Broadcasting is a fundamental primitive in local area networks (LANs).Operations of many data link protocols, for example, ARP (Address Resolution Protocol) and IGMP (Internet Group Management Protocol), must rely on this LAN primitive. To develop the broadcasting service in mobile ad hoc wireless LANs (WLANs) is a challenge. This is because a mobile ad hoc WLAN is a multi-hop wireless network in which messages may travel along several links from the source to the destination via a certain path. Additionally, there is no fixed network topology because of host moving. Furthermore, the broadcast nature of a radio channel makes a packet be transmitted by a node to be able to reach all neighbors. Therefore, the total number of transmissions (forward nodes) is generally used as the cost criterion for broadcasting. The problem of finding the minimum number of forward nodes in a static radio network is NP-complete. Almost all previous works, therefore, for broadcasting in the WLAN are focusing on finding approximation approaches in a, rather than, environment. In this paper, we propose a novel distributed protocol in WLANs to significantly reduce or eliminate the communication overhead in addition to maintaining positions of neighboring nodes. The important features of our proposed protocol are the adaptability to dynamic network topology change and the localized and parameterless behavior. The reduction in communication overhead for broadcasting operation is measured experimentally. From the simulation results, our protocol not only has the similar performance as the approximation approaches in the static network, but also outperforms existing ones in the adaptability to host moving.

For a subset of vertices S in a graph G, if v ∈ S and w ∈ V - S, then the vertex w is an external private neighbor of v (with respect to S) if the only neighbor of w in S is v. A dominating set S is a private dominating set if each v ∈ S has an external private neighbor. Bollóbas and Cockayne (Graph theoretic parameters concerning domination, independence and irredundance. J. Graph Theory 3 (1979) 241-250) showed that every graph without isolated vertices has a minimum dominating set which is also a private dominating set. We define a graph G to be a private domination graph if every minimum dominating set of G is a private dominating set. We give a constructive characterization of private domination trees.

Given a digraph D = (V, A), with vertex set V and arc set A, a set S ⊆ V is a dominating set if for every vertex v in V \ S, there are a vertex u in S and an arc (u, v) from u to v. In this chapter we consider the counterparts in directed graphs of independent, dominating, independent dominating, and total dominating sets in undirected graphs.

This paper presents the G-network, a new topological design which is a suitable architecture for point-to-point communication and interconnection networks, We show that the G-network has the following desirable characteristics: Efficient routing, small number of links, and fault tolerance. The performance of the G-network is compared to that of the Barrel Shifter and Illiac Mesh networks.

A set S of vertices in a graph G is a paired dominating set if every vertex of G is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching (not necessarily as an induced subgraph). The minimum cardinality of a paired dominating set of G is the paired domination number of G. This chapter presents a survey of the major results on paired domination with an emphasis on bounds for the paired domination number.

An odd open dominating set of a graph is a subset of the graph’s vertices with the property that the open neighborhood of each vertex in the graph contains an odd number of vertices in the subset. An odd closed r-dominating set is a subset of the graph’s vertices with the property that the closed r-ball centered at each vertex in the graph contains an odd number of vertices in the subset. We first prove that the n-fold direct product of simple graphs has an odd open dominating set if and only if each factor has an odd open dominating set. Secondly, we prove that the n-fold strong product of simple graphs has an odd closed r-dominating set if and only if each factor has an odd closed r-dominating set.

Thesis--Auburn University, 2008. / Abstract. Vita. Includes bibliographical references (p. 45-48).

Understanding RNA molecules is important to genomics research. Recently researchers at the Courant Institute of Mathematical Sciences used graph theory to model RNA molecules and provided a database of trees representing possible secondary RNA structures. In this thesis we use domination parameters to predict which trees are more likely to exist in nature as RNA structures. This approach appears to have promise in graph theory applications in genomics research.

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