Summary: The atom-bond connectivity \((ABC)\) index of a graph \(G=(V,E)\) is defined as \(ABC(G)=\sum_{v_iv_j\in E}\sqrt{[d(v_i)+d(v_j)-2]/[d(v_i)d(v_j)]}\), where \(d(v_i)\) denotes the degree of the vertex \(v_i\) of \(G\). This recently introduced molecular structure descriptor found interesting applications in the study of the heats of formation of alkanes. Let \(C(\pi)\) and \(\mathcal T(\pi)\) denote the sets of connected graphs and trees with degree sequence \(\pi\), respectively. L. Gan et al. [MATCH Commun. Math. Comput. Chem. 68, No. 1, 137–145 (2012; Zbl 1289.05046)] and R. Xing and B. Zhou [Filomat 26, No. 4, 683–688 (2012; Zbl 1289.05234)] proved that the so-called greedy tree is a tree with minimal ABC index in \(\mathcal T(\pi)\). In this paper we introduce the breadth-first searching graphs (BFS-graphs for short), and prove that, for any degree sequence \(\pi\) there exists a BFS-graph with minimal ABC index in \(C(\pi)\). This result is applicable to obtain the minimal value or lower bounds of ABC index of connected graphs.