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My google search turned up much information about what people are doing with inductive graphs, but no definitions. So I ask you, StackExchange, what is an inductive graph? When I think of induction, I think of recursion. But this must be a wrong line of thinking, because cannot all graphs be constructed recursively? Thanks for clearing up my confusion.

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up vote 3 down vote accepted

A graph $G$ is $d$-inductive if the vertices of $G$ can be numbered so that each vertex has at most $d$ edges to higher-numbered vertices.

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So does the set of verticies need to have a partial order? –  Athan Clark Jul 5 at 4:11
    
I think we're talking about numbering the vertices of an $n$-vertex graph with $1,2,3,\dots,n$, each vertex getting a different number. –  Gerry Myerson Jul 5 at 4:14
    
I see. However, the strict ordering is mandatory? Or could you simply limit the total number of edges any vertex has? –  Athan Clark Jul 5 at 4:25
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If no vertex has degree exceeding $r$, then (no matter how you number the vertices) each vertex has at most $r$ edges to higher-numbered vertices, so the graph is guaranteed to be (trivially) $r$-inductive. The concept "$d$-inductive" would appear to be interesting only if there are vertices of degree bigger than $d$. –  Gerry Myerson Jul 5 at 4:28
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To be fair, I just used search engines as well. But let me know if either of these ring a bell:

Wikipedia thinks that inductive graphs may also be known as degenerate graphs:

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On the other hand, the book The Theory of Graphs defines an inductive graph on page 13:

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Do these line up with the sorts of inductive graphs you've been looking at?

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