is_valid_degree_sequence_erdos_gallai

is_valid_degree_sequence_erdos_gallai(deg_sequence)

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.

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,

A strong index k is any index where and the value 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 [R226].

The ZZ condition says that for the sequence d if

then d is graphical. This was shown in Theorem 6 in [R226].

References

[R225]

A. Tripathi and S. Vijay. “A note on a theorem of Erdős & Gallai”, Discrete Mathematics, 265, pp. 417-420 (2003).

[R226]

(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]