11 research outputs found
On the minimum leaf number of cubic graphs
The \emph{minimum leaf number} of a connected graph is
defined as the minimum number of leaves of the spanning trees of . We
present new results concerning the minimum leaf number of cubic graphs: we show
that if is a connected cubic graph of order , then , improving on the best known result in [Inf. Process.
Lett. 105 (2008) 164-169] and proving the conjecture in [Electron. J. Graph
Theory and Applications 5 (2017) 207-211]. We further prove that if is also
2-connected, then , improving on the best
known bound in [Math. Program., Ser. A 144 (2014) 227-245]. We also present new
conjectures concerning the minimum leaf number of several types of cubic graphs
and examples showing that the bounds of the conjectures are best possible.Comment: 17 page
Spanning trees in graphs of minimum degree 4 or 5
AbstractFor a connected simple graph G let L(G) denote the maximum number of leaves in any spanning tree of G. Linial conjectured that if G has N vertices and minimum degree k, then L(G)⊞((k â 2)⧸(k + 1))N + ck where ck depends on k. We prove that if k = 4, L(G) 25N + 85; if k = 5, L(G) ⊞ 12N + 2. We give examples showing that these bounds are sharp
Kernel(s) for Problems with No Kernel: On Out-Trees with Many Leaves
The {\sc -Leaf Out-Branching} problem is to find an out-branching, that is a rooted oriented spanning tree, with at least leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms. Here, we take a kernelization based approach to the {\sc -Leaf-Out-Branching} problem. We give the first polynomial kernel for {\sc Rooted -Leaf-Out-Branching}, a variant of {\sc -Leaf-Out-Branching} where the root of the tree searched for is also a part of the input. Our kernel has cubic size and is obtained using extremal combinatorics. For the {\sc -Leaf-Out-Branching} problem, we show that no polynomial kernel is possible unless the polynomial hierarchy collapses to third level by applying a recent breakthrough result by Bodlaender et al. (ICALP 2008) in a non-trivial fashion. However, our positive results for {\sc Rooted -Leaf-Out-Branching} immediately imply that the seemingly intractable {\sc -Leaf-Out-Branching} problem admits a data reduction to independent kernels. These two results, tractability and intractability side by side, are the first ones separating {\it many-to-one kernelization} from {\it Turing kernelization}. This answers affirmatively an open problem regarding ``cheat kernelization'' raised by Mike Fellows and Jiong Guo independently.publishedVersio
2-Approximation algorithm for finding a spanning tree with maximum number of leaves
We study the problem of finding a spanning tree with maximum number of leaves. We present a simple 2-approximation algorithm for the problem, improving on the previous best performance ratio of 3 achieved by algorithms of Ravi and Lu. Our algorithm can be implemented to run in linear time using simple data structures. We also study the variant of the problem in which a given subset of vertices are required to be leaves in the tree. We provide a 5/2-approximation algorithm for this version of the proble
The Full Degree Spanning Tree Problem
Given a graph G, we study the problem of finding a spanning tree T that maximizes the number of vertices of full degree; that is, the number of vertices whose degree in T equals its degree in G. We prove a few general bounds and then analyze this parameter on various classes of graphs including grid graphs, hypercubes, and random regular graphs. We also explore a related problem that focuses on maximizing the number of leaves in a spanning tree of a graph
Bounds on distances for spanning trees of graphs.
Master of Science in Applied Mathematics, University of KwaZulu-Natal, Westville, 2018.In graph theory, there are several techniques known in literature for constructing
spanning trees. Some of these techniques yield spanning trees with many leaves. We
will use these constructed spanning trees to bound several distance parameters.
The cardinality of the vertex set of graph G is called the order, n(G) or n. The
cardinality of the edge set of graph G is called the size, m(G) or m. The minimum
degree of G, (G) or , is the minimum degree among the degrees of the vertices of
G: A spanning tree T of a graph G is a subgraph that is a tree which includes all
the vertices of G. The distance d(u; v) between two vertices u and v is the length
of a shortest u-v path of G. The eccentricity, ec (v), of a vertex v 2 V (G) is the
maximum distance from it to any other vertex in G. The diameter, diam(G) or d,
is the maximum eccentricity amongst all vertices of G. The radius, rad(G), is the
minimum eccentricity among all vertices of G. The average distance of a graph G,
(G), is the expected distance between a randomly chosen pair of distinct vertices.
We investigate how each constructed spanning tree can be used to bound diam-
eter, radius or average distance in terms of order, size and minimum degree. The
techniques to be considered include the radius-preserving spanning trees by Erd}os et
al, the Ding et al technique, and the Dankelmann and Entringer technique. Finally,
we use the Kleitman and West dead leaves technique to construct spanning trees
with many leaves for various values of the minimum degree k (for k = 3; 4 and
k > 4) and order n. We then use the leaf number to bound diameter.Date (2018) taken as per title page
Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)
The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..