# Questions tagged [graph-theory]

Graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects.

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### Graphs with pairwise distance of log*n

I'm looking for example of graphs where some vertices will have $\log^*n$ distance, where $n$ is the number of nodes in the graph. Not even sure how it should look like. Here $\log^* x$ denotes the ...
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### Efficient graph isomorphism for similar graph queries

Given the graph G1, G2 and G3, we want to perform isomorphism test F between G1 and G2 as well as G1 and G3. If G2 and G3 are very similar such that G3 is formed by deleting one node and inserting one ...
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### Find all perfect matchings in a general undirected graph

Is there an algorithm which can compute all perfect matchings of a general undirected graph (assuming they exist)?
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### Determine if a graph has exactly 1 cycle using a SAT solver

I have a connected undirected graph whose edges are either enabled or disabled. I want to create a set of clauses that are SAT iff all enabled edges are part of a single loop. If I assert that each ...
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### Computing the existence of a path in a code execution graph

I have a need for an algorithm which I can express as a reachability problem in a graph. Note that I'd appreciate any advices with respect to better wording this question. Also please tell me if this ...
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### Min Cut with Vertices

I have an undirected graph G with a set of vertices and edges. Each vertex has a weight w. Let's assume we have all vertices connected with some paths. I'm looking ...
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### Graph automorphism with prescribed values

Consider a graph $G$ with vertices labeled $1,...,n$ and edge weights $w_{ij}$. Recall an automorphism of G is a permutation $\sigma$ of the vertex labels such that $w_{\sigma(i),\sigma(j)}=w_{ij}$ ...
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### How many samples are needed to reconstruct a path?

Consider an input set of vertices $V$ and vertices $s,t\in V$. The goal is to learn some unknown shortest path from $s$ to $t$; the set of edges of the graph is hidden at first and there may be ...
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### The set of weight functions for which the assignment problem has non-trivial solutions

The standard assignment problem is specified with a square matrix ${\bf W}$ of weights (values, costs): $$V_{\cal P} = \sum_i w(i, b(i)) = \sum_{(i, j) \in {\cal P}} w_{ij},$$ where $\cal P$ is a ...
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### Graph path problem [duplicate]

I am trying to solve one graph traversing problem which might be classical to guys who are familiar with the topic. However, I am not. I have directed graph where nodes are cities and plane can fly ...
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### Finding a Hamiltonian cycle from perfect matching of a bipartite graph

A disjoint vertex cycle cover of G can be found by a perfect matching on the bipartite graph, H, constructed from the original graph, G, by forming two parts G (L) and its copy G(R) with original ...
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### Finding the maximum no. of people who get along in a group [closed]

Suppose that there are 15 people in a room. Assume that each person gets along with other people in the room (but not everyone). (Note that the "feeling is mutual" between any two people who are ...
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### Minimum cut with nonlinear objective function

Let $G$ be an undirected graph. The classic minimum (cardinality) cut problem asks for a cut $C\subseteq E(G)$, such that $|C|$ is minimum. Let us generalize it the following way: let $f$ be a ...
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### Densest k subgraph problem for outerplanar graphs?

The densest k subgraph problem aims to find a subgraph $H$ of a graph $G$ with exactly $k$ vertices that maximizes the number of edges $|E(H)|$. Does anyone know if there exists a polynomial-time ...
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### Counting quotient graphs, but not exactly

All graphs considered will be directed graphs $G=(V,E)$, with $E \subseteq V \times V$ (so possibly with self-loops). For $k \in \mathbb{N}_{\geq 1}$, I will write $[k]$ the set $\{1,\ldots,k\}$. A $k$...
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### Number of simple paths between two vertices in a DAG

Let $G = (N, A)$ be a connected acyclic digraph (DAG). Furthermore, let $s \in N$ and $t \in N$ be two vertices on this graph, such that $t$ is reachable from $s$. My problem is: how many simple $s-t$...
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### Counting sum of parities of cycle covers in cubic, planar, bipartite graphs

Let $G$ be a cubic (i.e. every degree exactly three), planar, bipartite graph. By Hall's theorem its edges can be partitioned into three perfect matchings. Take any such partition $M_0,M_1,M_2$ and ...
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### Generating a random connected bipartite graph

A (n, m, k)-bipartite graph is a bipartite graphs with: independent sets of size $\{n, m\}$ a total of $k \geq n+m-1$ edges We want an algorithm to generate a (n, m, k)-bipartite selected uniformly ...
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### What's the fastest known algorithm for finding the diameter of a graph?

Given a positively weighted graph what's the fastest algorithm for finding the diameter for that graph?
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### Viola's Reduction of 3XOR to listing triangles

Apparently this was due to Pătraşcu, but in this report on the ECCC server, Viola states that 3XOR can be reduced to listing triangles. Assume that given a graph in adjacency list format, with $m$ ...
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### Star seperators to explain computational complexity of algorithms on a class of graphs?

A lot of NP-hard optimization problems on graphs which are perfect become solvable in polynomial time. Unfortunately, the class of graphs that arise in my problem are not perfect. The graphs can be ...
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### maximize edges minus vertices in a weighted graph

for a given weighted vertices and edges graph, we want to find the maximum subgraph. the maximum subgraph is made of some vertices and some edges of the given graph which sum of the edges minus sum of ...