T-irreducible extension for union of paths and cycles

A graph $H$ with $n+1$ nodes is an extension of a graph $G$ with $n$ nodes if each maximal subgraph of $H$ contains $G$. Trivial extension of a graph $G$ is the connection of graph $G$ and the singleton graph (i.e. we add one node to the graph $G$ and this node join with each node of $G$). T-irreducible extension of graph $G$ is an extension of the graph $G$ which is obtained by removing maximal set of edges from the trivial extension of $G$. One of T-irreducible extensions is constructed for an arbitrary union of cycles and paths.