Spanning trees with small degrees and few leaves

We give an Ore-type condition sufficient for a graph G to have a spanning tree with small degrees and with few leaves.Comment: Accepted for publication in Applied Mathematics Letter

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

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

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