Christina M. Mynhardt

Construction of trees and graphs with equal domination parameters

We provide a simple constructive characterization for trees with equal domination and independent domination numbers, and for trees with equal domination and total domination numbers. We also consider a general framework for constructive characterizations for other equality problems.

Vertices contained in all or in no minimum total dominating set of a tree

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. We characterize the set of vertices of a tree that are contained in all, or in no, minimum total dominating sets of the tree.

Total dominating functions in trees: minimality and convexity

A total dominating function (TDF) of a graph G = (V, E) is a function f: V [0, 1] such that for each v ϵ V, ΣuϵN(v) f(u) ≥ 1 (where N(v) denotes the set of neighbors of vertex v). Convex combinations of TDFs are also TDFs. However, convex combinations of minimal TDFs (i.e., MTDFs) are not necessarily minimal. In this paper we discuss the existence in trees of a universal MTDF (i.e., an MTDF whose convex combinations with any other MTDF are also minimal).

Secure domination and secure total domination in graphs

A secure (total) dominating set of a graph G = (V; E) is a (total) dominating set X V with the property that for each u 2 V X, there exists x 2 X adjacent to u such that (X fxg) [ fug is (total) dominating. The smallest cardinality of a secure (total) dominating set is the secure (total) domination number s(G) ( st(G)). We characterize graphs with equal total and secure total domination numbers. We show that if G has minimum degree at least two, then st(G) s(G). We also show that st(G) is at most twice the clique covering number of G, and less than three times the independence number. With the exception of the independence number bound, these bounds are sharp.

Irredundance and Maximum Degree in Graphs

It is proved that the smallest cardinality among the maximal irredundant sets in an n–vertex graph with maximum degree Δ(≥2) is at least 2n/3Δ. This substantially improves a bound by Bollobas and Cockayne [1]. The class of graphs which attain this bound is characterised.

Universal minimal total dominating functions in graphs

A total dominating function (TDF) of a graph G = (V, E) is a function f: V → [0, 1] such that for each ν ϵ V, ΣuϵN(v) f(u) ⩾ 1 [where N(v) denotes the open neighborhood of vertex v]. Integer-valued TDFs are precisely characteristic functions of total dominating sets of G. Convex combinations of two TDFs are themselves TDFs but convex combinations of minimal TDFs (MTDFs) are not necessarily minimal. This paper is concerned with the existence of a universal MTDF in a graph, i.e., a MTDF g such that convex combinations of g and any other MTDF are themselves minimal.

On the domination number of prisms of graphs

For a permutation π of the vertex set of a graph G, the graph πG is obtained from two disjoint copies G1 and G2 of G by joining each v in G1 to π(v) in G2. Hence if π = 1, then πG = K2 × G, the prism of G. Clearly, γ(G) ≤ γ(πG) ≤ 2γ(G). We study graphs for which γ(K2 × G) = 2γ(G), those for which γ(πG) = 2γ(G) for at least one permutation π of V (G) and those for which γ(πG) = 2γ(G) for each permutation π of V (G).

The sequence of upper and lower domination, independence and irredundance numbers of a graph

Abstract Necessary and sufficient conditions are established for the existence of a graph whose upper and lower domination, independence and irredundance numbers are six given positive integers. This result shows that the only relationships between these six parameters which hold for all graphs and which do not involve other graph theoretical parameters, are already known.

On the product of upper irredundance numbers of a graph and its complement

Abstract A set X of vertices of a graph is irredundant if the closed neighbourhood of each x ϵ X is not contained in the union of closed neighbourhoods of the vertices of X − {x}. The upper irredundance number, IR(G) is the largest number of vertices in any irredundant set of G. We prove that for any p-vertex graph G, IR(G) ⋅ IR( G ) ⩽⌈ p(p+2) 4 ⌉ and exhibit all graphs which attain this bound.

A characterization of (γ, i)-trees

We prove a Harnack inequality for Dirichlet eigenfunctions of abelian homogeneous graphs and their convex subgraphs. We derive lower bounds for Dirichlet eigenvalues using the Harnack inequality. We also consider a randomization problem in connection with combinatorial games using Dirichlet eigenvalues.