CAC - Cactus

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In the mathematical field of graph theory, a spanning tree T of a connected, undirected graph
G is a tree composed of all the vertices and some (or perhaps all) of the edges of G. Informally,
a spanning tree of G is a selection of edges of G that form a tree spanning every vertex. That is,
every vertex lies in the tree, but no cycles (or loops) are formed. On the other hand, every
bridge of G must belong to T (a bridge is an edge whose deletion increases the number of
connected components).
A spanning tree of a connected graph G can also be defined as a maximal set of edges of G that
contains no cycle, or as a minimal set of edges that connect all vertices. - Wikipedia
In graph theory, a cactus (sometimes called a cactus tree) is a connected graph in which any
two simple cycles have at most one vertex in common. Equivalently, every edge in such a graph
belongs to at most one simple cycle. Equivalently, every block (maximal subgraph without a
cut-vertex) is an edge or a cycle. - Wikipedia

In the mathematical field of graph theory, a spanning tree T of a connected, undirected graph G is a tree composed of all the vertices and some (or perhaps all) of the edges of G. Informally, a spanning tree of G is a selection of edges of G that form a tree spanning every vertex. That is, every vertex lies in the tree, but no cycles (or loops) are formed. On the other hand, every bridge of G must belong to T (a bridge is an edge whose deletion increases the number of connected components).

A spanning tree of a connected graph G can also be defined as a maximal set of edges of G that contains no cycle, or as a minimal set of edges that connect all vertices. - Wikipedia

In graph theory, a cactus (sometimes called a cactus tree) is a connected graph in which any two simple cycles have at most one vertex in common. Equivalently, every edge in such a graph belongs to at most one simple cycle. Equivalently, every block (maximal subgraph without a cut-vertex) is an edge or a cycle. - Wikipedia

 

 

cactus graph

 

Your task in this problem is to count the number of ways you can convert a cactus graph to a spanning tree.

Input

 

The first line of input will be the number of test cases. Each test case will start with a two numbers N and E where N is the number of vertices of the cactus graph, vertices are numbered from 1 to N, 3 <= N <= 81 and E is the number of edges in the graph, 2 <= E <= 120. The next E lines each one will have two numbers v and w and that means there is an edge between vertix v and w.

 

Output

 

For each test case print “Case C: X” without quotes where C is the case number starting with 1 and X is the number of ways you can convert the given cactus graph to a spanning tree.

 

Example

Input:
2
3 3
1 2
2 3
1 3
5 5
1 2
2 3
2 4
3 4
4 5

Output:
Case 1: 3
Case 2: 3

hide comments
sonmi451: 2021-02-09 01:03:23

I struggled for this way too long, so I hope I am able to help some desperate souls:
1) it is really true, you need unsigned long long or BigInteger for java (consider the input values for N and E!)
2) if the given graph is a spanning tree, return 1
3) if there are no ways to create a valid spanning tree, return 0
Without this knowledge, this task is 2% coding and 98% wondering why these output definitions are so low.

zoro_swordsman: 2021-02-03 17:59:42

hey guys, i don't know what should I print in the case of an invalid input, for example, N<3 or E>120
i always get the report "wrong answer"!

sagsango: 2020-03-12 22:55:06

long long is giving WA.
Use unsigned long long to get AC.

nadstratosfer: 2019-12-23 22:34:03

Res = 1 if the graph is already a MST. WTF?! To each of you who WA'd on this stupidity and yet didn't bother to let others know, I wish you overtime on New Years eve.

Edelweiss: 2013-05-15 12:22:39

I'm sorry to have been suspicious of test data. I'm really really sorry... please excuse me... I'll do anything you want from mee... I've also miscalculated the maximum of the result :P
Ashamed...

Last edit: 2013-11-12 11:37:11
Jacob Plachta: 2013-05-14 16:18:24

@Federico: Ah, thanks - I miscalculated and thought I'd only need a signed long long.

Federico Lebrón: 2013-05-14 15:01:15

If it's any help, my bug was printing unsigned long long as %lld instead of %llu :s

hossamyosef: 2013-05-14 07:59:57

this problem had been tested by about 30 teams at our contest!!

hossamyosef: 2013-05-14 07:59:57

test data is correct !

Francky: 2013-05-14 07:59:57

As psetter don't give a quick answer, I hide the problem.
@psetter : please answer, or fix the problem. Add a comment and the problem will be visible.


Added by:hossamyosef
Date:2013-05-13
Time limit:3s
Source limit:50000B
Memory limit:1536MB
Cluster: Cube (Intel G860)
Languages:All except: ASM64
Resource:FCIS/ASU Local Contest 2013