  A set $S$ of vertices in a graph $G = (V, E)$ is called {\em cycle
independent} if the induced subgraph $\langle S\rangle$ is acyclic, and called
{\em odd-cycle indepdendet} if $\langle S\rangle$ is bipartite. A set $S$ is
{\em cycle dominating} (resp. {\em odd-cycle dominating}) if for every vertex
$u \in V \setminus S$ there exists a vertex $v \in S$ such that $u$ and $v$ are
contained in a (resp. odd cycle) cycle in $\langle S \setminus \{u\}\rangle$. A
set $S$ is {\em cycle irredundant} (resp. odd-cycle irredundant) if for every
vertex $v \in S$ there exists a vertex $u \in V \setminus S$ such that $u$ and
$v$ are in a (resp. odd cycle) cycle of $\langle S \setminus \{u\}\rangle$, but
$u$ is not in a cycle of $\langle S \cup \{u\} \setminus \{v\}\rangle$. In this
paper we present these new concepts, which relate in a natural way to
independence, domination and irredundance in graphs. In particular, we
construct analogs to the domination inequality chain for these new concepts.
