Questions tagged [graph-theory]

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

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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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1answer
166 views

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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1answer
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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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1answer
75 views

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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110 views

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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81 views

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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1answer
83 views

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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90 views

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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1answer
629 views

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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1answer
158 views

Minimum cost cut with discount - what is the complexity?

Consider an undirected graph $G=(V,E)$ with non-negative edge costs. Given an integer $k$ with $0\leq k\leq |E|$, let us call an edge set $C\subseteq E$ a $k$-discounted cut, if the following hold: $...
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1answer
160 views

Does Hadwiger conjecture imply that NP = coNP?

(Disclaimer: I suspect the answer is no, but I fail to see why) Here is a nice picture by David Epstein (taken from Wikipedia) illustrating Hadwiger's conjecture: The point is that if in a given ...
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40 views

Reference on generalization of plane graph duality between bonds and simple cycles

I've also asked this question on Mathoverflow, but it hasn't gotten an answer after several months: https://mathoverflow.net/questions/316132/reference-on-generalization-of-plane-graph-duality-between-...
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128 views

Shortest s-t path when is allowed to ignore k weights

Given an undirected graph $G$ with $n$ vertices and $m$ edges, with non-negative weights on the edges, what's the best algorithm that computes the shortest path from $s$ to $t$, where you are allowed ...
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1answer
295 views

Naive definition of treewidth

Treewidth has arguably pretty involved definition. Recently I was thinking about a problem and turns out it easy to solve it for graphs with small ``naive treewidth''. Naive treewidth is defined as ...
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143 views

Is there a standard name for this way of modifying graphs?

Let $G = (V, E)$ be an undirected graph. Let me take an edge $\{x, y\}$ (in blue in the drawing) such that $x$ and $y$ have other incident edges. Among the incident edges we choose one edge $e_x = \{...
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1answer
60 views

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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256 views

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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172 views

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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1answer
152 views

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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124 views

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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1answer
149 views

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 ...
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1answer
145 views

Separating words and graph isomorphism

I wonder if there are any known implications of Babai's recent quasi-polynomial time algorithm for Graph Isomorphism to separating words by DFA's. In both cases the ultimate goal is to differentiate ...
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52 views

Is there an efficient way to reduce a set of graphs under isomorphism?

Suppose we have a set $S$ of (fixed-size) graphs, many of which are isomorphic to each other. How do you find a minimum-size set $M$ of graphs such that every $g\in S$ is isomorphic to some graph in $...
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1answer
153 views

Is balanced Hamiltonian cycle NP complete on maximal plane graphs?

I know that the Hamiltonian cycle is NP complete on the class of maximal plane graphs. If we instead ask about balanced Hamiltonian cycles (i.e. same number of faces on both sides) on maximal plane ...
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1answer
82 views

Pulling a graph across a partition

I am looking for the name for a particular graph property, if it has been studied, and efficient algorithms for computing it, if they exist. I realise that this may be a well known property that I am ...
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1answer
86 views

Cryptography protocols using graph problem instances

I personally am only aware of basic examples of public key cryptography and I haven't studied cryptography yet. I'm curious if there are circumstances in cryptography where using problem instances ...
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132 views

Crime prevention using graph theory and machine learning

I am looking for a way to the model the incidence of crime among a network of individuals. Part of it will use machine learning, and part of it will have to resort to some graph theoretic ...
9
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1answer
319 views

Is there an algorithm that finds the forbidden minors?

The Robertson–Seymour theorem says that any minor-closed family $\mathcal G$ of graphs can be characterized by finitely many forbidden minors. Is there an algorithm that for an input $\mathcal G$ ...
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151 views

Random unbalanced bipartite graphs are good small set expanders

My question is about small set expansion properties of random unbalanced bipartite graphs. Fix a positive $\delta<1/2$, and a positive integers $n,m,d$. Let us call a bipartite graph $\mathcal{G}$...
4
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1answer
79 views

Complexity of finding Exact Size Cut-Sets in Bipartite Graphs

I am interested in the problem of deciding if a cut-set of a given size $k$ (i.e. the number of edges crossing the partitions is $k$) exists in a given bipartite graph (both the graph and $k$ are part ...
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1answer
46 views

Problem property name where an optimal solution in a graph can be used as a solution in any subgraph

Suppose one is given a graph optimization problem where the optimal solution $S$ for the problem on graph $G$ can be used as a solution for any subgraph of $G$. In other words, given $S$ is an optimal ...
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1answer
121 views

Counting/Enumerating Minimal Edge Covers

A Minimal Edge Cover is an Edge Cover such that no other Edge Cover is a proper subset of it. Questions Which is the complexity of counting Minimal Edge Covers? Do we know any non-trivial ...
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1answer
882 views

Number of 4 cycles

Let $C_4$ be a cycle with four vertices. For an arbitrary graph $G$ with $n$ vertices and m edges say $m>n\sqrt n$, how many $C_4$s exist? Is there a lower bound for this?
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51 views

Reduction of irregular graphs, to regular graphs, while preserving hamiltonicity

I am wondering if this is a topic that has had research done... If I could reduce irregular graphs to regular graphs (including replacing redundant node clusters with dummy nodes), while ensuring ...
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1answer
191 views

Could chromatic number be easy to calculate when colouring is hard for some graph class?

Similar question was asked before, but there was an error in it so it was left unanswered Graph class with easy chromatic number, but NP-hard coloring Is there any infinite set of graphs $C$ such as: ...
4
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1answer
162 views

Is graph connectivity definable in existential MSO with vertices and edges?

Can $\exists$MSO$_2$ express graph connectivity? Monadic SO (MSO) is the fragment of second-order logic in which the second-order quantifiers range over relations of arity 1 only. $\exists$MSO is the ...
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99 views

Graphs with minimal-size induced subgraphs

I consider undirected graphs $G = (V, E)$ for which I write $\text{n}(G) := |V|$ the number of vertices and $\text{m}(G) := |E|$ the number of edges. For $d \in \mathbb{N}$, I say that $G$ is $d$-...
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116 views

A variant of hitting set: finding a matching to hit all edge sets

In general, the hitting set problem is given a family $\cal S$ of sub-sets, $\{S_1, \cdots, S_h\}$, and a universal set $U = \bigcup_{i\in [1,h]} S_i$. It asks for a minimum set $H \subseteq U$ ...
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1answer
186 views

Minimum graph cut with constraints

Given an undirected acyclic raph $G = \{V,E\}$, with each edge $e$ having weight $c_e$in the range $[-\infty, +\infty] $, I want to compute a partition of the graph into $N$ disjoint sets $G_i, i=1,......
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1answer
207 views

Paths of length $p$ in a Graph, $p$ a prime

I came across the following decision problem, for which I wondered whether anybody of you came across a similar problem and can give me some insight on its nature/complexity. Given as input a ...
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1answer
114 views

Finding Cheapest n-Path [closed]

Given a weighted directed acyclic graph, how can I find the cheapest path from an Origin Vertex to a Destination Vertex which ...
4
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1answer
278 views

Fast algorithm to find a maximum connected subgraph of k vertices

Given an undirected graph $G = (V, E)$ and a function $f: 2^V \to \mathbb{R}^+,$ where $2^V$ is the set of all subsets of $V$. Find a connected subgraph $T = (V_T, E_T)$ of k vertices such that $f(V_T)...
6
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0answers
102 views

Ramsey number for path in a colored tournament

I am looking about the following Ramseyish theorem with additional structure on tournament (see [1]). For all number of colors $k$, there exists $n=T(k)$ such that all tournaments of size $n$ ...
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2answers
106 views

Looking for an algorithm to construct a graph from two subgraphs

I am looking for an algorithm to construct a graph from two subgraphs. The problem is as following: Given two graphs g1(V, E) and g2(V, E), find a graph G(V, E) where V(g1) ⊆ V(G), V(g2) ⊆ V(G), E(g1)...
3
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1answer
125 views

Mutually exclusive replacement paths, an existing problem?

A replacement path is a simple path allocated to an edge $e \in G$ that connects the endpoints of $e$ in $G \setminus \{e\}$. The problem An undirected graph $G$ is given and the task is to allocate ...
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1answer
61 views

How to make any graph 2-degenerate?

I have to show a PPT(polynomial time reduction) from 'Colorful graph Motif' to '2-Degenerate Steiner Tree'. As input graph should be 2-degenerate, but here is normal graph G (that is, basically an ...
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49 views

Complexity of extending $P_4 $-partition of cubic graphs

This is a question I posted on MathOverflow before but never got an answer. I am cross-posting it here. Surprising phenomena occurs when we want to extend a partial solution of some easy problems. We ...
4
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2answers
159 views

Check if graph stays connected after edge swap

Checking whether a (simple, undirected) graph is connected can be done in linear time in the number of edges. What I am looking for is a more efficient way of checking whether it stays connected after ...
3
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1answer
296 views

What exactly is Lawler's modification to Yen's algorithm and how does it work?

I recently read about Yen's algorithm, I understand the algorithm and it seems correct, however Wikipedia mentions that there exists "Lawler's modification" to the algorithm, which is described as ...