On Erd˝os-Gallai and Havel-Hakimi algorithms

04 if Hn odd
05 L = False
06 return L
07 L = True
08 return L
On Erdős-Gallai and Havel-Hakimi algorithms 7
The running time of this algorithm is Θ(n) in all cases. Figure 1 characterizes
the efficiency of Parity-test.
(1) is only a necessary condition, therefore Parity-Test is only an approximate
(filtering) algorithm.
3.2 Binomial test
Our second test is based on the second necessary condition of Erdős-Gallai
theorem. It received the given name since we estimate the number of the inner
edges of the head of b using a binomial coefficient. Let Ti = bi+1 +· · ·+bn (i =
1, . . . , n).
Lemma 3 If n ≥ 1 and b is an n-graphical sequence, then
Hi ≤ i(i − 1) + Ti (i = 1, . . . , n − 1). (3)
Proof. The left side of (3) represents the degree requirement of the head of b.
On the right side of (3) i(i − 1) is an upper bound of the inner degree capacity
of the head, while Ti is an upper bound of the degree capacity of the tail,
belonging to the index i.
The following program is based on Lemma 3.
Input. n: number of the vertices (n ≥ 1);
b = (b1, . . . , bn): an n-regular even sequence;
H = (H1, . . . , Hn): Hi the sum of the first i elements of b;
H0: auxiliary variable, helping to compute the elements of H.
Output. L: logical variable (L = False signals, that b is surely not graphical,
while L = True shows, that the test could not decide, whether b is graphical).
Working variables. i: cycle variable;
T = (T1, . . . , Tn): Ti the sum of the last n − i elements of b;
T0: auxiliary variable, helping to compute the elements of T.
Binomial-Test(n, b, H, L)
01 T0 = 0
02 for i = 1 to n − 1