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It is commonly known that directed graphs are defined as a double $G_d:=(V,E)$ such that $E \subseteq V^2$, and that undirected graphs $G_u:=(V,E)$ such that $E \subseteq \left\{ \{a,b\}\Big\vert a \in V \land b\in V \right\}$.

Why? Why does one not just define, say, an undirected graph to be a directed graph wherein each edge merely happens to have a symmetric counterpart? Alternatively, why not say directed graphs are but undirected graphs so that one could write $v \in e \in E$ to say a vertex lies on an edge?

Currently very frustrated with the lack of standardisation of Graph Theory at the moment.

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    $\begingroup$ Undirected graphs are a special case of directed graphs, as you note. But they are a special case of such enormous importance that it's usually better to think of them as having undirected edges than to think of them having pairs of symmetric edges. $\endgroup$ Commented yesterday

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Because it would not be correct. For example, the complete graph $K_3$ has three edges, not six edges as your suggested definition would give it. Fundamental concepts like "Euler circuit" (using every edge exactly once) would become trivial if every edge was replaced by two opposing directed edges. We would get so fed up with talking about adding or removing some number of "pairs of opposing edges" that we would eventually realise everything would become much easier if we made "pair of opposing edges" the basic unit - and then we would get back to the standard definition.

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