Fastest algorithm for a 'circular linked list builder'

Question

I have two questions:

1) What is the fastest algorithm to put this list in a "connecting" order?
2) Is this an existing problem/algorithm, and what name does it have?

The nodes are always connected in a circular fashion, so my starting ID does not matter.

Given this list:

id node1 node2
A  4     9
B  6     1
C  1     3
D  3    10
E  7     2
F  4     2
G  9     8
H 10     5
I  7     5
J  6     8

node1 & node2 are not in a specific order so id A can be 4 - 9 as well as 9 - 4.

The output should be this (it doesn't matter if it starts with A, as long as the output is a chain).

node ids: 4 - 9 - 8 - 6 - 1 - 3 - 10 - 5 - 7 - 2 - 4
ids:        A   G   J   B   C   D    H   I   E   F

(I am writing my code in C#. But pseudo code in any language will do)

Answer 1

I believe that you're looking for Eulerian path