Induced cycles vertex number and (1,2)-domination in cubic graphs

A (1,2)-dominating set in a graph $G$ is a set $S$ such that every vertex outside $S$ has at least one neighbor in $S$, and each vertex in $S$ has at least two neighbors in $S$. The (1,2)-domination number, ${\gamma }_{1,2}\left(G\right)$, is the minimum size of such a set, while $c_{ind}\left(G\right)$ is the cardinality of the largest vertex set in $G$ that induces one or more cycles. In this paper, we initiate the study of a relationship between these two concepts and discuss how establishing such a connection can contribute to solving a conjecture on the lower bound of $c_{ind}\left(G\right)$ for cubic graphs. We also establish an upper bound on $c_{ind}\left(G\right)$ for cubic graphs and characterize graphs that achieve this bound.