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Inverse Problem for Degree Sequences

Let G = (V,E) be a simple undirected graph, meaning that it has no self-loops (edges of the form (v,v)) and no multiple edges between the same pair of vertices.

For such a graph we can define its degree sequence deg, where each element represents the degree of a vertex.

For example, the graph shown below has degree sequence

deg = (3,1,1,1)

Example graph with degree sequence (3,1,1,1)

In this exercise we consider the inverse problem: given a degree sequence deg, determine whether it is possible to construct a simple undirected graph with that sequence.

For each of the following statements, determine whether it is True (T) or False (F).

i) deg = (2,2) generates more than one graph.
ii) deg = (2,2,2) generates exactly one graph.
iii) deg = (2,2,2,2,2,2) generates exactly one graph.

Alternatives

a) TTF
b) FTT
c) TFT
d) FTF
e) None of the above

Original idea by: Luis Alberto Vásquez Vargas

Comments

  1. Joao MeidanisMarch 8, 2026 at 8:41 PM

    Good question, I modified it a bit and took it.

    ReplyDelete
  2. Luis VásquezMarch 9, 2026 at 11:07 AM

    Thanks, Professor. I was trying to use isomorphism terms without having to get verbose or intricate so I've chosen a simpler way, but you got a very elegant, simpler and balanced statement (not too formal, not too vague or ambiguos)

    ReplyDelete

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