‎‎Let $G=(V‎, ‎E)$ be a simple graph with vertex set $V$ and edge set $E$‎. ‎A {em mixed Roman dominating function} (MRDF) of $G$ is a function $f:Vcup Erightarrow {0,1,2}$ satisfying the condition that every element $xin Vcup E$ for which $f(x)=0$ is adjacent‎‎or incident to at least one element $yin Vcup E$ for which $f(y)=2$‎. ‎The weight of an‎‎MRDF $f$ is $sum _{xin Vcup E} f(x)$‎. ‎The mi...

In this paper, we introduce and initiate the study of outer-connected semitotal domination in graphs. Given a graph G without isolated vertices, set S vertices is dominating if every vertex outside adjacent to distance at most 2 units from another S. A an either = V (G) or ̸= satisfying property that subgraph induced by \ connected. The smallest cardinality ̃γt2(G) number ofG. First, determine sp...