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

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

Max-Leaves Spanning Tree is APX-hard for Cubic Graphs

We consider the problem of finding a spanning tree with maximum number of leaves (MaxLeaf). A 2-approximation algorithm is known for this problem, and a 3/2-approximation algorithm when restricted to graphs where every vertex has degree 3 (cubic graphs). MaxLeaf is known to be APX-hard in general, and NP-hard for cubic graphs. We show that the problem is also APX-hard for cubic graphs. The APX-hardness of the related problem Minimum Connected Dominating Set for cubic graphs follows

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

Spanning trees without adjacent vertices of degree 2

Albertson, Berman, Hutchinson, and Thomassen showed in 1990 that there exist highly connected graphs in which every spanning tree contains vertices of degree 2. Using a result of Alon and Wormald, we show that there exists a natural number $d$ such that every graph of minimum degree at least $d$ contains a spanning tree without adjacent vertices of degree 2. Moreover, we prove that every graph with minimum degree at least 3 has a spanning tree without three consecutive vertices of degree 2

Streaming Complexity of Spanning Tree Computation

The semi-streaming model is a variant of the streaming model frequently used for the computation of graph problems. It allows the edges of an n-node input graph to be read sequentially in p passes using Õ(n) space. If the list of edges includes deletions, then the model is called the turnstile model; otherwise it is called the insertion-only model. In both models, some graph problems, such as spanning trees, k-connectivity, densest subgraph, degeneracy, cut-sparsifier, and (Δ+1)-coloring, can be exactly solved or (1+ε)-approximated in a single pass; while other graph problems, such as triangle detection and unweighted all-pairs shortest paths, are known to require Ω̃(n) passes to compute. For many fundamental graph problems, the tractability in these models is open. In this paper, we study the tractability of computing some standard spanning trees, including BFS, DFS, and maximum-leaf spanning trees. Our results, in both the insertion-only and the turnstile models, are as follows. Maximum-Leaf Spanning Trees: This problem is known to be APX-complete with inapproximability constant ρ ∈ [245/244, 2). By constructing an ε-MLST sparsifier, we show that for every constant ε > 0, MLST can be approximated in a single pass to within a factor of 1+ε w.h.p. (albeit in super-polynomial time for ε ≤ ρ-1 assuming P ≠ NP) and can be approximated in polynomial time in a single pass to within a factor of ρ_n+ε w.h.p., where ρ_n is the supremum constant that MLST cannot be approximated to within using polynomial time and Õ(n) space. In the insertion-only model, these algorithms can be deterministic. BFS Trees: It is known that BFS trees require ω(1) passes to compute, but the naïve approach needs O(n) passes. We devise a new randomized algorithm that reduces the pass complexity to O(√n), and it offers a smooth tradeoff between pass complexity and space usage. This gives a polynomial separation between single-source and all-pairs shortest paths for unweighted graphs. DFS Trees: It is unknown whether DFS trees require more than one pass. The current best algorithm by Khan and Mehta [STACS 2019] takes Õ(h) passes, where h is the height of computed DFS trees. Note that h can be as large as Ω(m/n) for n-node m-edge graphs. Our contribution is twofold. First, we provide a simple alternative proof of this result, via a new connection to sparse certificates for k-node-connectivity. Second, we present a randomized algorithm that reduces the pass complexity to O(√n), and it also offers a smooth tradeoff between pass complexity and space usage.ISSN:1868-896

Extremal Infinite Graph Theory

We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 figure