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Let (G) and e(G) denote the domination number and exponential domination number of graph G, respectively. Henning et al., in [Hered-itary equality of domination and exponential domination, Discuss. Math. Graph Theory 38 (2018) 275-285] gave a conjecture: There is a finite set F of graphs such that a graph G satisfies (H) = e(H) for every induced subgraph H of G if and only if G is F-free. In this paper, we study the conjecture for subcubic graphs. We characterize the class F by minimal forbidden induced subgraphs and prove that the conjecture holds for subcubic graphs. [ABSTRACT FROM AUTHOR]