A dominating set \(S \subseteq V\) is said to be an external staircase corona dominating set (ESCD-set), if every vertex in \(\left\langle { S } \right\rangle\) is either a pendent vertex or a support vertex and \(\left\langle { V - S } \right\rangle = rK_{2}\) . The minimum cardinality of an external staircase corona dominating set is called the external staircase corona domination number (ESCD-number) of H and is denoted by \(\gamma_{ESC} \left( H \right).\) In this article, we initiate a study of this parameter.