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Is it true that the heaviest edge in a directed graph can not be in the MST of that graph? I don't think it is true because we might end up with a heaviest edge that is not part of a cycle. Can any …

EDIT: This question is different from the other in a sense that unlike it this one goes into specifics and is intended to solve the problem. In the previous post, the only answer was a hint. In this p …

I got it myself... The idea is to set the flows of all edges to 0 one by one. It will take M iterations to do so. Once all edges are set to zero it becomes clear that the highest number of augmenting …

Can you help me with this problem ? Given an undirected graph $G$ and an integer $n$, prove that determining whether the graph has wheel on $n$ vertices $W_{n}$ (a wheel $W_{i}$ is such that $i$ n …

In the following two threads I specified the question in the wrong way (easier to solve that way). Proving that finding wheel subgraphs is NP-complete Reducing from Hamiltonian Cycle problem to the …

Let's say we a have flow network with $m$ edges and integer capacities. Prove that there exists a sequence of at most $m$ augmenting paths that yield the maximum flow. A good way to start thinking …

We are given a graph $G=(V,E)$ with positive edge weights $w_{i}$ and numerical {0,1,-1} labels $l$ for all vertices . We know that $G$ has a subset $G'$ with all vertices labeled 0. The problem is to …

Assume there exists some algorithm that solves vertex cover problem in time polynomial in terms of $n$ and exponential for $k$ with the run time that looks like this $O(k^2 55^k n^3)$. Can we claim th …