Edge-Vertex Dominating Sets and Edge-Vertex Domination Polynomials of Cycles
Affiliation(s)
1 Department of Mathematics, Nesamony Memorial Christian College, Marthandam, India. 2 Department of Mathematics, Mar Ephraem College of Engineering & Technology, Marthandam, India.
1 Department of Mathematics, Nesamony Memorial Christian College, Marthandam, India. 2 Department of Mathematics, Mar Ephraem College of Engineering & Technology, Marthandam, India.
ABSTRACT
Let G = (V, E) be a simple graph. A set S
E(G) is an edge-vertex
dominating set of G (or simply an ev-dominating set), if for all vertices v 
V(G); there exists an
edge e
S such that e dominates v. Let 
denote the family of all ev-dominating sets of 
with cardinality i. Let 
. In this paper, we
obtain a recursive formula for 
. Using this
recursive formula, we construct the polynomial,
, which we call edge-vertex domination polynomial of 
(or simply an ev-domination polynomial of 
) and obtain some
properties of this polynomial.
Let G = (V, E) be a simple graph. A set S
E(G) is an edge-vertex
dominating set of G (or simply an ev-dominating set), if for all vertices v V(G); there exists an
edge eS such that e dominates v. Let denote the family of all ev-dominating sets of with cardinality i. Let . In this paper, we
obtain a recursive formula for . Using this
recursive formula, we construct the polynomial, , which we call edge-vertex domination polynomial of (or simply an ev-domination polynomial of ) and obtain some
properties of this polynomial.
Cite this paper
Vijayan, A. and Sherin Beula, J. (2015) Edge-Vertex Dominating Sets and Edge-Vertex Domination Polynomials of Cycles. Open Journal of Discrete Mathematics, 5, 74-87. doi: 10.4236/ojdm.2015.54007.
Vijayan, A. and Sherin Beula, J. (2015) Edge-Vertex Dominating Sets and Edge-Vertex Domination Polynomials of Cycles. Open Journal of Discrete Mathematics, 5, 74-87. doi: 10.4236/ojdm.2015.54007.
References
[1] Sampath Kumar, E. and Kamath, S.S. (1992) Mixed Domination in Graphs. Sankhya: The Indian Journal of Statistics, 54, 399-402. [2] Alikhani, S. and Peng, Y.-H. (2009) Dominating Sets and Domination Polynomials of Paths. International Journal of Mathematics and Mathematical Sciences, 2009, Article ID: 542040.
http://dx.doi.org/10.1155/2009/542040 [3] Alikhani, S. and Peng, Y.-H. (2009) Dominating Sets and Domination Polynomials of Cycles. [4] Vijayan, A. and Sherin Beula, J. (2014) ev-Dominating Sets and ev-Domination Polynomials of Paths. International Organization of Scientific Research Journal of Mathematics, 10, 7-17. [5] Vijayan, A. and Lal Gipson, K. (2013) Dominating Sets and Domination Polynomials of Square of Paths. Open Journal of Discrete Mathematics, 3, 60-69. [6] Chartand, G. and Zhang, P. (2005) Introduction to Graph Theory. McGraw-Hill, Boston. [7] Alikhani, S. and Peng, Y.-H. (2009) Introduction to Domination Polynomial of a Graph.
[1] Sampath Kumar, E. and Kamath, S.S. (1992) Mixed Domination in Graphs. Sankhya: The Indian Journal of Statistics, 54, 399-402. [2] Alikhani, S. and Peng, Y.-H. (2009) Dominating Sets and Domination Polynomials of Paths. International Journal of Mathematics and Mathematical Sciences, 2009, Article ID: 542040.
http://dx.doi.org/10.1155/2009/542040 [3] Alikhani, S. and Peng, Y.-H. (2009) Dominating Sets and Domination Polynomials of Cycles. [4] Vijayan, A. and Sherin Beula, J. (2014) ev-Dominating Sets and ev-Domination Polynomials of Paths. International Organization of Scientific Research Journal of Mathematics, 10, 7-17. [5] Vijayan, A. and Lal Gipson, K. (2013) Dominating Sets and Domination Polynomials of Square of Paths. Open Journal of Discrete Mathematics, 3, 60-69. [6] Chartand, G. and Zhang, P. (2005) Introduction to Graph Theory. McGraw-Hill, Boston. [7] Alikhani, S. and Peng, Y.-H. (2009) Introduction to Domination Polynomial of a Graph.