Branch Vertices Research Articles

Spanning Trees with Few Branch Vertices

A branch vertex in a tree is a vertex of degree at least three. We prove that, for all $s\geq 1$, every connected graph on $n$ vertices with minimum degree at least $(\frac{1}{s+3}+o(1))n$ contains...

Closed-loop randomized kinodynamic path planning for an autonomous underwater vehicle

A randomized kinodynamic path planning algorithm based on the incremental sampling-based method is proposed here as the state-of-the-art in this field applicable in an autonomous underwater vehicle. Designing a feasible path for this vehicle from an initial position and velocity to a target position and velocity in three-dimensional spaces by considering the kinematic constraints such as obstacles avoidance and dynamic constraints such as hard bounds and non-holonomic characteristic of AUV are the main motivation of this research. For this purpose, a closed-loop rapidly-exploring random tree (CL-RRT) algorithm is presented. This CL-RRT consists of three tightly coupled components: a RRT algorithm, three fuzzy proportional-derivative controllers for heading and diving control and a six degree-of-freedom nonlinear AUV model. The branches of CL-RRT are expanded in the configuration space by considering the kinodynamic constraints of AUV. The feasibility of each branch and random offspring vertex in the CL-RRT is checked against the mentioned constraints of AUV. Next, if the planned branch is feasible by the AUV, then the control signals and related vertex are recorded through the path planner to design the final path. This proposed algorithm is implemented on a single board computer (SBC) through the xPC Target and then four test-cases are designed in 3D space. The results of the processor-in-the-loop tests are compared by the conventional RRT and indicate that the proposed CL-RRT not only in a rapid manner plans an initial path, but also the planned path is feasible by the AUV.

An exact and heuristic approach for the d-minimum branch vertices problem

Given a connected graph \(G=(V,E)\), the d-Minimum Branch Vertices (d-MBV) problem consists in finding a spanning tree of G with the minimum number of vertices with degree strictly greater than d. We developed a Miller–Tucker–Zemlin based formulation with valid inequalities for this problem. The results obtained for different values of d show the effectiveness of the proposed method, which has solved several instances faster than previous methods. Also, an heuristic is proposed for this problem, that was tested on several instances of the Minimum Branch Vertices problem, which is the d-MBV problem, when \(d = 2\).

General multiplicative Zagreb indices of trees

We study trees of given order, given number of pendant vertices, segments and branching vertices. We present upper and lower bounds on the general multiplicative Zagreb indices for trees of given order and number of pendant vertices/segments/branching vertices. Bounds on the first and the second multiplicative Zagreb indices are corollaries of the general results. We also characterize trees having the largest and the smallest indices.

On the zeroth-order general Randić index, variable sum exdeg index and trees having vertices with prescribed degree

The zeroth-order general Randić index (usually denoted by [Formula: see text]) and variable sum exdeg index (denoted by [Formula: see text]) of a graph [Formula: see text] are defined as [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] is degree of the vertex [Formula: see text], [Formula: see text] is a positive real number different from 1 and [Formula: see text] is a real number other than [Formula: see text] and [Formula: see text]. A segment of a tree is a path [Formula: see text], whose terminal vertices are branching or/and pendent, and all non-terminal vertices (if exist) of [Formula: see text] have degree 2. For [Formula: see text], let [Formula: see text], [Formula: see text], [Formula: see text] be the collections of all [Formula: see text]-vertex trees having [Formula: see text] pendent vertices, [Formula: see text] segments, [Formula: see text] branching vertices, respectively. In this paper, all the trees with extremum (maximum and minimum) zeroth-order general Randić index and variable sum exdeg index are determined from the collections [Formula: see text], [Formula: see text], [Formula: see text]. The obtained extremal trees for the collection [Formula: see text] are also extremal trees for the collection of all [Formula: see text]-vertex trees having fixed number of vertices with degree 2 (because the number of segments of a tree [Formula: see text] can be determined from the number of vertices of [Formula: see text] having degree 2 and vice versa).

Extremal values of Merrifield-Simmons index for trees with two branching vertices

Extremal values of Merrifield-Simmons index for trees with two branching vertices

On the reduced second Zagreb index of trees

The current note is devoted to investigate the trees, which maximize or minimize the reduced second Zagreb index among all [Formula: see text]-vertex trees with fixed number of segments. This note also involves development of some results, which may be used to characterize the extremal trees with respect to the aforementioned index among all [Formula: see text]-vertex trees having fixed number of branching vertices.

Decomposition methods based on articulation vertices for degree-dependent spanning tree problems

Decomposition methods for optimal spanning trees on graphs are explored in this work. The attention is focused on optimization problems where the objective function depends only on the degrees of the nodes of the tree. In particular, we deal with the Minimum Leaves problem, the Minimum Branch Vertices problem and the Minimum Degree Sum problem. The decomposition is carried out by identifying the articulation vertices of the graph and then its blocks, solving certain subproblems on the blocks and then bringing together the optimal sub-solutions following adequate procedures. Computational results obtained using similar Integer Programming formulations for both the original and the decomposed problems show the advantage of the proposed methods on decomposable graphs.

On ordinary and signless Laplacian spectral radius of graphs with fixed number of branch vertices

The unique graphs with maximum spectral radius and signless Laplacian spectral radius are determined among trees, unicyclic graphs and bicyclic graphs respectively with fixed numbers of vertices and branch vertices (vertices of degree at least 3) using a unified approach.

A branch-and-cut algorithm for the minimum branch vertices spanning tree problem

Given a connected undirected graph G=(V,E), the Minimum Branch Vertices Problem (MBVP) asks for a spanning tree of G with the minimum number of vertices having degree greater than two in the tree. These are called branch vertices. This problem, with applications in the context of optical networks, is known to be NP-hard. We model the MBVP as an integer linear program, with undirected variables, we derive valid inequalities and we prove that some of these are facet defining. We then develop a hybrid formulation containing undirected and directed variables. Both models are solved with branch-and-cut. Comparative computational results show the superiority of the hybrid formulation.

First Eigenvalue of Nonsingular Mixed Unicyclic Graphs with Fixed Number of Branch Vertices

Mixed graphs are graphs whose edges may be directed or undirected. The first eigenvalue of a mixed graph is the least nonzero eigenvalue of its Laplacian matrix. We determine the unique mixed graphs with minimum first eigenvalue over all nonsingular mixed unicyclic graphs with fixed number of branch vertices, and the unique graph with minimum least signless Laplacian eigenvalue over all nonbipartite unicyclic graphs with fixed number of branch vertices.

An effective decomposition approach and heuristics to generate spanning trees with a small number of branch vertices

Given a graph $G=(V,E)$, the minimum branch vertices problem consists in finding a spanning tree $T=(V,E')$ of $G$ minimizing the number of vertices with degree greater than two. We consider a simple combinatorial lower bound for the problem, from which we propose a decomposition approach. The motivation is to break down the problem into several smaller subproblems which are more tractable computationally, and then recombine the obtained solutions to generate a solution to the original problem. We also propose effective constructive heuristics to the problem which take into consideration the problem's structure in order to obtain good feasible solutions. Computational results show that our decomposition approach is very fast and can drastically reduce the size of the subproblems to be solved. This allows a branch and cut algorithm to perform much better than when used over the full original problem. The results also show that the proposed constructive heuristics are highly efficient and generate very good quality solutions, outperforming other heuristics available in the literature in several situations.

Spanning trees whose stems have a bounded number of branch vertices

Spanning trees whose stems have a bounded number of branch vertices

On non-traceable, non-hypotraceable, arachnoid graphs

Motivated by questions concerning optical networks, in 2003 Gargano, Hammar, Hell, Stacho, and Vaccaro defined the notions of spanning spiders and arachnoid graphs. A spider is a tree with at most one branch (vertex of degree at least 3). The spider is centred at the branch vertex (if there is any, otherwise it is centred at any of the vertices). A graph is arachnoid if it has a spanning spider centred at any of its vertices. Traceable graphs are obviously arachnoid, and Gargano et al. observed that hypotraceable graphs (non-traceable graphs with the property that all vertex-deleted subgraphs are traceable) are also easily seen to be arachnoid. However, they did not find any other arachnoid graphs, and asked the question whether they exist. The main goal of this paper is to answer this question in the affirmative, moreover, we show that for any prescribed graph H, there exists a non-traceable, non-hypotraceable, arachnoid graph that contains H as an induced subgraph.

The single equality [formula omitted] does not imply the quasinormality of weighted shifts on rootless directed trees

It is proved that each bounded injective bilateral weighted shift W satisfying the equality W⁎nWn=(W⁎W)n for some integer n⩾2 is quasinormal. For any integer n⩾2, an example of a bounded non-quasinormal weighted shift A on a rootless directed tree with one branching vertex which satisfies the equality A⁎nAn=(A⁎A)n is constructed. It is also shown that such an example can be constructed in the class of composition operators in L2-spaces over σ-finite measure spaces.

A Structure Theorem for Strong Immersions

AbstractA graph H is strongly immersed in G if H is obtained from G by a sequence of vertex splittings (i.e., lifting some pairs of incident edges and removing the vertex) and edge removals. Equivalently, vertices of H are mapped to distinct vertices of G (branch vertices) and edges of H are mapped to pairwise edge‐disjoint paths in G, each of them joining the branch vertices corresponding to the ends of the edge and not containing any other branch vertices. We describe the structure of graphs avoiding a fixed graph as a strong immersion. The theorem roughly states that a graph which excludes a fixed graph as a strong immersion has a tree‐like decomposition into pieces glued together on small edge cuts such that each piece of the decomposition has a path‐like linear decomposition isolating the high degree vertices.

Tabu Search Heuristics for Two Bounded Degree Spanning Tree Problems

In this work, we address two variants of the spanning tree problem. The first problem deals with finding a spanning tree such that the number of branch vertices, namely vertices with degrees of at least three, is minimum (MBV) and the second one involves constructing a spanning tree with minimum sum of branch vertex degrees (MDS). We suggest an attribute based Tabu Search (TS) heuristic approach for both problems. According to extensive computational experiments done on standard test sets, we observe that the TS heuristics outperform the best known heuristics from the literature.

Strong Immersions and Maximum Degree

A graph $H$ is strongly immersed in $G$ if $G$ is obtained from $H$ by a sequence of vertex splittings (i.e., lifting some pairs of incident edges and removing the vertex) and edge removals. Equivalently, vertices of $H$ are mapped to distinct vertices of $G$ (branch vertices), and edges of $H$ are mapped to pairwise edge-disjoint paths in $G$, each of them joining the branch vertices corresponding to the ends of the edge and not containing any other branch vertices. We show that there exists a function $d\colon N\to N$ such that for all graphs $H$ and $G$, if $G$ contains a strong immersion of the star $K_{1,d(\Delta(H))|V(H)|}$ whose branch vertices are $\Delta(H)$-edge-connected to one another, then $H$ is strongly immersed in $G$. This has a number of structural consequences for graphs avoiding a strong immersion of $H$. In particular, a class $\mathcal{G}$ of simple 4-edge-connected graphs contains all graphs of maximum degree 4 as strong immersions if and only if $\mathcal{G}$ has either unbounded maximum degree or unbounded tree-width.

Relations, models and a memetic approach for three degree-dependent spanning tree problems

In this paper we take into account three different spanning tree problems with degree-dependent objective functions. The main application of these problems is in the field of optical network design. In particular, we propose the classical Minimum Leaves Spanning Tree problem as a relevant problem in this field and show its relations with the Minimum Branch Vertices and the Minimum Degree Sum Problems. We present a unified memetic algorithm for the three problems and show its effectiveness on a wide range of test instances.

An edge-swap heuristic for generating spanning trees with minimum number of branch vertices

This paper presents a new edge-swap heuristic for generating spanning trees with a minimum number of branch vertices, i.e. vertices of degree greater than two. This problem was introduced in Gargano et al. (Lect Notes Comput Sci 2380:355–365, 2002) and has been called the minimum branch vertices problem by Cerulli et al. (Comput Optim Appl 42:353–370, 2009). The heuristic starts with a random spanning tree and iteratively reduces the number of branch vertices by swapping tree edges with edges not currently in the tree. It can be easily implemented as a multi-start heuristic. We report on extensive computational experiments comparing single-start and multi-start variants on our heuristic with other heuristics previously proposed in the literature.