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

K. Reji Kumar; Gary MacGillivray

Discussiones Mathematicae Graph Theory (2010)

- Volume: 30, Issue: 3, page 407-423
- ISSN: 2083-5892

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topK. Reji Kumar, and Gary MacGillivray. "Structure of the set of all minimal total dominating functions of some classes of graphs." Discussiones Mathematicae Graph Theory 30.3 (2010): 407-423. <http://eudml.org/doc/270796>.

@article{K2010,

abstract = {In this paper we study some of the structural properties of the set of all minimal total dominating functions ($_T$) of cycles and paths and introduce the idea of function reducible graphs and function separable graphs. It is proved that a function reducible graph is a function separable graph. We shall also see how the idea of function reducibility is used to study the structure of $_T(G)$ for some classes of graphs.},

author = {K. Reji Kumar, Gary MacGillivray},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {minimal total dominating functions (MTDFs); convex combination of MTDFs; basic minimal total dominating functions (BMTDFs); simplex; polytope; simplicial complex; function separable graphs; function reducible graphs},

language = {eng},

number = {3},

pages = {407-423},

title = {Structure of the set of all minimal total dominating functions of some classes of graphs},

url = {http://eudml.org/doc/270796},

volume = {30},

year = {2010},

}

TY - JOUR

AU - K. Reji Kumar

AU - Gary MacGillivray

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

JO - Discussiones Mathematicae Graph Theory

PY - 2010

VL - 30

IS - 3

SP - 407

EP - 423

AB - In this paper we study some of the structural properties of the set of all minimal total dominating functions ($_T$) of cycles and paths and introduce the idea of function reducible graphs and function separable graphs. It is proved that a function reducible graph is a function separable graph. We shall also see how the idea of function reducibility is used to study the structure of $_T(G)$ for some classes of graphs.

LA - eng

KW - minimal total dominating functions (MTDFs); convex combination of MTDFs; basic minimal total dominating functions (BMTDFs); simplex; polytope; simplicial complex; function separable graphs; function reducible graphs

UR - http://eudml.org/doc/270796

ER -

## References

top- [1] B. Grünbaum, Convex Polytopes (Interscience Publishers, 1967).
- [2] E.J. Cockayne and C.M. Mynhardt, A characterization of universal minimal total dominating functions in trees, Discrete Math. 141 (1995) 75-84, doi: 10.1016/0012-365X(93)E0192-7. Zbl0824.05034
- [3] E.J. Cockayne, C.M. Mynhardt and B. Yu, Universal minimal total dominating functions in graphs, Networks 24 (1994) 83-90, doi: 10.1002/net.3230240205. Zbl0804.90122
- [4] E.J. Cockayne, C.M. Mynhardt and B. Yu, Total dominating functions in trees: Minimality and convexity, J. Graph Theory 19 (1995) 83-92, doi: 10.1002/jgt.3190190109. Zbl0819.05035
- [5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, 1998). Zbl0890.05002
- [6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs - Advanced Topics (Marcel Dekker, Inc., New York, 1998). Zbl0883.00011
- [7] K. Reji Kumar, Studies in Graph Theory - Dominating functions, Ph.D Thesis (Manonmaniam Sundaranar University, Tirunelveli, India, 2004).
- [8] K. Reji Kumar, G. MacGillivray and R.B. Bapat, Topological properties of the set of all minimal total dominating functions of a graph, manuscript. Zbl1244.05167
- [9] D.B. West, Graph Theory : An introductory course (Prentice Hall, New York, 2002).

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