# A characterisation of universal minimal total dominating functions in trees

@article{Cockayne1995ACO, title={A characterisation of universal minimal total dominating functions in trees}, author={Ernest J. Cockayne and Christina M. Mynhardt}, journal={Discret. Math.}, year={1995}, volume={141}, pages={75-84} }

## 9 Citations

### Basic Maximal Total Strong Dominating Functions

- Mathematics
- 2020

Let G = (V,E) be a simple graph. A subset D of V (G) is called a total strong dominating set of G, if for every u ∈ V (G), there exists a v ∈ D such that u and v are adjacent and deg(v) ≥ deg(u). The…

### EXTREMUM AGGREGATES OF MINIMAL 0-DOMINATING FUNCTIONS OF GRAPHS

- Mathematics
- 1996

Abstract A 0-dominating function 0DF of a graph G = (V,E) is a function f: V → [0,1] such that Σ xeN(v) f(x) ≥ 1 for each ν e V with f(v) = 0. The aggregate of a 0DF f is defined by ag(f) = ΣveV f(v)…

### The algorithmic complexity of certain functional variations of total domination in graphs

- MathematicsAustralas. J Comb.
- 2004

It is shown that the decision problem corresponding to the computation of the total minus domination number of a graph is NP-complete, even when restricted to bipartite graphs or chordal graphs.

### Some results on universal minimal total dominating functions

- Mathematics
- 2001

In this paper, we introduce the concepts of redundant constraint and exceptional vertex which play an important role in the characterization of universal minimal total dominating functions (universal…

### Structure of the set of all minimal total dominating functions of some classes of graphs

- MathematicsDiscuss. Math. Graph Theory
- 2010

It is proved that a function reducible graph is a function separable graph and the idea of function reducibility is used to study the structure of FT (G) for some classes of graphs.

### Convexity of Minimal Total Dominating Functions Of Quadratic Residue Cayley Graphs

- Mathematics
- 2012

Nathanson [17] paved the way for the emergence of a new class of graphs, namely,Arithmetic Graphs by introducing the concepts of Number Theory, particularly, the Theory of congruences in Graph…

### A survey of selected recent results on total domination in graphs

- MathematicsDiscret. Math.
- 2009

### Fractional Domination, Fractional Packings, and Fractional Isomorphisms of Graphs

- Psychology
- 2005

Except where reference is made to the work of others, the work described in this dissertation is my own or was done in collaboration with my advisory committee. This dissertation does not include…

## References

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### Universal minimal total dominating functions in graphs

- MathematicsNetworks
- 1994

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.

### Total dominating functions in trees: Minimality and convexity

- MathematicsJ. Graph Theory
- 1995

The existence in trees of a universal MTDF (i.e., an MTDF whose convex combinations with any other MTDF are also minimal) is discussed.

### Convexity of minimal total dominating functions in graphs

- MathematicsJ. Graph Theory
- 1995

A sufficient condition for an MTDF to be universal is given which generalizes previous results and is found that graphs obtained by the operation from paths, cycles, complete graphs, wheels, and caterpillar graphs have a universal MTDF.