| Summary: | Let $$G = (V(G),E(G))$$ be a graph and $$f:V(G) \to \{ 0,1,2\} $$ be a function where for every vertex $$v \in V(G)$$ with $$f(v) = 0,$$ there is a vertex $$u \in {N_G}(v),$$ where $$f(u) = 2.$$ Then $$f$$ is a Roman dominating function or a $$RDF$$ of $$G.$$ The weight of $$f$$ is $$f(V(G)) = \sum\nolimits_{v \in V(G)} f(v).$$ The minimum weight of all $$RDF$$ is called the Roman domination number of $$G,$$ denoted by $${\gamma _R}(G).$$ Let $$G$$ be a graph with $$V(G) = \{ {v_1},{v_2},\ldots,{v_n}\} $$ and G' be a copy of $$G$$ with $$V({G'}) = \{ v_1',v_2',\ldots,v_n'\} .$$ Then a functigraph $$G$$ with function $$\sigma :V(G) \to V({G'})$$ is denoted by $$C(G,\sigma ),$$ its vertices and edges are $$V(C(G,\sigma )) = V(G) \cup V({G'})$$ and $$E(C(G,\sigma )) = E(G) \cup E({G'}) \cup \{ v\sim {v'} | v\in V(G),{v'} \in V({G'}),\sigma (v) = {v'}\} ,$$ respectively. This paper deals with the Roman domination number of the functigraph and its complement. We present a general bound $${\gamma _R}(G) \le {\gamma _R}(C(G,\sigma )) \le 2{\gamma _R}(G),$$ where $$\sigma :V(G) \to V({G'})$$ is a permutation. Also, the Roman domination number of some special graphs are considered. We obtain a general bound of $${\gamma _R}(\overline {C(G,\sigma )} $$ and we show that this bound is sharp.
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