Branch Vertices Research Articles

Cutting-plane-based algorithms for two branch vertices related spanning tree problems

A branch vertex is a vertex with degree larger than or equal to three. This paper addresses two spanning tree problems in an undirected, simple graph. The first one is to find a spanning tree that minimizes the number of branch vertices (MBV), and the second one is to find a spanning tree that minimizes the degree sum of branch vertices (MDS). These two problems arise in the design of wavelength-division networks (WDN), when the cost of equipments for enabling multicast communication is to be minimized. After investigating the relations of MBV and MDS with the problem of minimizing the number of leaves in a spanning tree, new formulations based on ILP are proposed for MBV and MDS, along with two cutting plane algorithms for addressing them. A repair function is also introduced for deriving feasible solutions from the candidate trees returned at each iteration of the cutting plane algorithm, as well as a Tabu search procedure for further quality improvement. The resulting hybrid approach is shown to outperform pure ILP formulations and heuristic approaches published earlier.

Exact solution for branch vertices constrained spanning problems

Given a connected graph G, a vertex v of G is said to be a branch vertex if its degree is strictly greater than 2. The Minimum Branch Vertices Spanning Tree problem (MBVST) consists in nding a spanning tree of G with the minimum number of branch vertices. This problem has been well studied in the literature and has several applications specially for routing in optical networks. However, this kind of applications do not explicitly impose a sub-graph as solution. A more flexible structure called hierarchy is proposed. Hierarchy, which can be seen as a generalization of trees, is de ned as a homomorphism of a tree in a graph. Since minimizing the number of branch vertices in a hierarchy does not make sense, we propose to search the minimum cost spanning hierarchy such that the number of branch vertices is less than or equal to an integer R. We introduce the Branch Vertices Constrained Minimum Spanning Hierarchy (BVCMSH) problem which is NP-hard. The Integer Linear Program (ILP) formulation of this new problem is given. To evaluate the di erence of cost between trees and hierarchies, we confront the BVCMSH problem to the Branch Vertices Constrained Minimum Spanning Tree (BVCMST) problem by comparing its exact solutions. It appears from this comparison that when R 2, the hierarchies improve the average cost from more than 8% when jVGj = 20 and from more than 12% when jVGj = 30.

Lower and upper bounds for the spanning tree with minimum branch vertices

We study a variant of the spanning tree problem where we require that, for a given connected graph, the spanning tree to be found has the minimum number of branch vertices (that is vertices of the tree whose degree is greater than two). We provide four different formulations of the problem and compare different relaxations of them, namely Lagrangian relaxation, continuous relaxation, mixed integer-continuous relaxation. We approach the solution of the Lagrangian dual both by means of a standard subgradient method and an ad-hoc finite ascent algorithm based on updating one multiplier at the time. We provide numerical result comparison of all the considered relaxations on a wide set of benchmark instances. A useful follow-up of tackling the Lagrangian dual is the possibility of getting a feasible solution for the original problem with no extra costs. We evaluate the quality of the resulting upper bound by comparison either with the optimal solution, whenever available, or with the feasible solution provided by some existing heuristic algorithms.

Mysteries around the graph Laplacian eigenvalue 4

We describe our current understanding on the phase transition phenomenon associated with the graph Laplacian eigenvalue λ=4 on trees: eigenvectors for λ<4 oscillate semi-globally while those for λ>4 are concentrated around junctions. For starlike trees, we obtain a complete understanding of this phenomenon. For general graphs, we prove the number of λ>4 is bounded from above by the number of vertices with degrees higher than 2; and if a graph contains a branching path, then the eigencomponents for λ>4 decay exponentially from the branching vertex toward the leaf.

Spanning Trees with a Bounded Number of Branch Vertices in a Claw-Free Graph

Let k be a non-negative integer. A branch vertex of a tree is a vertex of degree at least three. We show two sufficient conditions for a connected claw-free graph to have a spanning tree with a bounded number of branch vertices: (i) A connected claw-free graph has a spanning tree with at most k branch vertices if its independence number is at most 2k + 2. (ii) A connected claw-free graph of order n has a spanning tree with at most one branch vertex if the degree sum of any five independent vertices is at least n ? 2. These conditions are best possible. A related conjecture also is proposed.

Subdivisions ofK5in Graphs Embedded on Surfaces With Face-Width at Least 5

We prove that if G is a 5-connected graph embedded on a surface Σ (other than the sphere) with face-width at least 5, then G contains a subdivision of K5. This is a special case of a conjecture of P. Seymour, that every 5-connected nonplanar graph contains a subdivision of K5. Moreover, we prove that if G is 6-connected and embedded with face-width at least 5, then for every v ∈ V(G), G contains a subdivision of K5 whose branch vertices are v and four neighbors of v.

Unbounded subnormal weighted shifts on directed trees. II

Criteria for subnormality of unbounded injective weighted shifts on leafless directed trees with one branching vertex are proposed. The case of classical weighted shifts is discussed. The relevance of an inductive limit approach to subnormality of weighted shifts on directed trees is revealed.

A non-hyponormal operator generating Stieltjes moment sequences

A linear operator S in a complex Hilbert space H for which the set D∞(S) of its C∞-vectors is dense in H and {‖Snf‖2}n=0∞ is a Stieltjes moment sequence for every f∈D∞(S) is said to generate Stieltjes moment sequences. It is shown that there exists a closed non-hyponormal operator S which generates Stieltjes moment sequences. What is more, D∞(S) is a core of any power Sn of S. This is established with the help of a weighted shift on a directed tree with one branching vertex. The main tool in the construction comes from the theory of indeterminate Stieltjes moment sequences. As a consequence, it is shown that there exists a non-hyponormal composition operator in an L2-space (over a σ-finite measure space) which is injective, paranormal and which generates Stieltjes moment sequences. The independence assertion of Barry Simonʼs theorem which parameterizes von Neumann extensions of a closed real symmetric operator with deficiency indices (1,1) is shown to be false.

New heuristics for two bounded-degree spanning tree problems

A vertex v of a connected graph G=(V,E) is called a branch vertex if its degree is greater than two. Pertaining to branch vertices, this paper studies two optimization problems having roots in the domain of optical networks. The first one, referred to as MBV, seeks a spanning tree T of G with the minimum number of branch vertices, whereas the second problem, referred to as MDS, seeks a spanning tree T of G with the minimum degree sum of branch vertices. Both MBV and MDS are NP-Hard. Two heuristics approaches are presented for each problem. The first approach is a problem specific heuristic, whereas the latter one is a hybrid ant-colony optimization algorithm. Computational results show the effectiveness of our proposed approaches.

Weighted shifts on directed trees

A new class of (not necessarily bounded) operators related to (mainly innite) directed trees is introduced and investigated. Operators in question are to be considered as a generalization of classical weighted shifts, on the one hand, and of weighted adjacency operators, on the other; they are called weighted shifts on directed trees. The basic properties of such op- erators, including closedness, adjoints, polar decomposition and moduli are studied. Circularity and the Fredholmness of weighted shifts on directed trees are discussed. The relationships between domains of a weighted shift on a directed tree and its adjoint are described. Hyponormality, cohyponormality, subnormality and complete hyperexpansivity of such operators are entirely characterized in terms of their weights. Related questions that arose during the study of the topic are solved as well. Particular trees with one branching vertex are intensively studied mostly in the context of subnormality and com- plete hyperexpansivity of weighted shifts on them. A strict connection of the latter with k-step backward extendibility of subnormal as well as completely hyperexpansive unilateral classical weighted shifts is established. Models of subnormal and completely hyperexpansive weighted shifts on these particular trees are constructed. Various illustrative examples of weighted shifts on di- rected trees with the prescribed properties are furnished. Many of them are simpler than those previously found on occasion of investigating analogical properties of other classes of operators.

Spanning Trees: A Survey

In this paper, we give a survey of spanning trees. We mainly deal with spanning trees having some particular properties concerning a hamiltonian properties, for example, spanning trees with bounded degree, with bounded number of leaves, or with bounded number of branch vertices. Moreover, we also study spanning trees with some other properties, motivated from optimization aspects or application for some problems.

Implications among linkage properties in graphs

AbstractGiven a graph H with vertices w1, …, wm, a graph G with at least m vertices is H‐linked if for every choice of vertices v1, …, vm in G, there is a subdivision of H in G such that vi is the branch vertex representing wi (for all i ). This concept generalizes the notions of k‐linked, k‐connected, and k‐ordered graphs. For graphs H1 and H2 with the same order that are not contained in stars, the property of being H1‐linked implies that of being H2‐linked if and only if H2⊆H1. The implication also holds when H1 is obtained from H2 by replacing an edge xy with an edge from y to a new vertex x′. Other instances of nonimplication are obtained, using a lemma that the number of vertices appearing in minimum vertex covers of a graph G is at most the vertex cover number plus the size of a maximum matching. © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 327‐337, 2009

Ore‐type degree conditions for a graph to be H‐linked

AbstractGiven a fixed multigraph H with V(H) = {h1,…, hm}, we say that a graph G is H‐linked if for every choice of m vertices v1, …, vm in G, there exists a subdivision of H in G such that for every i, vi is the branch vertex representing hi. This generalizes the notion of k‐linked graphs (as well as some other notions). For a family ${\cal H}$ of graphs, a graph G is ${\cal H}$‐linked if G is H‐linked for every $H\in {\cal H}$. In this article, we estimate the minimum integer r = r(n, k, d) such that each n‐vertex graph with $\sigma_{2}(G)\ge {r}$ is ${\cal H}$‐linked, where ${\cal H}$ is the family of simple graphs with k edges and minimum degree at least $d \ge 2$. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 14–26, 2008

Bounded-degree spanning tree problems: models and new algorithms

Given a connected graph G, a vertex v of G is said to be a branch vertex if its degree is greater than 2. We consider two problems arising in the context of optical networks: Finding a spanning tree of G with the minimum number of branch vertices and Finding a spanning tree of G such that the degree sum of the branch vertices is minimized. For these NP-hard problems, heuristics, that give good quality solutions, do not exist in the literature. In this paper we analyze the relation between the problems, provide a single commodity flow formulation to solve the problems by means of a solver and develop different heuristic strategies to compute feasible solutions that are compared with the exact ones. Our extensive computational results show the algorithms to be very fast and effective.

Minimum degree conditions for [formula omitted]-linked graphs

For a fixed multigraph H with vertices w 1 , … , w m , a graph G is H- linked if for every choice of vertices v 1 , … , v m in G, there exists a subdivision of H in G such that v i is the branch vertex representing w i (for all i). This generalizes the notions of k -linked, k -connected, and k -ordered graphs. Given a connected multigraph H with k edges and minimum degree at least two and n ⩾ 7.5 k , we determine the least integer d such that every n-vertex simple graph with minimum degree at least d is H-linked. This value D ( H , n ) appears to equal the least integer d ′ such that every n-vertex graph with minimum degree at least d ′ is b ( H ) -connected, where b ( H ) is the maximum number of edges in a bipartite subgraph of H.

Neighborhood unions and extremal spanning trees

We generalize a known sufficient condition for the traceability of a graph to a condition for the existence of a spanning tree with a bounded number of leaves. Both of the conditions involve neighborhood unions. Further, we present two results on spanning spiders (trees with a single branching vertex). We pose a number of open questions concerning extremal spanning trees.

A Hard X-Ray View of Scorpius X-1 with INTEGRAL : Nonthermal Emission?

We present here simultaneous INTEGRAL/RXTE observations of Sco X-1, and in particular a study of the hard X-ray emission of the source and its correlation with the position in the Z-track of the X-ray color-color diagram. We find that the hard X-ray (above about 30 keV) emission of Sco X-1 is dominated by a power-law component with a photon index of ~3. The flux in the power-law component slightly decreases when the source moves in the color-color diagram in the sense of increasing inferred mass accretion rate from the horizontal branch to the normal branch/flaring branch vertex. It becomes not significantly detectable in the flaring branch, where its flux has decreased by about an order of magnitude. These results present close analogies to the behavior of GX 17+2, one of so-called Sco-like Z sources. Finally, the hard power law in the spectrum of Sco X-1 does not show any evidence of a high energy cutoff up to 100 - 200 keV, strongly suggesting a non-thermal origin of this component.

A continuous flaring- to normal-branch transition in Scorpius X-1

We report the first resolved rapid transition from a Flaring Branch Oscillation to a Normal Branch Oscillation in the RXTE data of the Z source Sco X-1. The transition took place on a time scale of ~100 s and was clearly associated to the Normal Branch-Flaring Branch vertex in the color–color diagram. We discuss the results in the context of possible association of the Normal Branch Oscillation with other oscillations known both in Neutron-Star and Black-Hole systems, concentrating on the similarities with the narrow 4–6 Hz oscillations observed at high flux in Black-Hole Candidates.

On Minimum Degree Implying That a Graph is H‐Linked

Given a fixed multigraph $H$, possibly containing loops, with $V(H) = \{h_1,\ldots,h_m\}$, we say that a graph $G$ is $H$-linked if for every choice of $m$ vertices $v_1,\ldots,v_m$ in $G$, there exists a subdivision of $H$ in $G$ such that $v_i$ is the branch vertex representing $h_i$ (for all $i$). This generalizes the concept of $k$-linked graphs (as well as a number of other well-known path or cycle properties). In this paper we determine a sharp lower bound on $\delta(G)$ (which depends upon $H$) such that each graph $G$ on at least $10(|V(H)|+|E(H)|)$ vertices satisfying this bound is $H$-linked.

An extremal problem for H‐linked graphs

AbstractWe introduce the notion of H‐linked graphs, where H is a fixed multigraph with vertices w1,…,wm. A graph G is H‐linked if for every choice of vertices υ1,…, υm in G, there exists a subdivision of H in G such that υi is the branch vertex representing wi (for all i). This generalizes the notions of k‐linked, k‐connected, and k‐ordered graphs. Given k and n ≥ 5k+6, we determine the least integer d such that, for every loopless graph H with k edges and minimum degree at least two, every n‐vertex graph with minimum degree at least d is H‐linked. This value D1(k,n) appears to equal the least integer d′ such that every n‐vertex graph with minimum degree at least d′ is k‐connected. On the way to the proof, we extend a theorem by Kierstead et al. on the least integer d˝ such that every n‐vertex graph with minimum degree at least d˝ is k‐ordered. © 2005 Wiley Periodicals, Inc.