Edge-disjoint Spanning Trees Research Articles

Constructing edge-disjoint spanning trees in several cube-based networks with applications to edge fault-tolerant communication

Constructing edge-disjoint spanning trees in several cube-based networks with applications to edge fault-tolerant communication

On Spanning Tree Packing Number of the Complement of Generalized Petersen Graph and Cocktail Party Graph

For any graph G, the spanning tree packing number of \(\sigma\) (G), is the maximum number of edge-disjoint spanning trees contained in G. In this study, we determined the maximum number of edge-disjoint spanning trees of the generalized petersen graph and cocktail graph.

Complete family reduction and spanning connectivity in line graphs

Complete families of connected graphs, introduced by Catlin in the 1980s, have been known useful in the study of certain graphical properties that are closed under taking contractions. We show that given any complete family C of connected graphs such that C contains graphs with sufficiently many edge-disjoint spanning trees, for any real number a and b with 0<a<1, there exists a finite obstacle family F=F(a,b,C) such that for any simple graph G on n vertices satisfying the Ore-type degree conditionmin⁡{dG(u)+dG(v):u,v∈V(G) and uv∉E(G)}≥an+b, either G∈C or G can be contracted to a member in F. This result is applied to the study of spanning connectivity of line graphs. The spanning connectivity is the largest integer s such that for any k with 0≤k≤s and for any u,v∈V(G) with u≠v, G has a spanning subgraph H consisting of k internally disjoint (u,v)-paths. Z. Ryjáček and P. Vrána in [J. Graph Theory, 66 (2011) 152-173] prove that a fascinating conjecture of Thomassen on hamiltonian line graphs is equivalent to that every essentially 4-edge-connected graph has a 2-spanning-connected line graph. We prove that for any essentially 3-edge-connected graph G and any positive integer s, if G satisfies an Ore degree condition lower bounded by an arbitrary linear function in the number of vertices, then L(G) is s-spanning-connected with only finitely many contraction obstacles. When s=3, we determine a finite graph family J′(n) such that for every simple graph G on n≥156 vertices with κ(L(G))≥3 and satisfying d(u)+d(v)≥2(n−6)5 for any pair of nonadjacent vertices u and v, we have either κ⁎(L(G))≥3 or G is contractible to a member in J′(n).

Extremal Graphs for a Spectral Inequality on Edge-Disjoint Spanning Trees

Liu, Hong, Gu, and Lai proved if the second largest eigenvalue of the adjacency matrix of graph $G$ with minimum degree $\delta \ge 2m+2 \ge 4$ satisfies $\lambda_2(G) &lt; \delta - \frac{2m+1}{\delta+1}$, then $G$ contains at least $m+1$ edge-disjoint spanning trees, which verified a generalization of a conjecture by Cioabă and Wong. We show this bound is essentially the best possible by constructing $d$-regular graphs $\mathcal{G}_{m,d}$ for all $d \ge 2m+2 \ge 4$ with at most $m$ edge-disjoint spanning trees and $\lambda_2(\mathcal{G}_{m,d}) &lt; d-\frac{2m+1}{d+3}$. As a corollary, we show that a spectral inequality on graph rigidity by Cioabă, Dewar, and Gu is essentially tight.

An Algorithm to Construct Completely Independent Spanning Trees in Line Graphs

Abstract In the past few years, much importance and attention have been attached to completely independent spanning trees (CISTs). Many results, such as edge-disjoint Hamilton cycles, traceability, number of spanning trees, structural properties, topological indices, etc., have been obtained on line graphs, and researchers have applied the line graphs of some interconnection networks such as generalized hypercubes, augmented cubes, crossed cubes, etc., into data center networks, such as SWCube, AQLCube, BCDC, etc. At the meanwhile, few results of CISTs are reported on the line graphs. In this paper, we establish the relation of edge-disjoint spanning trees in an interconnection network $G$’ with its line graph $G$ by proposing a general algorithm for the first time. By this method, more CISTs can be obtained comparing with results in the literature. Then, the decrease of diameter is discussed and simulation experiments are shown on the line graphs of hypercubes.

Spectral conditions for graph rigidity in the Euclidean plane

Rigidity is the property of a structure that does not flex. It is well studied in discrete geometry and mechanics, and has applications in material science, engineering and biological sciences. A bar-and-joint framework is a pair (G,p) of graph G together with a map p of the vertices of G into a Euclidean space. We view the edges of (G,p) as bars and the vertices as universal joints. The vertices can move continuously as long as the distances between pairs of adjacent vertices are preserved. The framework is rigid if any such motion preserves the distances between all pairs of vertices. In 1970, Laman obtained a combinatorial characterization of rigid graphs in the Euclidean plane. In 1982, Lovász and Yemini discovered a new characterization and proved that every 6-connected graph is rigid. Combined with a combinatorial characterization of global rigidity given by Jackson and Jordán in 2009, it is actually proved that every 6-connected graph is globally rigid. Consequently, if the algebraic connectivity of a graph is greater than 5, then it is globally rigid. In this paper, we improve this bound and show that for a graph G with minimum degree δ≥6, if its algebraic connectivity is greater than 2+1δ−1, then G is rigid and if its algebraic connectivity is greater than 2+2δ−1, then G is globally rigid. Our results imply that every connected regular Ramanujan graph with degree at least 8 is globally rigid. We also prove a more general result giving a sufficient spectral condition for the existence of k edge-disjoint spanning rigid subgraphs. The same condition implies that a graph contains k edge-disjoint spanning 2-connected subgraphs. This result extends previous spectral conditions for packing edge-disjoint spanning trees.

Edge-disjoint spanning trees and forests of graphs

The spanning tree packing number of a graph G is the maximum number of edge-disjoint spanning trees in G, and the arboricity of G is the minimum number of edge-disjoint forests needed to partition the edge set of G. In this paper, we give bounds on the spanning tree packing number and the arboricity of graphs in terms of effective resistances. As applications, we show that equiarboreal graphs are uniformly dense, and determine the maximum number of edge-disjoint spanning c-forests of equiarboreal graphs, including edge-transitive graphs, 1-walk-regular graphs, distance-regular graphs and so on. The arboricity of a regular graph can be derived from our effective resistance bounds directly.

Game edge-connectivity of graphs

Recently, the authors introduced a game invariant of graphs, called a game connectivity. In this paper, we investigate the edge version of the invariant, called a game edge-connectivity, by introducing a new combinatorial game on a graph, called a graph edge-cutting game. The game is one of generalizations of a classical combinatorial game, named the Shannon switching game. As an analog of the study of the Shannon switching game, we have a complete characterization of graphs with game edge-connectivity infinity in terms of the number of edge-disjoint spanning trees. As a corollary of the above, any graph with edge-connectivity at least 4 has game edge-connectivity infinity. Thus, since determining the game edge-connectivity of a given cubic graph is the most interesting problem, we give a useful tool to estimate the game edge-connectivity of cubic graphs. Other than those above, we study many fundamental topics of the game edge-connectivity of graphs, and propose several open problems.

EASBVN: efficient approximation scheme for broadcasting in vehicular networks

Vehicular ad-hoc networks (VANETs) suffer from dis-connectivity due to very high mobility, node sparseness, and lossy link. Thus, the broadcasting algorithms in the vehicular multi-hop scenario have to operate in all extreme scenarios as dense, sparse, and disconnected topologies that are changeable dynamically due to different traffic conditions. The store-and-carry-forward scheme for disconnected topologies with dynamic nodes can be model as a Steiner-tree problem due to link weight change dynamically. The Steiner tree has to be recomputed to obtain the new Steiner tree, making it more difficult for broadcasting algorithms in VANETs. We show that broadcasting in VANETs (i.e., given graph $$G(V_1, E_1)$$ with the terminal set, $$T_1 \subseteq V1$$ ), modeled as to find as many (maximum) disjoint Steiner trees as possible that are disjoint on the non-terminal nodes and the edges are APX-hard even for small $$T_1$$ . If all the nodes in G are terminal nodes, i.e., $$T_1=V_1$$ , the trouble is finding a maximum cardinality set of edge-disjoint spanning trees and encountering a considerable lot of edge-disjoint spanning tree. We then propose an $$O(\log n)$$ -Approximation Algorithm for Broadcasting in VANETs via bipartite graph ( $$G=((T_1,V_1/T_1),E_1)$$ ) and using projection. We compared the performance of the proposed scheme (i.e., EASBVN) with existing state-of-the-scheme and found it to be better than the state-of-the-art schemes (i.e., DV-CAST, UVCAST, DPPEMB, MoZo, and MobiVNDN) in terms of end-to-end delay and re-healing time.

Distributed Calculation of Edge-Disjoint Spanning Trees for Robustifying Distributed Algorithms Against Man-in-the-Middle Attacks

In this paper, we provide a distributed methodology to allow a network of agents, tasked to execute a distributed algorithm, to overcome Man-in-the-Middle (MITM) attacks that aim at steering the result of the algorithm toward inconsistent values or dangerous configurations. We want the agents to be able to restore the correct result of the algorithm despite the attacks. To this end, we provide a distributed algorithm to let the set of agents, interconnected by an undirected network topology, construct several edge-disjoint spanning trees by assigning labels to their incident edges. The ultimate objective is to use these spanning trees to run multiple instances of the same distributed algorithm in parallel, in order to detect MITM attacks or other faulty or malicious link behavior (e.g., when the instances yield different results) and to restore the correct result (when the majority of instances is unaffected). The proposed algorithm is lightweight and asynchronous, and is based on iterated depth-first visits on the graph. We complement this paper with a thorough analysis of the performance of the proposed algorithms.

Network strength games: the core and the nucleolus

The maximum number of edge-disjoint spanning trees in a network has been used as a measure of the strength of a network. It gives the number of disjoint ways that the network can be fully connected. This suggests a game theoretic analysis that shows the relative importance of the different links to form a strong network. We introduce the Network strength game as a cooperative game defined on a graph $$G=(V,E)$$. The player set is the edge-set E and the value of a coalition $$S \subseteq E$$ is the maximum number of disjoint spanning trees included in S. We study the core of this game, and we give a polynomial combinatorial algorithm to compute the nucleolus when the core is non-empty.

Edge connectivity, packing spanning trees, and eigenvalues of graphs

ABSTRACT Let be the set of simple graphs (or multigraphs) G such that for each there exists at least two non-empty disjoint proper subsets satisfying and edge connectivity for . A multigraph is a graph with possible multiple edges, but no loops. Let be the maximum number of edge-disjoint spanning trees of a graph G. Motivated by a question of Seymour on the relationship between eigenvalues of a graph G and bounds of , we mainly give the relationship between the third largest (signless Laplacian) eigenvalue and the bounds of and of a simple graph or a multigraph , respectively.

Constructing spanning trees in augmented cubes

The spanning trees T1,T2,…,Tk of G are edge-disjoint spanning trees (EDSTs) if they are pairwise edge-disjoint. In addition to it if they are pairwise internally vertex disjoint then they are called completely independent spanning trees (CISTs) in G. In networks, EDSTs and CISTs are useful to increase fault-tolerance, bandwidth, and security. The possible geometric configurations in which hundreds or even thousands of processors may be linked together are examined to find the geometry that best supports computations. A much-studied topology is the hypercube and its variants. The n-dimensional augmented cube, denoted as AQn, a variation of the hypercube possesses several embeddable properties that the hypercube and its other variations do not possess. Wang et al. (2017) asked to derive an algorithm that constructs edge-disjoint spanning trees in an augmented cube. In this paper, construction of n−1 edge-disjoint spanning trees of the augmented cube AQn (n≥3) is given. The result is optimal with respect to the number of edge-disjoint spanning trees. Pai and Chang (2016) provided an approach for constructing two CISTs in several hypercube-variant networks with diameter 2n−1. They asked to design algorithms to construct more than two CISTs in high dimensional hypercube-variant networks with a smaller diameter. For AQn (n≥6), we construct four completely independent spanning trees of which two trees are with diameters 2n−5 and two trees are with diameters 2n−3.

Edge-disjoint spanning trees and the number of maximum state circles of a graph

Motivated by the connection with the genus of unoriented alternating links, Jin et al. (Acta Math Appl Sin Engl Ser, 2015) introduced the number of maximum state circles of a plane graph G, denoted by \(s_{\max }(G)\), and proved that \(s_{\max }(G)=\max \{e(H)+2c(H)-v(H)|\) H is a spanning subgraph of \(G\}\), where e(H), c(H) and v(H) denote the size, the number of connected components and the order of H, respectively. In this paper, we show that for any (not necessarily planar) graph G, \(s_{\max }(G)\) can be achieved by the spanning subgraph H of G whose each connected component is a maximal subgraph of G with two edge-disjoint spanning trees. Such a spanning subgraph is proved to be unique and we present a polynomial-time algorithm to find such a spanning subgraph for any graph G.

Almost disjoint spanning trees: Relaxing the conditions for completely independent spanning trees

The search of spanning trees with interesting disjunction properties has led to the introduction of edge-disjoint spanning trees, independent spanning trees and more recently completely independent spanning trees. We group together these notions by defining (i,j)-disjoint spanning trees, where i (j, respectively) is the number of vertices (edges, respectively) that are shared by more than one tree. We illustrate how (i,j)-disjoint spanning trees provide some nuances between the existence of disjoint connected dominating sets and completely independent spanning trees. We prove that determining if there exist two (i,j)-disjoint spanning trees in a graph G is NP-complete, for every two positive integers i and j. Moreover we prove that for square of graphs, k-connected interval graphs, complete graphs and several grids, there exist (i,j)-disjoint spanning trees for interesting values of i and j.

Tree robustness of a graph

The spanning tree packing number of a graph G, denoted by STP(G), is the maximum number of edge-disjoint spanning trees contained in G. Let denotes the set of all spanning trees of G. We say that a spanning tree T of G is of type () if the maximum number of edge-disjoint spanning trees contained in (subgraph obtained by deleting all edges of T) is , in other words . The set of all ’s types of spanning trees gives a partition of . We introduce st-robustness of a graph as a new connectivity measure that evaluates the connectivity of a graph in case of deleting some of its spanning trees. st-robustness of a graph G, denoted by STR(G), is defined as the expected value of , when T ranges over all spanning trees of G. In this paper we study this parameter and evaluate it in the case of complete graphs.

Group Connectivity, Strongly Z_m-Connectivity, and Edge Disjoint Spanning Trees

Let $\mathbb Z_m$ be the cyclic group of order $m \geq 3$. A graph $G$ is $\mathbb Z_m$-connected if $G$ has an orientation $D$ such that for any mapping $b: V(G) \mapsto \mathbb Z_m$ with $\sum_{v\in V(G)}b(v)=0$, there exists a mapping $f:E(G) \mapsto \mathbb Z_m -\{0\}$ satisfying $\sum_{e\in E_D^+(v)} f(e) - \sum_{e\in E_D^-(v)} f(e) = b(v)$ in $\mathbb Z_m$ for any $v \in V(G)$; and a graph $G$ is strongly $\mathbb Z_m$-connected if, for any mapping $\theta: V(G)\rightarrow \mathbb Z_m$ with $\sum_{v\in V(G)}\theta(v) = |E(G)|$ in $\mathbb Z_m$, there is an orientation $D$ such that $d_D^+(v)=\theta(v)$ in $\mathbb Z_m$ for each $v \in V(G)$. In this paper, we study the relation between $\mathbb Z_m$-connected graphs and strongly $\mathbb Z_m$-connected graphs and show that a graph $G$ is $\mathbb Z_m$-connected if and only if $(m-2)G$ is strongly $\mathbb Z_m$-connected, where $(m-2)G$ is the graph obtained from $G$ by replacing each edge in $G$ with $m-2$ parallel edges. We also show that if $G$ is...

Edge-Disjoint Spanning Trees, Edge Connectivity, and Eigenvalues in Graphs

Let λ2G and i¾?G denote the second largest eigenvalue and the maximum number of edge-disjoint spanning trees of a graph G, respectively. Motivated by a question of Seymour on the relationship between eigenvalues of a graph G and bounds of i¾?G, Cioaba and Wong conjectured that for any integers d,ki¾?2 and a d-regular graph G, if λ2G<d-2k-1d+1, then i¾?Gi¾?k. They proved the conjecture for k=2,3, and presented evidence for the cases when ki¾?4. Thus the conjecture remains open for ki¾?4. We propose a more general conjecture that for a graph G with minimum degree i¾?i¾?2ki¾?4, if λ2G<i¾?-2k-1i¾?+1, then i¾?Gi¾?k. In this article, we prove that for a graph G with minimum degree i¾?, each of the following holds.iFor k∈{2,3}, if i¾?i¾?2k and λ2G<i¾?-2k-1i¾?+1, then i¾?Gi¾?k.iiFor ki¾?4, if i¾?i¾?2k and λ2G<i¾?-3k-1i¾?+1, then i¾?Gi¾?k.Our results sharpen theorems of Cioaba and Wong and give a partial solution to Cioaba and Wong's conjecture and Seymour's problem. We also prove that for a graph G with minimum degree i¾?i¾?ki¾?2, if λ2G<i¾?-2k-1i¾?+1, then the edge connectivity is at least k, which generalizes a former result of Cioaba. As corollaries, we investigate the Laplacian and signless Laplacian eigenvalue conditions on i¾?G and edge connectivity.

Note on the spanning-tree packing number of lexicographic product graphs

The spanning-tree packing number of a graph G is the maximum number of edge-disjoint spanning trees contained in G. In this paper, we obtain a sharp lower bound for the spanning-tree packing number of lexicographic product graphs.

Degree sequence realizations with given packing and covering of spanning trees

Designing networks in which every processor has a given number of connections often leads to graphic degree sequence realization models. A nonincreasing sequence d=(d1,d2,…,dn) is graphic if there is a simple graph G with degree sequence d. The spanning tree packing number of graph G, denoted by τ(G), is the maximum number of edge-disjoint spanning trees in G. The arboricity of graph G, denoted by a(G), is the minimum number of spanning trees whose union covers E(G). In this paper, it is proved that, given a graphic sequence d=d1≥d2≥⋯≥dn and integers k2≥k1>0, there exists a simple graph G with degree sequence d satisfying k1≤τ(G)≤a(G)≤k2 if and only if dn≥k1 and 2k1(n−1)≤∑i=1ndi≤2k2(n−|I|−1)+2∑i∈Idi, where I={i:di<k2}. As corollaries, for any integer k>0, we obtain a characterization of graphic sequences with at least one realization G satisfying a(G)≤k, and a characterization of graphic sequences with at least one realization G satisfying τ(G)=a(G)=k.