The authors study the extremal Betti numbers of the \(m\)th symbolic power of a square-free monomial ideal, \(I\subset R\), of dimension \(2\). Such an ideal can be identified with the Stanley-Reisner ideal of a \(1\)-dimensional simplicial complex, which, in turn, we can associate with a simple graph. The main results of this paper are collected in Theorems 3.4 to 3.6 and consist of a complete description of the extremal Betti numbers (positions and values) of the symmetric powers of \(I\), in terms of graph invariants. As a corollary, the authors give a classification, in terms of \(m\) and \(G\), of the rings \(R/I^{(m)}\) which are pseudo-Gorenstein (Corollary 3.7). In section 4, the focus is on the level property of \(R/I^{(2)}\). It is shown that this ring is level if and only if \(G\) is either a complete graph, or a complete bipartite graph, or a Moore graph of girth \(5\) (Theorem 4.6).