Consider the following game on an undirected graph G. There are two players, a red color player R and a blue color player B. Initially all edges of G are uncolored. The two players alternately color an uncolored edge of G with their color until all edges are colored. The goal of B is that in the end, the blue-colored edges form a connected spanning subgraph of G. A connected spanning subgraph of G is a connected subgraph that contains all the vertexes of graph G. The goal of R is to prevent B from achieving his goal.
Assume that R starts the game. Suppose that both players play in the smartest way. Your task is to find out whether B will win the game.
Input:
Each test case begins with a line of two integers n ( 1 <= n <= 10) and m (0 <= m <= 30), indicating the number of vertexes and edges in the graph. All vertexes are numbered from 0 to n-1.
Then m lines follow. Each line contains two integers p and q ( 0 <= p, q < n) , indicating there is an edge between vertex p and vertex q.
Output:
For each test case print a line which is either “YES” or “NO” indicating B will win the game or not.
Example:
3 4
0 1
1 2
2 0
0 2
Output: Yes
My idea:
If we can find two disjoint spanning trees of the graph, then player B wins the game. Otherwise, A wins.
‘Two disjoint spanning trees’ means the edge sets of the two trees are disjoint
I wonder if you can prove or disprove my idea
Your idea is correct. Find a proof here:
http://www.cadmo.ethz.ch/education/lectures/FS08/graph_algo/solution01.pdf
If you search for “connectivity game” or “maker breaker games” you should find some more interesting problems and algorithms.