Total outer-connected dominating sets and total outer-connected domination polynomial of complete bipartite graph K2,n

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Document Type : Research Paper

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1 Department of Mathematics, Scott Christian College (Autonomous), Nagercoil - 629 003, Kanyakumari District, Tamilnadu, India. (Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli-627 012

2 Department of Mathematics, Scott Christian College(Autonomous),Nagercoil-629003.Affilliated to Manonmanium Sundaranar University,Abishekapatti,Tirunelveli-627012

Abstract

Let G = (V, E) be a simple graph. A set D⊆V (G) is a total outer−connected dominating set of G if D is total dominating, and the induced sub-graph G[V (G) − D] is a connected graph. Let K2,n be the complete bipartite graph and D ̃tc (K2,n,i) denote the family of all total outer-connected dominating sets of K2,n with cardinality i. Let d ̃tc (K2,n,i)=|D ̃tc (K2,n,i)|. In this paper, we obtain recursive formula for d ̃tc (K2,n,i). Using this recursive formula, we construct the polynomial, D~tc (K2,n,x)=∑i=22+nd ̃tc (K2,n,i)xi  which we call total outer−connected domination polynomial of K2,n and obtain some  properties of this polynomial.

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References

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Journal of Hyperstructures

Articles in Press, Accepted Manuscript
Available Online from 26 October 2025

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