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

• We investigate the relationship between the (1,2)-domination number γ 1 , 2 ( G ) and the induced cycles vertex number c ind ( G ) in cubic graphs. • We establish a sharp upper bound on c ind ( G ) for cubic graphs and characterize all graphs attaining this bound. • We introduce a new family of graphs, called trilobites , and conjecture that trilobites are the only graphs for which γ 1 , 2 ( G ) > c ind ( G ) . The conjecture is supported by several sufficient conditions ensuring γ 1 , 2 ( G ) ≤ c ind ( G ) . • The truth of this conjecture would imply the validity of a conjecture by Henning et al. (2016), which states that every cubic graph of order n satisfies c ind ( G ) ≥ n 2 . 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, γ 1 , 2 ( G ) , is the minimum size of such a set, while c i n d ( G ) 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 i n d ( G ) for cubic graphs. We also establish an upper bound on c i n d ( G ) for cubic graphs and characterize graphs that achieve this bound.