is_valid_degree_sequence_erdos_gallai¶
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is_valid_degree_sequence_erdos_gallai
(deg_sequence)[source]¶ 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 – True if deg_sequence is graphical and False if not. Return type: bool Notes
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,
k∑i=1di≤k(k−1)+n∑j=k+1minA 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 [2].
The ZZ condition says that for the sequence d if
|d| >= \frac{(\max(d) + \min(d) + 1)^2}{4*\min(d)}then d is graphical. This was shown in Theorem 6 in [2].
References
[1] A. Tripathi and S. Vijay. “A note on a theorem of Erdős & Gallai”, Discrete Mathematics, 265, pp. 417-420 (2003).