A polynomial algorithm to compute the minimum degree spanning trees of directed acyclic graphs with applications to the broadcast problem

In this paper, we focus on the directed minimum degree spanning tree problem and the minimum time broadcast problem. Firstly, we propose a polynomial time algorithm for the minimum degree spanning tree problem in directed acyclic graphs. The algorithm starts with an arbitrary spanning tree, and iteratively reduces the number of vertices of maximum degree. We can prove that the algorithm must reduce a vertex of the maximum degree for each phase, and finally result in an optimal tree. The algorithm terminates in O(mnlogn) time, where m and n are the numbers of edges and vertices of the graph, respectively. Moreover, we apply the new algorithm to the minimum time broadcast problem. Two consequences for directed acyclic graphs are: (1) the problem under the vertex-disjoint paths mode can be approximated within a factor of of the optimum in O(mnlogn)-time; (2) the problem under the edge-disjoint paths mode can be approximated within a factor of O(Δ^*/logΔ^*) of the optimum in O(mnlogn)-time, where Δ^* is the minimum k such that there is a spanning tree of the graph with maximum degree k.     

A fast local search method for minimum energy broadcast in wireless ad hoc networks

Local search methods are often used to reduce the power consumption of broadcast routing in wireless networks. For a classic method, sweep, the best available time complexity result is O(|V|⁴). We present an O(|V|²)-time method, which exhaustively removes unnecessary transmissions yielding a solution comparable to that of sweep.     

Broadcasting from multiple originators

We begin an investigation of broadcasting from multiple originators, a variant of broadcasting in which any k vertices may be the originators of a message in a network of n vertices. The requirement is that the message be distributed to all n vertices in minimum time. A minimumk-originator broadcast graph is a graph on n vertices with the fewest edges such that any subset of k vertices can broadcast in minimum time. B_k(n) is the number of edges in such a graph. In this paper, we present asymptotic upper and lower bounds on B_k(n). We also present an exact result for the case when . We also give an upper bound on the number of edges in a relaxed version of this problem in which one additional time unit is allowed for the broadcast.     

Approximation algorithms for minimum broadcast schedule problem in wireless sensor networks

A wireless sensor network usually consists of a large number of sensor nodes deployed in a field. One of the major communication operations is to broadcast a message from one node to the rest of the others. In this paper, we adopt the conflict-free communication model and study how to compute a transmission schedule that determines when and where a node should forward the message so that all nodes could receive the message in minimum time. We give two approximation algorithms for this NP-hard problem that have better theoretically guaranteed performances than the existing algorithms. The proposed approach could be applied to some other similar problems.     

Minimum broadcast tree decompositions

Recursive constructions for decomposing the complete directed graph D_n into minimum broadcast trees of order n are given, thereby showing the existence of such decompositions for all n. Such decompositions can be used for a routing system in a network where every participant has the ability to broadcast a message to the group; as each arc is used in only one tree, a participant’s further actions upon receipt of a message depend only on its sender, and so all routing information can be stored locally rather than in the message itself.     

The minimum semidefinite rank of a triangle-free graph

We employ a result of Moshe Rosenfeld to show that the minimum semidefinite rank of a triangle-free graph with no isolated vertex must be at least half the number of its vertices. We define a Rosenfeld graph to be such a graph that achieves equality in this bound, and we explore the structure of these special graphs. Their structure turns out to be intimately connected with the zero-nonzero patterns of the unitary matrices. Finally, we suggest an exploration of the connection between the girth of a graph and its minimum semidefinite rank, and provide a conjecture in this direction.     

A planar hypohamiltonian graph with 48 vertices

We present a planar hypohamiltonian graph on 48 vertices, and derive some consequences. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 338–342, 2007     

On minimum rank and zero forcing sets of a graph

For a graph G on n vertices and a field F, the minimum rank of G over F, written as mr^F(G), is the smallest possible rank over all n×n symmetric matrices over F whose (i,j)th entry (for ) is nonzero whenever ij is an edge in G and is zero otherwise. The maximum nullity of G over F is M^F(G)=n-mr^F(G). The minimum rank problem of a graph G is to determine mr^F(G) (or equivalently, M^F(G)). This problem has received considerable attention over the years. In [F. Barioli, W. Barrett, S. Butler, S.M. Cioab?, D. Cvetkovi?, S.M. Fallat, C. Godsil, W. Haemers, L. Hogben, R. Mikkelson, S. Narayan, O. Pryporova, I. Sciriha, W. So, D. Stevanovi?, H. van der Holst, K.V. Meulen, A.W. Wehe, AIM Minimum Rank-Special Graphs Work Group, Zero forcing sets and the minimum rank of graphs, Linear Algebra Appl. 428 (2008) 1628-1648], a new graph parameter Z(G), the zero forcing number, was introduced to bound M^F(G) from above. The authors posted an attractive question: What is the class of graphs G for which Z(G)=M^F(G) for some field F? This paper focuses on exploring the above question.     

A scale-free graph model based on bipartite graphs

Most biological networks have some common properties, on which models have to fit. The main one is that those networks are scale-free, that is that the distribution of the vertex degrees follows a power-law. Among the existing models, the ones which fit those characteristics best are based on a time evolution which makes impossible the analytic calculation of the number of motifs in the network. Focusing on applications, this calculation is very important to decompose networks in a modular manner, as proposed by Milo et al.. On the contrary, models whose construction does not depend on time, miss one or several properties of real networks or are not computationally tractable. In this paper, we propose a new random graph model that satisfies the global features of biological networks and the non-time-dependency condition. It is based on a bipartite graph structure, which has a biological interpretation in metabolic networks.     

An upper bound for the restrained domination number of a graph with minimum degree at least two in terms of order and minimum degree

Let G=(V,E) be a graph. A set S⊆V is a restrained dominating set if every vertex in V−S is adjacent to a vertex in S and to a vertex in V−S. The restrained domination number of G, denoted γ_r(G), is the smallest cardinality of a restrained dominating set of G. We will show that if G is a connected graph of order n and minimum degree δ and not isomorphic to one of nine exceptional graphs, then .     

Vertex‐disjoint cycles containing specified vertices in a bipartite graph

A minimum degree condition is given for a bipartite graph to contain a 2‐factor each component of which contains a previously specified vertex. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 145–166, 2004     

ON THE MINIMUM FEASIBLE GRAPH FOR FOUR SETS

ＯＮＴＨＥＭＩＮＩＭＵＭＦＥＡＳＩＢＬＥＧＲＡＰＨＦＯＲＦＯＵＲＳＥＴＳＸＵＹＩＮＦＥＮＧＡＮＤＦＵＸＩＡＯＢＩＮＧＡｂｓｔｒａｃｔ：ＧｉｖｅｎａｃｏｍｐｌｅｔｅｇｒａｐｈｗｉｔｈｖｅｒｔｅｘｓｅｔＸａｎｄｓｕｂｓｅｔｓＸ＿１，Ｘ＿２，．．．，Ｘ＿ｎ...     

Mixed integer formulations for the probabilistic minimum energy broadcast problem in wireless networks

In this paper we study a new variant of the minimum energy broadcast (MEB) problem, namely the probabilistic MEB (PMEB). The objective of the classic MEB problem is to assign transmission powers to the nodes of a wireless network is such a way that the total energy dissipated on the network is minimized, while a connected broadcasting structure is guaranteed by the assigned transmission powers. In the new variant of the problem treated in this paper, node failure is taken into account, aiming at providing solutions with a chosen reliability level for the broadcasting structure. Three mixed integer linear programming formulations for the new problem are presented, together with efficient formulation-dependent methods for their solution. Computational results are proposed and discussed. One method emerges as the most promising one under realistic settings. It is able to handle problems with up to fifty nodes.     

Long paths through four vertices in a 2‐connected graph

Let G be a 2‐connected graph, let u and v be distinct vertices in V(G), and let X be a set of at most four vertices lying on a common (u, v)‐path in G. If deg(x) ≥ d for all x ∈ V(G) \ {u, v}, then there is a (u, v)‐path P in G with X ⊂ V(P) and |E(P)| ≥ d. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 55–65, 2000     

The number of vertices of degree 5 in a contraction-critically 5-connected graph

An edge of a 5-connected graph is said to be 5-contractible if the contraction of the edge results in a 5-connected graph. A 5-connected graph with no 5-contractible edge is said to be contraction-critically 5-connected. Let V(G) and V₅(G) denote the vertex set of a graph G and the set of degree 5 vertices of G, respectively. We prove that each contraction-critically 5-connected graph G has at least |V(G)|/2 vertices of degree 5. We also show that there is a sequence of contraction-critically 5-connected graphs {G_i} such that lim_i→∞|V₅(G_i)|/|V(G_i)|=1/2.