# Tagged Questions

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### Non-Approximate Dynamic All-Pairs Shortest Path algorithm for Undirected, Unweighted Graphs?

I am looking for an algorithm involving adding unweighted edges to an empty, undirected graph (with vertices) and then for each, updating the table of shortest paths. An example is if we have ...
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### Finding a node that is minimal w.r.t. maximum distance to any other node [closed]

Given a graph and an edge in the graph, I want to find a point on this edge that makes the maximum distance from all the nodes to this point minimized. Is there any graph-theory based algorithm that ...
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### Finding independent sets so that all nodes are hit frequently

I have a problem, and I appreciate it if you could share your thoughts. Assume that I have a graph. Assume that I have $k$ iterations. I want to find only one independent set (IS) in the graph in ...
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### Max K-Cut Randomized Algorithm

The simplest heuristic for MAX k-CUT is just to randomly partition V into k sets. If Pb denotes the (random) partition produced and P∗ denotes the optimum partition then it is easy to see that ...
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### Research in Graph Theory versus Graph Algorithms

I have a very generic question to ask. It is related to research. I'm interested in Graph theory. I have done a course in it. I have done some topics related to both graph theory as a point of ...
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### Given a graph, finding if a node has three adjacents from a node subset $N$

Given a graph $G = (V,E)$, assume that we have two disjoint vertex sets $N = \{n_1, n_2 ...\} \subset V$ and $P = \{p_1, p_2, ...\} \subset V$ such that $N \bigcup P \neq V$. I want to find if there ...
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### Choose $n/2$ vertices and guarantee $3/4$ of edges are accounted for proof

Give a polynomial-time algorithm that finds ceil(V/2) vertices that collectively account for at least three-fourths (3/4) of the edges in an arbitrary undirected graph. The algorithm I have come up ...
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### Given a complete, weighted and undirected graph $G$, complexity of finding a path with a specific cost

Given a fully connected graph $G$, suppose that we are searching for a simple path $P$ with a specific cost $c$. Is answering to that problem yes or no equivalent to subset-sum problem? What would ...
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### Is Hamiltonian path NP-hard on graphs of diameter 2?

Let $G$ be a graph of diameter 2 ($\forall u,v\in V: d(u,v)\leq2$). Can we decide if $G$ has Hamiltonian path in poly time? What about digraphs? Perhaps some motivation is in place: the ...
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### Concrete and simple applications for bipartite graphs [closed]

I am looking for concrete and simple problems that may be solved using bipartite graphs or bipartite graph properties. Any idea along with explanations are welcome.
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### Recognizing interval graphs--“equivalent intervals”

I was reading a paper for recognizing interval graphs. Here is an excerpt from the paper: Each interval graph has a corresponding interval model in which two intervals overlap if and only if ...
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### What is the difference between maximal flow and maximum flow?

What is the difference between maximal flow and maximum flow. I am reading these terms while working on Ford Fulkerson algorithms and they are quite confusing. I tried on internet, but couldn't get a ...
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### number of edges in a graph

I got a problem related to graph theory - Consider an undirected graph ܩ where self-loops are not allowed. The vertex set of G is {(i,j):1<=i,j <=12}. There is an edge between (a, b) and (c, ...
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### Reconstruct directed graph from list of ancestors for each node

I have a problem that I encountered that boils down to the following: Considered this directed graph I found on Google: I have the following information available to me ...
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### Prim's Minimum Spanning Tree implementation $O(mn)$ or $O(m+n \log n)$?

I am reading Prim's MST for the first time and wanted to implement the fast version of it . $m$ - The number of edges in the graph $n$ - The number of vertices in the graph Here's the algorithm ...
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### Algorithm to determine a minimal cost graph [closed]

I'm trying to solve this problem: Given a collection of cities and the number of commuters between cities, design a network of roads for minimal cost where cost includes the cost of building the ...
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### How to implement graph search to solve Sudoku puzzle

My teacher pointed out to us during lectures that we could use Graph Search to help us solve Sudoku puzzles which has left me puzzled . I dont see how this is possible as Graph Search is mostly ...
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### Minimizing the following objective function with matrices [closed]

I am trying to work out centrality in a network using Freeman's network centrality. I have an in degree of 83 and an out degree of 110. I want to work out the network centrality using my out degree ...
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### Meyniel's theorem + finding a Hamiltonian path for a specific graph family

Let's say we have a directed graph $G = (V, E)$ for which $(v, w) \in E$ and/or $(w,v) \in E$ holds true for all $v, w \in V$. My feeling is that this graph most definitely is Hamiltonian, and I want ...
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### Why is determining the size of a maximum independent set or a clique in P?

I read that determining the size of the maximum independent set (and also a clique of maximum size) is in P. The versions that find the actual solution are known to be NP-hard. With respect to ...
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### How many times an empty 4-cycle can be counted in an undirected graph?

I have an undirected graph where each node is labelled with an integer key and I'm asked to detect every simple 4-cycle, which can be seen as an empty square (i.e. the two opposite nodes of the cycle ...
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### Proving the correctness of an algorithm, which computes the connectivity of a directed graph

Let $G=(V,E)$ be a directed graph. The connectivity of a graph is the defined as the cardinality of a smallest separator of $G$. A separator of $G$ is a subset $U$ of $V$, such that $G-U$ is not ...
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### Degree conditions sufficient for Hall's theorem

Let $G=(L,R,E)$ be a bipartite graph, are there conditions on the degree of the vertices under which the condition of Hall's theorem is surely satisfied? (meaning a perfect matching exists in the ...