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

Abstract

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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.

How to cite

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K. 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

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  1. [1] B. Grünbaum, Convex Polytopes (Interscience Publishers, 1967). 
  2. [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. [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. [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. [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. [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. [7] K. Reji Kumar, Studies in Graph Theory - Dominating functions, Ph.D Thesis (Manonmaniam Sundaranar University, Tirunelveli, India, 2004). 
  8. [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. [9] D.B. West, Graph Theory : An introductory course (Prentice Hall, New York, 2002). 

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