaves in a digraph is a fixed parameter tractable problem. A similar result holds for finding rooted spanning trees with many leaves in digraphs from a wide family $\cal L$ that includes all strong and acyclic digraphs. This settles completely an open question of Fellows and solves another one for digraphs in $\cal L$. Our algorithms are based on the following combinatorial result which can be viewed as a generalization of many results for a `spanning tree with many leaves' in the undirected case, and which is interesting on its own: If a digraph $D\in \cal L$ of order $n$ with minimum in-degree at least 3 contains a rooted spanning tree, then $D$ contains one with at least $(n/2)^{1/5}-1$ leaves

Kernelization for Finding Lineal Topologies (Depth-First Spanning Trees) with Many or Few Leaves

For a given graph $G$, a depth-first search (DFS) tree $T$ of $G$ is an $r$-rooted spanning tree such that every edge of $G$ is either an edge of $T$ or is between a \textit{descendant} and an \textit{ancestor} in $T$. A graph $G$ together with a DFS tree is called a \textit{lineal topology} $\mathcal{T} = (G, r, T)$. Sam et al. (2023) initiated study of the parameterized complexity of the \textsc{Min-LLT} and \textsc{Max-LLT} problems which ask, given a graph $G$ and an integer $k\geq 0$, whether $G$ has a DFS tree with at most $k$ and at least $k$ leaves, respectively. Particularly, they showed that for the dual parameterization, where the tasks are to find DFS trees with at least $n-k$ and at most $n-k$ leaves, respectively, these problems are fixed-parameter tractable when parameterized by $k$. However, the proofs were based on Courcelle's theorem, thereby making the running times a tower of exponentials. We prove that both problems admit polynomial kernels with \Oh(k^3) vertices. In particular, this implies FPT algorithms running in k^{\Oh(k)}\cdot n^{O(1)} time. We achieve these results by making use of a \Oh(k)-sized vertex cover structure associated with each problem. This also allows us to demonstrate polynomial kernels for \textsc{Min-LLT} and \textsc{Max-LLT} for the structural parameterization by the vertex cover number.Comment: 16 pages, accepted for presentation at FCT 202