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In this paper we consider pushdown graphs, i.e. infinite graphs that can bedescribed as transition graphs of deterministic real-time pushdown automata. Weconsider the case where some vertices are designated as being final and webuilt, in a breadth-first manner, a marking of edges that lead to such vertices(i.e., for every vertex that can reach a final one, we mark all out-going edgeslaying on some shortest path to a final vertex).  Our main result is that the edge-marked version of a pushdown graph mayitself no longer be a pushdown graph, as we prove that this enrich graph mayhave an undecidable MSO theory.  In this paper we consider pushdown graphs, i.e. infinite graphs that can bedescribed as transition graphs of deterministic real-time pushdown automata. Weconsider the case where some vertices are designated as being final and webuild, in a breadth-first manner, a marking of edges that lead to such vertices(i.e., for every vertex that can reach a final one, we mark all out-going edgeslaying on some shortest path to a final vertex).  Our main result is that the edge-marked version of a pushdown graph mayitself no longer be a pushdown graph, as we prove that the MSO theory of thisenriched graph may be undecidable.

Marking Shortest Paths On Pushdown Graphs Does Not Preserve MSO Decidability