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Returns True if deg_sequence can be realized by a simple graph.

The validation is done using the Erdős-Gallai theorem [EG1960].

Parameters :

deg_sequence : list

A list of integers

Returns :

valid : bool

True if deg_sequence is graphical and False if not.


This implementation uses an equivalent form of the Erdős-Gallai criterion. Worst-case run time is: O(n) where n is the length of the sequence.

Specifically, a sequence d is graphical if and only if the sum of the sequence is even and for all strong indices k in the sequence,

\[\sum_{i=1}^{k} d_i \leq k(k-1) + \sum_{j=k+1}^{n} \min(d_i,k) = k(n-1) - ( k \sum_{j=0}^{k-1} n_j - \sum_{j=0}^{k-1} j n_j )\]

A strong index k is any index where \(d_k \geq k\) and the value \(n_j\) is the number of occurrences of j in d. The maximal strong index is called the Durfee index.

This particular rearrangement comes from the proof of Theorem 3 in [R245].

The ZZ condition says that for the sequence d if

\[\begin{split}|d| >= \frac{(\max(d) + \min(d) + 1)^2}{4*\min(d)}\end{split}\]

then d is graphical. This was shown in Theorem 6 in [R245].


[R244]A. Tripathi and S. Vijay. “A note on a theorem of Erdős & Gallai”, Discrete Mathematics, 265, pp. 417-420 (2003).
[R245](1, 2, 3) I.E. Zverovich and V.E. Zverovich. “Contributions to the theory of graphic sequences”, Discrete Mathematics, 105, pp. 292-303 (1992).

[EG1960], [choudum1986]