Given an equidimensional algebraic set \(X\), its dual graph is defined on the vertices corresponding to the irreducible components of \(X\) with edges connecting components that intersect in codimension one. Hartshorne’s connectedness theorem says that the graph is connected if the coordinate ring of \(X\) is Cohen-Macaulay [R. Hartshorne, Am. J. Math. 84, 497–508 (1962; Zbl 0108.16602)]. The authors extend Hartshorne’s result showing that the graph is \(r\)-connected if \(X\) is a Gorenstein subspace arrangement, where \(r\) is the Castelnuovo-Mumford regularity of \(X\). For coordinate arrangements, this yields an algebraic extension of Balinski’s theorem for simplicial polytopes. The authors also provide a bound for the diameter of the graph in the case that \(X\) is an arrangement of lines (no three of them meeting in the same point) which is canonically embedded in \(\mathbb{P}^n\). For coordinate arrangements, this yields an algebraic expansion on the combinatorial result by K. Adiprasito and B. Benedetti [Math. Oper. Res. 39, No. 4, 1340–1348 (2014; Zbl 1319.52017)] that the Hirsch conjecture holds for flag normal simplicial complexes. Moreover, the authors provide a graph as an example which is not the dual graph of any simplicial complex (no matter the dimension).