Let G= (V, E) be a simple graph with vertex set V and edge set E. A mixed Roman dominating function (MRDF) of G is a function f: V∪ E→ { 0 , 1 , 2 } satisfying the condition every element x∈ V∪ E for which f(x) = 0 is adjacent or incident to at least one element y∈ V∪ E for which f(y) = 2. The weight of a MRDF f is ω(f) = ∑ x∈V∪Ef(x). The mixed Roman domination number of G is the minimum weight of a mixed Roman dominating function of G. In this paper, we initiate the study of the mixed Roman domination number and we present bounds for this parameter. We characterize the graphs attaining an upper bound and the graphs having small mixed Roman domination numbers.

A set S of vertices in a graph G=(V,E) is a total dominating set of G if every vertex of V is adjacent to some vertex in S. In this thesis, we consider total dominating sets of minimum cardinality which have the additional property that distinct vertices of V are totally dominated by distinct subsets of the total dominating set.

Master of Science / Department of Mathematics / Sarah Reznikoff / A (1, ≤ 2)-identifying code is a subset of the vertex set C of a graph such that each pair of vertices intersects C in a distinct way. This has useful applications in locating errors in multiprocessor networks and threat monitoring. At the time of writing, there is no simply-stated rule that will indicate if a graph is (1, ≤ 2)-identifiable. As such, we discuss properties that must be satisfied by a valid (1, ≤ 2)-identifying code, characteristics of a graph which preclude the existence of a (1, ≤ 2)-identifying code, and relationships between the maximum degree and order of (1, ≤ 2)-identifiable graphs. Additionally, we show that (1, ≤ 2)-identifiable graphs have no forbidden induced subgraphs and provide a list of (1, ≤ 2)-identifiable graphs with minimum (1, ≤ 2)-identifying codes indicated.

This thesis considers a novel problem of obtaining global channel state information (CSI) at every node in an ad-hoc wireless network. A class of protocols for dissemination and estimation are developed which attempt to minimize the staleness of the estimates throughout the network. This thesis also provides an optimal protocol for CSI dissemination in networks with complete graph topology and a near optimal protocol in networks having incomplete graph topology. In networks with complete graph topology, the protocol for CSI dissemination is shown to have a resemblance to finding Eulerian tours in complete graphs. For networks having incomplete graph topology, a lower bound on maximum staleness is given and a near optimal algorithm based on finding minimum connected dominating sets and proper scheduling is described in this thesis.

This thesis considers a novel problem of obtaining global channel state information (CSI) at every node in an ad-hoc wireless network. A class of protocols for dissemination and estimation are developed which attempt to minimize the staleness of the estimates throughout the network. This thesis also provides an optimal protocol for CSI dissemination in networks with complete graph topology and a near optimal protocol in networks having incomplete graph topology. In networks with complete graph topology, the protocol for CSI dissemination is shown to have a resemblance to finding Eulerian tours in complete graphs. For networks having incomplete graph topology, a lower bound on maximum staleness is given and a near optimal algorithm based on finding minimum connected dominating sets and proper scheduling is described in this thesis.

This thesis considers a novel problem of obtaining global channel state information (CSI) at every node in an ad-hoc wireless network. A class of protocols for dissemination and estimation are developed which attempt to minimize the staleness of the estimates throughout the network. This thesis also provides an optimal protocol for CSI dissemination in networks with complete graph topology and a near optimal protocol in networks having incomplete graph topology. In networks with complete graph topology, the protocol for CSI dissemination is shown to have a resemblance to finding Eulerian tours in complete graphs. For networks having incomplete graph topology, a lower bound on maximum staleness is given and a near optimal algorithm based on finding minimum connected dominating sets and proper scheduling is described in this thesis.

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.

Let G be a graph and f be a function that maps V to {0,1,2, ..., diam(G)}. Let V+ be the set of all vertices such that f(v) is positive. If for every vertex v not in V+ there exists a vertex w in V+ such that the distance between v and w is at most f(w), then f is called a dominating broadcast of G. The cost of the broadcast f is the sum of the values f(v) over all vertices v in V. The minimum cost of a dominating broadcast is called the broadcast domination number of G. A subset S of V is a multipacking if, for every v in V and for every integer k which is at least 1 and at most rad(G), the set S contains at most k vertices at distance at most k from v. The multipacking number of G is the maximum cardinality of a multipacking of G. In the first part of the thesis, we describe how linear programming can be used to give a cubic algorithm to find the broadcast domination number and multipacking number of strongly chordal graphs. Next, we restrict attention to trees, and describe linear time algorithms to compute these numbers. Finally, we introduce k-broadcast domination and k-multipacking, develop the basic theory and give a bound for the 2-broadcast domination number of a tree in terms of its order. / Graduate

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.