Chromatic Transversal Roman Domination in Graphs

For a graph $G$ with chromatic number $k$, a dominating set $S$ of $G$ is called a chromatic-transversal dominating set (ctd-set) if $S$ intersects every color class of any $k$-coloring of $G$.  The minimum cardinality of a ctd-set of $G$ is called the {\em chromatic transversal domination number} of $G$ and is denoted by $\gamma_{ct}(G)$.  A {\em Roman dominating function} (RDF) in a graph $G$ is a function $f : V(G) \to \{0, 1, 2\}$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) = 2$.  The weight of a Roman dominating function is the value $w(f) = \sum_{u \in V} f(u)$.  The minimum weight of a Roman dominating function of a graph $G$ is called the {\em Roman domination number} of $G$ and is denoted by $\gamma_R(G)$.  The concept of {\em chromatic transversal domination} is extended to Roman domination as follows:   For a graph $G$ with chromatic number $k$, a {\em Roman dominating function} $f$ is called a {\em chromatic-transversal Roman dominating function} (CTRDF) if the set of all vertices $v$ with $f(v) > 0$ intersects every color class of any $k$-coloring of $G$.  The minimum weight of a chromatic-transversal Roman dominating function of a graph $G$ is called the {\em chromatic-transversal Roman domination number} of $G$ and is denoted by $\gamma_{ctR}(G)$.  In this paper a study of this parameter is initiated.