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Enumerating degree sequences in digraphs and a cycle–cocycle reversing system. (English) Zbl 1114.05024
Summary: We give some new enumerations of indegree sequences of orientations of a graph using the Tutte polynomial. Then we introduce some discrete dynamical systems in digraphs consisting in reversing cycles, cocycles, or both, which extend the edge firing game (reversing sinks) by considering all orientations (reversing cocycles) and by introducing duality (reversing cycles). We show that indegree sequences can represent the configurations of these systems, and we enumerate equivalence classes of these systems. In particular, concerning the cycle–cocycle reversing system, we show that its configurations are in bijection with indegree sequences of orientations having a given vertex (quasi-sink of the system) reachable from any other. We also briefly discuss its generalization to oriented matroids, and relate structural and enumerative properties of its configurations to those of the sandpile model or chip firing game.

05C07 Vertex degrees
05C20 Directed graphs (digraphs), tournaments
52C40 Oriented matroids in discrete geometry
Full Text: DOI
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