I want to understand the Dart of a graph. Many authors and also in the book Graphs on surfaces and their applications by Sergei K. Lando, Alexander K. Zvonkin used this term Dart of a Graph/edges. I don't know the meaning of that. I searched online. I didn't find any satisfactory answer. Any kind of help would be appreciated. Thank you in advance.
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1$\begingroup$ According to this paper sciencedirect.com/science/article/pii/S0095895606000682, a dart is the same as an arc. A similar (but not exactly the same) use of dart can be found in planarity.org/Klein_basic_graph_definitions.pdf. $\endgroup$– Ira GesselCommented Dec 3, 2021 at 6:59
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$\begingroup$ I saw this article planarity.org/Klein_basic_graph_definitions.pdf. They consider all the points of $E \times \lbrace -1,1\rbrace$, but in the mentioned book they consider half-edge. That creates confusion. $\endgroup$– Lokenath KunduCommented Dec 3, 2021 at 8:49
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The term 'dart' is often used in the voltage graph construction from topological graph theory (also known as lifts or regular coverings). A couple of examples of this usage are the papers 'New record graphs in the degree-diameter problem' by Loz & Siran or 'Lifting graph automorphisms by voltage assignments' by Malnic, Nedela & Skoviera. In this construction the two different directions of each edge need to be considered separately, as they are associated with inverse elements from a group. This requirement doesn't hold for directed arcs when taking lifts of digraphs or mixed graphs, so a directed 2-cycle in a graph is treated somewhat differently to an edge viewed as a pair of darts; hence there is a small technical difference between a 'dart' and an 'arc'.
