Alice A. McRae

Minus domination in graphs

Abstract We introduce one of many classes of problems which can be defined in terms of 3-valued functions on the vertices of a graph G = (V,E) of the form |:V → {−1,0,1}. Such a function is said to be a minus dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every ν ϵ V, |(N[ν])⩾ 1, where N[ν] consists of ν and every vertex adjacent to ν. The weight of a minus dominating function is |(V) = Σ|(ν), over all vertices ν ϵ V. The minus domination number of a graph G, denoted γ−(G), equals the minimum weight of a minus dominating function of G. For every graph G, γ−(G)⩽γ(G) where γ(G) denotes the domination number of G. We show that if T is a tree of order n⩾4, then γ(T)−γ−(T)⩽(n−4)/5 and this bound is sharp. We attempt to classify graphs according to their minus domination numbers. For each integer n we determine the smallest order of a connected graph with minus domination number equal to n. Properties of the minus domination number of a graph are presented and a number of open questions are raised.

The algorithmic complexity of minus domination in graphs

A three-valued function f defined on the vertices of a graph G = (V, E), f : V → {−1, 0, 1}, is a minus dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every v ϵ V, f(N[v]⩾1), where N[v] consists of v and every vertex adjacent to v. The weight of a minus dominating function is f(V) = ∑ f(v), over all vertices v ϵ V. The minus domination number of a graph G, denoted γ−(G), equals the minimum weight of a minus dominating function of G. The upper minus domination number of a graph G, denoted Γ−(G), equals the maximum weight of a minimal minus dominating function of G. In this paper we present a variety of algorithmic results. We show that the decision problem corresponding to the problem of computing γ− (respectively, Γ−) is NP-complete even when restricted to bipartite graphs or chordal graphs. We also present a linear time algorithm for finding a minimum minus dominating function in an arbitrary tree.

Majority domination in graphs

Abstract A two-valued function f defined on the vertices of a graph G = (V, E), f: V → -1, 1, is a majority dominating function if the sum of its function values over at least half the closed neighborhoods is at least one. That is, for at least half the vertices v ϵ V, f(N[v]) ⩾ 1, where N[v] consists of v and every vertex adjacent to v. The weight of a majority dominating function is f(V) = ∑f(v), over all vertices v ϵ V. The majority domination number of a graph G, denoted γmaj(G), equals the minimum weight of a majority dominating function of G. In this paper we present properties of the majority domination number and establish its value for various classes of graphs. We show that the decision problem corresponding to the problem of computing γmaj(G) is NP-complete.

[1, 2]-sets in graphs

A subset S@?V in a graph G=(V,E) is a [j,k]-set if, for every vertex v@?V@?S, j@?|N(v)@?S|@?k for non-negative integers j and k, that is, every vertex v@?V@?S is adjacent to at least j but not more than k vertices in S. In this paper, we focus on small j and k, and relate the concept of [j,k]-sets to a host of other concepts in domination theory, including perfect domination, efficient domination, nearly perfect sets, 2-packings, and k-dependent sets. We also determine bounds on the cardinality of minimum [1, 2]-sets, and investigate extremal graphs achieving these bounds. This study has implications for restrained domination as well. Using a result for [1, 3]-sets, we show that, for any grid graph G, the restrained domination number is equal to the domination number of G.

Minus domination in regular graphs

Abstract A three-valued function f defined on the vertices of a graph G = ( V , E ), f : V → {−1, 0, 1}, is a minus dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every v ∈ V , f ( N [ v ]) ⩾ 1, where N [ v ] consists of v and every vertex adjacent to v . The weight of a minus dominating function is f ( V ) = Σ f ( v ), over all vertices v ∈ V . The minus domination number of a graph G , denoted γ − ( G ), equals the minimum weight of a minus dominating function of G . In this note, we establish a sharp lower bound on γ − ( G ) for regular graphs G .

Nearly perfect sets in graphs

Abstract In a graph G =( V , E ), a set of vertices S is nearly perfect if every vertex in V - S is adjacent to at most one vertex in S . Nearly perfect sets are closely related to 2-packings of graphs, strongly stable sets, dominating sets and efficient dominating sets. We say a nearly perfect set S is 1-minimal if for every vertex u in S , the set S - u is not nearly perfect. Similarly, a nearly perfect set S is 1-maximal if for every vertex u in V - S , S ∪ u is not a nearly perfect set. Lastly, we define n p ( G ) to be the minimum cardinality of a 1-maximal nearly perfect set, and N p ( G ) to be the maximum cardinality of a 1-minimal nearly perfect set. In this paper we calculate these parameters for some classes of graphs. We show that the decision problem for n p ( G ) is NP-complete; we give a linear algorithm for determining n p ( T ) for any tree T ; and we show that N p ( G ) can be calculated for any graph G in polynomial time.

Self-stabilizing algorithms for {k}-domination

In the self-stabilizing algorithmic paradigm for distributed computing each node has only a local view of the system, yet in a finite amount of time the system converges to a global state, satisfying some desired property. A function f : V (G) → {0, 1, 2, . . . , k} is a {k}- dominating function if Σj∈N[i] f(j) ≥ k for all i ∈ V (G). In this paper we present self-stabilizing algorithms for finding a minimal {k}-dominating function in an arbitrary graph. Our first algorithm covers the general case, where k is arbitrary. This algorithm requires an exponential number of moves, however we believe that its scheme is interesting on its own, because it can insure that when a node moves, its neighbors hold correct values in their variables. For the case that k = 2 we propose a linear time self-stabilizing algorithm.

Roman { 2 } -domination

In this paper, we initiate the study of a variant of Roman dominating functions. For a graph G = ( V , E ) , a Roman { 2 } -dominating function f : V ? { 0 , 1 , 2 } has the property that for every vertex v ? V with f ( v ) = 0 , either v is adjacent to a vertex assigned 2 under f , or v is adjacent to least two vertices assigned 1 under f . The weight of a Roman { 2 } -dominating function is the sum ? v ? V f ( v ) , and the minimum weight of a Roman { 2 } -dominating function f is the Roman { 2 } -domination number. First, we present bounds relating the Roman { 2 } -domination number to some other domination parameters. In particular, we show that the Roman { 2 } -domination number is bounded above by the 2-rainbow domination number. Moreover, we prove that equality between these two parameters holds for trees and cactus graphs with no even cycles. Finally, we show that associated decision problem for Roman { 2 } -domination is NP-complete, even for bipartite graphs.

Using maximality and minimality conditions to construct inequality chains

Abstract The following inequality chain has been extensively studied in the discrete mathematical literature: if⩽y⩽i⩽β⩽Γ⩽ IR , where ir and IR denote the lower and upper irredundance numbers of a graph, γ and Γ denote the lower and upper domination numbers of a graph, i denotes the independent domination number and β denotes the vertex independence number of a graph. More than one hundred papers have been published on aspects of this chain. In this paper we define a simple mechanism which explains why this inequality chain exists and how it is possible to define many similar chains of potentially arbitrary length.

Early participation of CS students in research

Research experiences are widely available to upper-division computer science students during the academic year and during summer. Co-op and internship opportunities are available to this group as well. Due to the fact that freshman and sophomore students do not have sufficient background, they are often left behind and are not involved in research activities. This paper shares some experiences with a program that was put in place through an NSF STEP grant that provides research opportunities to freshman, sophomore, and first year transfer students. The paper presents examples of projects in which computer science scholars were involved. We have learned that lower-division computer science students are excited about the opportunity to participate in research. Early participation in research helps build a strong community among the freshman and sophomore students, keeps these students engaged, and results in a higher retention rate.