In graph theory, a directed pseudograph can have multiple edges, and loops. My question is, a single vertex can have more than one loop? Or for each vertex, at most one loop is allowed?
Thanks.
Yes.
In general multiple edges between the same points in the same direction are allowed (aka parallel arcs). So are multiple loops on the same point, since loops are simply edges with the same start and end point.
Related
My problem is a generalization of a task solved by [Blossom algorithm] by Edmonds. The original task is the following: given a complete graph with weighted undirected edges, find a set of edges such that
1) every vertex of the graph is adjacent to only one edge from this set (i.e. vertices are grouped into pairs)
2) sum over weights of edges in this set is minimal.
Now, I would like to modify the first goal into
1') vertices are grouped into sets of 3 vertices (or in general, d vertices), and leave condition 2) unchanged.
My questions:
Do you know if this 'generalised' problem has a name?
Do you know about an algorithm solving it in number of steps being polynomial of number of vertices (like Blossom algorithm for an original problem)? I don't see a straightforward generalisation of Blossom algorithm, as it is based on looking for augmenting paths on a graph compressed to a bipartite graph (and uses here Hungarian algorithm). But augmenting paths do not seem to point to groups of vertices different than pairs.
Best regards,
Paweł
There are many theories about calculating of graph similarity such as vertex edge overlap, jacard, co-sine, edit distance, signature similarity, lambda distance, deltacon so on. These things are based on single edge of the graph. But there are many graphs having multiple edges in real world.
Given similar two graphs like above, how could we calculate graph similarity?
Using previous graph similarity, there are only 2-dimension vector and the entry is just scalar that is number, but in multiple edge's graph, the entry should be tuple. Because there are one more actions between nodes. For the previous method, it could be called who-knows-whom schem, but latter graph, it could be said who-knows-whom*-how*. I think the previous mothods could be used for the multiple edge's graph easily, so there aren't logic or methods about it.
Thanks in advance!
There is not "the" way yo compute graph similarity.
Depending on your data and problem, very different approaches may be good. In many cases, simply merging the two edges into one makes perfect sense. For example, if I have two roads of capacity x and y to go from A to B - for many analyses this is comparable to having just one rode, with the combined capacity.
Does anyone have an algorithm for this problem? :
"Suppose we have a directed graph with n vertices and we want to make it acyclic.
How many sets of edges can we pick so that reversing the edges inside any one of those sets will make the graph acyclic?"
I can find the loops inside a graph,but i'm wondering if reversing one of edges of a loop , will produce another loop.
I know that for coloring graph nodes, backtracking/brute force is a common solution. But I was wondering if using DFS I can also achieve a solution ?
Backtracking gives you the opportunity to go back and try other color possibility in order to paint all the nodes with N colors
DFS will start from one node and color it, then jump to its neighbor and color it in different color than its neighbors etc ...
I did a search about using this method but I didn't find an algorithm that uses this.
Question: Is using DFS possible for coloring graph nodes. If yes, is it more efficient than backtracking ?
Thank you
I believe there is some confusion when comparing backtracking and DFS wrt vertex coloring. DFS traversal for a graph gives full enumeration of its vertices in a sequence related to its structure. It does not, however, constitute a full enumeration for the vertex coloring problem, which would require taking into account the possible colors of the vertices.
Thus, if I understand correctly, what you have implemented is a greedy heuristic coloring for a graph directed by DFS.
On the other hand, a backtracking/brute force solution as you name it (such as [Randall-Brown 72]) will provide an exact solution for the minimum coloring problem since it considers every possible vertex coloring. Note that DFS traversal could be used to sort the vertices initially (topological sort) and feed that order to the exact solver.
I want to know what will be an efficient way to implement an undirected weighted graph. I want to perform Prims and Kruskal algorithms on it. I know about adjacency lists but wouldn't that waste memory; for eg. lets suppose I have two vertexes A and B connected by an edge with weight 'x', so I'll need to add two entries in the adjacency list:
A,B,x
B,A,x
Am I missing something?
Adjacency lists are the memory-efficient way of implementing graphs, rather than adjacency matrices.
Actually, you have two options here.
If you want less time and more memory, you should do what you've written.
If you want more time and less memory, you could implement your edges A,B,x where A>B. But then, you would spend a lot of time while getting the adjacent vertices of any vertex.
It's your call. But second bullet is not preferred if you're dealing with less than millions of nodes.
since the graph is undirected I guess you will need only one edge between the nodes A and B