We know that:
If we have N vertices
To build a connected undirected graph, you'll need at least N-1 edges.
Let M be the set of possible connected undirected graphs with N-1 edges.
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Can we prove or disprove that if there's an undirected connected graph with more than N-1 edges, it must contain one of the graphs in M? In other words, can we take one of the graphs in M and add edges to create this new graph?
(by "containing", I mean that it has all the edges of the other graph plus some more.)
Can we prove or disprove that if there's an undirected connected graph with more than N-1 edges, it must contain one of the graphs in M?
Assuming that undirected connected graph g with more than N-1 edges has N vertices, the answer is "yes".
You can prove it by constructing a Spanning Tree of g, which is a subgraph with N vertices and N-1 edges. The problem statea that M contains all such graphs, a spanning tree of g is a member of M. Since a spanning tree is constructed by removing edges from g, you can add these edges back, thus going from a member of M back to the original graph g.
No, this isn't necessarily the case. As an example, imagine a path graph with 2n nodes (and, therefore, 2n - 1 edges). Cut out the middle edge, splitting the graph into two connected components that are each path graphs. Both of these paths have n - 1 edges, but neither connected component is a connected graph with 2n - 2 edges.
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An undirected graph having 'n' number of edges, then find out number of vertices that graph have?
Since an edge is a connection between to vertices, the amount of vertices is at max 2n.
The amount of vertices is at minimum n+1. (This is pretty logical if you imagine that you have 2 edges - then you will at minimum have 3 vertices, because each edge must connect 2 vertices)
So if e = n, then n+1 <= v <= 2n
There is no exact formula for the number of vertices in terms of number of edges in the general case. However, if you take special cases, you can say more: if the graph is a tree, then the number of vertices is one more than the number of edges.
Having an undirected graph with v vertices and e edges with each edges having a non-negative weight of at most w, the task is to find(if possible!) a subset of edges that connects all vertices with minimum cost. Also this graph has a root node which in the sub-graph should only be directly connected to exactly k other vertices(so it most have exactly k edges connected to it in the sub-graph). The algorithm should work in O(e⋅log(v)⋅log(w)) time.
So after some thinking I thought that we need to find a minimum spanning tree since it gives us the Minimum-cost subgraph which takes O(e.log(v)) time.However for insuring that the root node has exactly k edges in the resulting tree I couldn't come up with any working idea.
I think that maybe changing the weights of k edges that are connect to the root may enforce their selection,since in the Kruskal's algorithm we first sort the edges, if we pick exactly k edges that are already connected to the edge root in the graph and change their weights to some minimum amount then in the tree construction phase, chances are that this k edges make it to the MST. There are still problems here:
K edges should be picked so that it would be possible for them to make it to an MST.
Why a complete, directed graph G on n vertices and m edges has m = n(n-1) edges. But I tried lots of examples showing this statement is false, which would be n(n-1)/2 But our professor gives true to this statement. Can someone explain to me the correctness of this statement?
I think you haven't completely understood the difference between directed and undirected graphs.
Remember in an undirected graph, the orientation of the edges doesn't matter.
But in directed graphs, the orientation of edges matter.
As an analogy, suppose the two cities, A and B are represented by two nodes of a graph, and the path joining them represents the edge. Now if the edge is un-directed, you can go from A to B or vice versa. However if it's directed, it means that it's a one way road, you can only go from A to B or vice versa (depending on the orientation).
Now to answer your question, in an undirected graph, the total number of edges would be
C(n,2) = (n*(n-1))/2.
But in a directed graph of n nodes, every edge can be doubled up. i.e, One from A to B and other from B to A.
Hence, the total number of edges = 2*C(n,2)
Which translates to n*(n-1).
You are talking about directed graph, so between two nodes A and B, you can have one directed edge from A to B and one from B to A.
Knowing that, you can find the expected result by induction.
So I have an exercise that I should prove or disprove that:
1) if e is a minimum weight edge in the connected graph G such that not all edges are necessarily distinct, then every minimum spanning tree of G contains e
2) Same as 1) but now all edge weights are distinct.
Ok so intuitively, I understand that for 1) since not all edge weights are distinct, then it's possible that a vertex has the path with edge e but also another edge e_1 such that if weight(e) = weight (e_1) then there is a spanning tree which does not contain the edge e since the graph is connected. Otherwise if both e_1 and e are in the minimum spanning tree, then there is a cycle
and for 2) since all edge weights are distinct, then of course the minimum spanning tree will contain the edge e since any algorithm will always choose the smaller path.
Any suggestions on how to prove these two though? induction? Not sure how to approach.
Actually in your first proof when you say that if both e and e_1 are in G, then there's a cycle, that's not true, because they're minimal edges, so there doesn't have to be a cycle, and you do need to include them both into the MST, because if |E| > 1 and |V| > 2 then they both have to be there.
Anyways, a counter example for the first one is a complete graph with all edges of the same weight as e, the MST will contain only |V|-1 edges, but you didn't include all the other edges of that same weight, hence you have a contradiction.
As for the second one, if all edges are distinct, then if you remove the minimum edge and want to reconstruct the MST, the only way to go about this is to add a an edge connecting the 2 disjoint sets that were broken up by that minimum-weight edge.
Now suppose that you didn't remove that minimum-weight edge, and added that other edge, now you've created a cycle, and since all edges are distinct the cycle-creating edge will be greater than all of them, hence if you remove any former MST edge from that cycle, it will only increase the size of the MST. Which means that pretty much all former MST edges are critical when all edges have distinct weights.
Does the opposite of Kruskal's algorithm for minimum spanning tree work for it? I mean, choosing the max weight (edge) every step?
Any other idea to find maximum spanning tree?
Yes, it does.
One method for computing the maximum weight spanning tree of a network G –
due to Kruskal – can be summarized as follows.
Sort the edges of G into decreasing order by weight. Let T be the set of edges comprising the maximum weight spanning tree. Set T = ∅.
Add the first edge to T.
Add the next edge to T if and only if it does not form a cycle in T. If
there are no remaining edges exit and report G to be disconnected.
If T has n−1 edges (where n is the number of vertices in G) stop and
output T . Otherwise go to step 3.
Source: https://web.archive.org/web/20141114045919/http://www.stats.ox.ac.uk/~konis/Rcourse/exercise1.pdf.
From Maximum Spanning Tree at Wolfram MathWorld:
"A maximum spanning tree is a spanning tree of a weighted graph having maximum weight. It can be computed by negating the weights for each edge and applying Kruskal's algorithm (Pemmaraju and Skiena, 2003, p. 336)."
If you invert the weight on every edge and minimize, do you get the maximum spanning tree? If that works you can use the same algorithm. Zero weights will be a problem, of course.
Although this thread is too old, I have another approach for finding the maximum spanning tree (MST) in a graph G=(V,E)
We can apply some sort Prim's algorithm for finding the MST. For that I have to define Cut Property for the maximum weighted edge.
Cut property: Let say at any point we have a set S which contains the vertices that are in MST( for now assume it is calculated somehow ). Now consider the set S/V ( vertices not in MST ):
Claim: The edge from S to S/V which has the maximum weight will always be in every MST.
Proof: Let's say that at a point when we are adding the vertices to our set S the maximum weighted edge from S to S/V is e=(u,v) where u is in S and v is in S/V. Now consider an MST which does not contain e. Add the edge e to the MST. It will create a cycle in the original MST. Traverse the cycle and find the vertices u' in S and v' in S/V such that u' is the last vertex in S after which we enter S/V and v' is the first vertex in S/V on the path in cycle from u to v.
Remove the edge e'=(u',v') and the resultant graph is still connected but the weight of e is greater than e' [ as e is the maximum weighted edge from S to S/V at this point] so this results in an MST which has sum of weights greater than original MST. So this is a contradiction. This means that edge e must be in every MST.
Algorithm to find MST:
Start from S={s} //s is the start vertex
while S does not contain all vertices
do
{
for each vertex s in S
add a vertex v from S/V such that weight of edge e=(s,v) is maximum
}
end while
Implementation:
we can implement using Max Heap/Priority Queue where the key is the maximum weight of the edge from a vertex in S to a vertex in S/V and value is the vertex itself. Adding a vertex in S is equal to Extract_Max from the Heap and at every Extract_Max change the key of the vertices adjacent to the vertex just added.
So it takes m Change_Key operations and n Extract_Max operations.
Extract_Min and Change_Key both can be implemented in O(log n). n is the number of vertices.
So This takes O(m log n) time. m is the number of edges in the graph.
Let me provide an improvement algorithm:
first construct an arbitrary tree (using BFS or DFS)
then pick an edge outside the tree, add to the tree, it will form a cycle, drop the smallest weight edge in the cycle.
continue doing this util all the rest edges are considered
Thus, we'll get the maximum spanning tree.
This tree satisfies any edge outside the tree, if added will form a cycle and the edge outside <= any edge weights in the cycle
In fact, this is a necessary and sufficient condition for a spanning tree to be maximum spanning tree.
Pf.
Necessary: It's obvious that this is necessary, or we could swap edge to make a tree with a larger sum of edge weights.
Sufficient: Suppose tree T1 satisfies this condition, and T2 is the maximum spanning tree.
Then for the edges T1 ∪ T2, there're T1-only edges, T2-only edges, T1 ∩ T2 edges, if we add a T1-only edge(x1, xk) to T2, we know it will form a cycle, and we claim, in this cycle there must exist one T2-only edge that has the same edge weights as (x1, xk). Then we can exchange these edges will produce a tree with one more edge in common with T2 and has the same sum of edge weights, repeating doing this we'll get T2. so T1 is also a maximum spanning tree.
Prove the claim:
suppose it's not true, in the cycle we must have a T2-only edge since T1 is a tree. If none of the T2-only edges has a value equal to that of (x1, xk), then each of T2-only edges makes a loop with tree T1, then T1 has a loop leads to a contradiction.
This algorithm taken from UTD professor R. Chandrasekaran's notes. You can refer here: Single Commodity Multi-terminal Flows
Negate the weight of original graph and compute minimum spanning tree on the negated graph will give the right answer. Here is why: For the same spanning tree in both graphs, the weighted sum of one graph is the negation of the other. So the minimum spanning tree of the negated graph should give the maximum spanning tree of the original one.
Only reversing the sorting order, and choosing a heavy edge in a vertex cut does not guarantee a Maximum Spanning Forest (Kruskal's algorithm generates forest, not tree). In case all edges have negative weights, the Max Spanning Forest obtained from reverse of kruskal, would still be a negative weight path. However the ideal answer is a forest of disconnected vertices. i.e. a forest of |V| singleton trees, or |V| components having total weight of 0 (not the least negative).
Change the weight in a reserved order(You can achieve this by taking a negative weight value and add a large number, whose purpose is to ensure non-negative) Then run your family geedy-based algorithm on the minimum spanning tree.