For any class K of compacta and any compactum X we say that: (a) X is confluently K-representable if X is homeomorphic to the inverse limit of an inverse sequence of members of K with confluent bonding mappings, and (b) X is confluently K-like provided that X admits, for every ε > 0, a confluent ε-mapping onto a member of K. The symbol double-struck L sign ℂ stands for the class of all locally connected compacta. It is proved in this paper that for each compactum X and each family K of graphs, X is confluently K-representable if and only if X is confluently K-like. We also show that for any compactum the properties of: (1) being confluently graph-representable, and (2) being 1-dimensional and confluently double-struck L sign ℂ-like, are equivalent. Consequently, all locally connected curves are confluently graph-representable. We also conclude that all confluently arc-like continua are homeomorphic to inverse limits of arcs with open bonding mappings, and all confluently tree-like continua are absolute retracts for hereditarily unicoherent continua.