Terminal Steiner tree problem : Complexity and Algorithms

Given a connected graph $G$ and a terminal set $R\subseteq V(G)$, the Steiner tree problem (ST) asks for a tree that spans all of $R$ with at most $r$ vertices from $V(G)\mathrm{\backslash }R$, for some integer $r\ge 0$. It is known from (Garey et al.,1977 ) that ST is NP-complete. A Steiner tree in which all terminal vertices are constrained to be leaves is called a terminal Steiner tree. Our study addresses the existence of a terminal Steiner tree, its complexity across various graph classes, black-box applications of the ST, and a fixed-parameter tractable (FPT) algorithm with respect to the number of terminals.