Jean E. Dunbar

Broadcasts in graphs

We say that a function f : V → {0, 1, ..., diam(G)} is a broadcast if for every vertex v ∈ V, f(v) ≤ e(v), where diam(G) denotes the diameter of G and e(v) denotes the eccentricity of v. The cost of a broadcast is the value f(V) = Σv∈V f(v). In this paper we introduce and study the minimum and maximum costs of several types of broadcasts in graphs, including dominating, independent and efficient broadcasts.

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.

The directed path partition conjecture

The Directed Path Partition Conjecture is the following: If D is a digraph that contains no path with more than λ vertices then, for every pair (a, b) of positive integers with λ = a + b, there exists a vertex partition (A, B) of D such that no path in D〈A〉 has more than a vertices and no path in D〈B〉 has more than b vertices.We develop methods for finding the desired partitions for various classes of digraphs.

The Path Partition Conjecture is true for claw-free graphs

The detour order of a graph G, denoted by @t(G), is the order of a longest path in G. The Path Partition Conjecture (PPC) is the following: If G is any graph and (a,b) any pair of positive integers such that @t(G)=a+b, then the vertex set of G has a partition (A,B) such that @t()=)=

A path(ological) partition problem

Let τ(G) denote the number of vertices in a longest path of the graph G and let k1 and k2 be positive integers such that τ(G) = k1+k2. The question at hand is whether the vertex set V (G) can be partitioned into two subsets V1 and V2 such that τ(G[V1]) ≤ k1 and τ(G[V2]) ≤ k2. We show that several classes of graphs have this partition property.

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 .

Gallai-type theorems and domination parameters

Abstract Let γ ( G ) denote the minimum cardinality of a dominating set of a graph G = ( V , E ). A longstanding upper bound for γ ( G ) is attributed to Berge: For any graph G with n vertices and maximum degree Δ ( G ), γ ( G ) ⩽ n − Δ ( G ). We characterise connected bipartite graphs which achieve this upper bound. For an arbitrary graph G we furnish two conditions which are necessary if γ ( G ) + Δ ( G ) = n and are sufficient to achieve n − 1 ⩽ γ ( G ) + Δ ( G ) ⩽ n . We further investigate graphs which satisfy similar equations for the independent domination number, i ( G ), and the irredundance number ir( G ). After showing that i ( G ) ⩽ n − Δ ( G ) for all graphs, we characterise bipartite graphs which achieve equality. Lastly, we show for the upper irredundance number, IR ( G ): For a graph G with n vertices and minimum degree δ ( G ), IR ( G ) ⩽ n - δ ( G ). Characterisations are given for classes of graphs which achieve this upper bound for the upper irredundance, upper domination and independence numbers of a graph.

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.

Cycles in k-traceable oriented graphs

A digraph of order at least k is termed k-traceable if each of its subdigraphs of order k is traceable. It turns out that several properties of tournaments-i.e., the 2-traceable oriented graphs-extend to k-traceable oriented graphs for small values of k. For instance, the authors together with O. Oellermann have recently shown that for k=2,3,4,5,6, all k-traceable oriented graphs are traceable. Moon [J.W. Moon, On subtournaments of a tournament, Canad. Math. Bull. 9(3) (1966) 297-301] observed that every nontrivial strong tournament T is vertex-pancyclic-i.e., through each vertex there is a cycle of every length from 3 up to the order of T. The present paper reports results pertaining to various cycle properties of strong k-traceable oriented graphs and explores the extent to which pancyclicity is retained by strong k-traceable oriented graphs. For each k>=2 there are infinitely many k-traceable oriented graphs-e.g. tournaments. However, we establish an upper bound (linear in k) on the order of k-traceable oriented graphs having a strong component with girth greater than 3. As an application of our findings, we show that the Path Partition Conjecture holds for 1-deficient oriented graphs having a strong component with girth at least 6. (A digraph is 1-deficient if its order is exactly one more than the order of its longest paths.)