Let S be a sequence of exactly one positive integers, n. what is the least number of 1s needed to be added to make it graphical?
To determine the least number of 1s needed to make a sequence graphical, we need to understand what a graphical sequence is. A sequence is said to be graphical if it can be represented as the degree sequence of a simple graph.
A degree sequence of a graph is a sequence of positive integers where each integer represents the degree (number of edges) of a vertex in the graph. For a sequence to be graphical, it must satisfy the graphical sequence condition: the sum of all the degrees in the sequence should be even.
To determine the least number of 1s needed, we first need to arrange the sequence in non-increasing order (from largest to smallest). Let's denote the sequence as S = [a1, a2, ..., an].
Now, we need to check if S satisfies the graphical sequence condition by calculating the sum of all the degrees (a1 + a2 + ... + an) and checking if it's even.
If the sum of the degrees is odd, it is not possible to make the sequence graphical.
If the sum of the degrees is even, we can start constructing the graphical sequence. We need to add the least number of 1s such that the sequence becomes graphical. To do this, we can follow the following steps:
1. If the largest element in the sequence (a1) is greater than or equal to n, subtract 1 from the first n elements of the sequence.
2. If the largest element is less than n, subtract 1 from the largest element and repeat Step 1.
Repeat these steps until the sequence becomes graphical (sum of degrees is even).
The number of 1s added during this process is the least number of 1s needed to make the sequence graphical.