Total Domination on Tree Operators

Abstract

Abstract Let G be a graph with vertex set V and edge set E , a set $$D\subseteq V$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>V</mml:mi> </mml:mrow> </mml:math> is a total dominating set if every vertex $$v\in V$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>v</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>V</mml:mi> </mml:mrow> </mml:math> has at least one neighbor in D . The minimum cardinality among all total dominating sets is called the total domination number, and it is denoted by $$\gamma _{t}(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . Given an arbitrary tree graph T , we consider some operators acting on this graph; $$\texttt {S}(T),\texttt {R}(T),\texttt {Q}(T)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $$\texttt {T}(T)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , and we give bounds of the total domination number of these new graphs using other parameters in the graph T . We also give the exact value of the total domination number in some of them.

Topics & Concepts