Summary: \textit{M. Fischermann} et al. [Discrete Math. 282, No. 1--3, 251--255 (2004; Zbl 1042.05068)] generalized bound polysemy to competition polysemy by using digraphs instead of posets. They provided a characterization of competition polysemic pairs and a characterization of the connected graphs \(G\) for which there exists a tree \(T\) such that \((G,T)\) is competition polysemic. In this paper we continue to study the competition polysemy and characterize the connected graphs \(G\) for which there exists a triangle-free unicyclic graph \(G'\) such that \((G,G')\) is competition polysemic. Furthermore, we generalize competition polysemy to \(m\)-competition polysemy and prove a characterization of \(m\)-competition polysemic pairs.