Questions tagged [graph-theory]

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

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7
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1answer
44 views

Complexity of finding an edge set yielding specified vertex degrees

I'm trying to figure out if the following two problems are known in general to be in P or NP-complete: Q1: Given a graph $G=(V,E)$ and integers $d_i,\,1\leq\,i\leq|V|$, does there exist a subset $E'...
3
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31 views

$NP$-Completeness of $\epsilon$-balanced graph parittioning for fixed $\epsilon$

Consider this graph partitioning problem: Let $G = (V, E)$ be a simple undirected graph and $0 \leq \epsilon \leq 1, M \geq 0$ be constants. Are there disjoint subsets $V_1, V_2$ with $V = V_1 \cup ...
9
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74 views

NP-Hardness of 4-cycle packing problem in complete bipartite digraph?

A directed complete bipartite graph is a bipartite graph where there is exactly one directed edge between any two vertices from its two different parts. In other words, it's an orientation of a ...
7
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106 views

Algebraic methods for testing planarity

Mac Lane's planarity criterion states that a graph is planar if and only if it has a cycle basis such that each edge is contained in at most two cycles, we call such a basis a 2-basis. A 2-basis of a ...
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18 views

Efficient Graph Affinity Matrix Computation

I need to compute an affinity matrix for an unweighted undirected graph of related musical artists for the purposes of spectral clustering. Now, the most obvious affinity measure to use is shortest ...
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39 views

Complexity of recognizing unit distance graphs

A Graph is Unit Distance Embeddable (UDE) if it can be embedded in the plane such that every edge has a length of 1. A minor of a UDE graph is also UDE so by the Graph Minor theorem, there must be a ...
6
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1answer
176 views

Complexity of finding the largest induced subgraph with all even degrees

What is the complexity of the following problem? Instance: Simple, undirected graph $G$, and a positive integer $k$. Question: Does $G$ have an induced subgraph on at least $k$ vertices, such that ...
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36 views

Scheduling and routing

Given: $k > 1 $ sales execs, each specializing in one of 4 lines of business (LOBs), where each exec works (sales and travel) at 7.5 hours / day $n > 1$ client sites. Constraints: Each ...
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15 views

Capacitated Vehicle Routing Problem with multiple pickups at some sites but constrained to max 1 pickup per vehicle per site

I am looking for literature or code for a variant of capacitated vehicle routing problem. The variance is that some sites have multiple items for pickup, but there is a constraint such that even if a ...
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128 views

How much does treewidth changes after removal of a path?

Let $G$ be a graph such that $\mathrm{tw}(G)=t$. Let $t' = \min\limits_{u,v \in V(G)} \max\limits_{P \text{ is a path from } u \text{ to } v} \mathrm{tw}(G - P)$. Then how small $t'$ can be? My ...
6
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2answers
151 views

Computing the edge orbits of a graph (and discussing definitions)

A (vertex) automorphism in a graph $G=(V,E)$ is a permutation $\sigma$ of the vertices that preserves adjacency, namely $\sigma(u) \sigma(v) \in E$ if and only if $uv \in E$. The automorphisms of a ...
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1answer
97 views

conversion to DAG

Can we reverse directions instead?
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168 views

Lighting up all elements of a poset by toggling upsets

I consider the following game on a finite poset $(P, <)$. At each point of the game, I have a set of elements $S$ of the poset which are "on", and all others are "off". Initially $S = \emptyset$. ...
2
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0answers
88 views

How hard is it to determine ex(n,G)?

Define the extremal Turán function $ex(n,G)$ of a graph $G$ as the most edges a graph on $n$ vertices can have without having a subgraph that is isomorphic to $G$. This function is known ...
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1answer
87 views

Finding simple fixed length paths in directed graphs

Is there an efficient algorithm to enumerate unique simple fixed-length paths (of size $k$) in directed graphs? What would be its time complexity?
0
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1answer
148 views

Find the shortest s-t trail(edge disjoint path) in a graph with negative weight edges

A walk in a graph is a finite or infinite sequence of edges which joins a sequence of vertices. A trail is a walk in which all edges are distinct. Note that a trial may visit a vertex multiple times ...
1
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1answer
114 views

Breaking cycles in network graph by adding nodes and rerouting edges

I have a quite "common" need : making a directed graph (with one or several cycles) a directed acyclic graph (DAG). But the way I want to achieve it is, I guess, way more specific : I would like to ...
9
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1answer
119 views

The source of the modular decomposition graph

When introducing graph modular decomposition, most authors use the 11-vertex graph, which I copy from wikipedia. The question is who is (are) the original designer of it. (I'm not asking who drew ...
0
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1answer
69 views

Dividing a complete graph into two cliques with maximal sum of edge weights

Problem: Considering a complete weighted graph $G$ with $n$ vertices, where $n\in2\mathbb Z$ is an even number, remove edges in such a way that you end up with two cliques of graph $G$, each having $\...
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101 views

Network Reliability Problem

Network reliability, in which we are given an undirected graph $G$ with a failure probability $p_e$ for each edge and we are asked to calculate the probability that the network becomes disconnected ...
9
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2answers
273 views

How long does it take to find a short cycle in a random graph?

Let $G \sim G(n, n^{-1/2})$ be a random graph on $\approx n^{3/2}$ edges. With very high probability, $G$ has many $4$-cycles. Our goal is to output any one of these $4$-cycles as quickly as ...
10
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2answers
180 views

“Relatives” of the shortest path problem

Consider a connected undirected graph with non-negative edge weights, and two distinguished vertices $s,t$. Below are some path problems that are all of the following form: find an $s-t$ path, such ...
3
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122 views

Enumerating Minimal (a,b) vertex separators in a DAG

A vertex subset $S \subseteq V$ is an $(a,b)$ separator for nonadjacent vertices $a$ and $b$ if the removal of $S$ from a graph $G$ separates $a$ and $b$ into distinct connected components. $S$ is a ...
2
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1answer
107 views

Relationship between $O(\log n)$ (bounded) treewidth and H-minor-free

What is the relationship between graphs which have $O(\log n)$ treewidth and $\mathcal{H}$-minor-free graphs? Are graphs which have $O(\log n)$ treewidth $\mathcal{H}$-minor-free? I know that graphs ...
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60 views

Generalization of k-Coloring: maximizing the number of vertices with no neighbours of same color

One can consider the following generalization of the $k$-Coloring problem: Let be given a graph $G$ and an two integers $k$ and $p$. A vertex $v$ of $G$ is properly colored if $v$ has no neighbour ...
7
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2answers
192 views

Lower bound on pebbling numbers

Out of curiosity, I tried finding the original paper showing that there are graphs that require $n/\log n$ pebbles in the sense of Hopcroft, Paul, and Valiant’s seminal paper “On Time Versus Space”. (...
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78 views

Expected value of a random experiment in a graph

I need to find the expected value of R in the random experiment below. $$ R = \frac{1}{K} \sum_{C \in \mathcal{H} } \ [\frac{1}{2} |V(C)| * (|V(C)| - 1) - |C|] $$ $\mathcal{H}$ is a partition on ...
2
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0answers
51 views

Is minimal cover under symmetric 3-deduction NP-complete?

Forgive me if this problem is known by another name, I do not know any references for it. Symmetric deduction. An equation $e \in E$ is a subset of variables $V$ such that knowing $|e| - 1$ of the ...
3
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0answers
110 views

Earliest forbidden subgraph characterisation

I wonder, what was the first non-trivial graph class for which there was a forbidden (induced) subgraph characterisation ? Of course, bipartite graph is one example but I am considering it as trivial ...
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1answer
59 views

Are there digraphs such that any two arborescences are arc-disjoint?

Let $D=(V,A)$ be a directed graph with root $r$. An $r$-arborescence of $D$ is a subgraph such that for any $v\in V-r$, there is exactly one directed path from $r$ to $v$. Hence an $r$-arborescence is ...
2
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0answers
72 views

Algorithm for computing the smallest subset of nodes to remove from a graph to make it a tree

I have encountered an interesting problem that I couldn't find any references to solve: Determine the smallest subset of nodes that need to be removed from an undirected graph to make it a tree. ...
0
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0answers
44 views

Matching of two weighted graphs allowing one-to-many mapping

I am looking for a heuristic for a graph matching problem as follows. Given two graphs: $A$ (consisting of nodes $a_i$) and $B$ (consisting of nodes $b_i$). Typically the size of $B$ is larger than ...
0
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0answers
59 views

s,t-Graphs representing infinite number of addition chains

I am looking at directed acyclic multi-graphs $G=(V,E)$ with a single source and sink with integer labeled arcs. Each vertex has exactly two inputs except $s$. Each vertex has at least one output ...
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50 views

Directed Acyclic Graph partition into minimum subgraphs with a constraint

I have this problem, not sure there is a name for it, wherein a Directed Acyclic Graph has different colored nodes. The idea is to partition it into minimum number of subgraphs with the following 2 ...
2
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0answers
76 views

Common techniques for the acyclic orientation problem under some special constraint?

An acyclic orientation of an undirected graph is an assignment of a direction to each edge(an orientation) that does not form any directed cycle and therefore generates a directed acyclic graph(DAG). ...
10
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1answer
215 views

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 ...
2
votes
1answer
69 views

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 ...
3
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0answers
74 views

Embed graph in $\ell_2$ space so that edge and non-edge distances are separated by a constant factor

Suppose I have an undirected unweighted graph $G = (V,E)$. Is there a way to compute points $x_v \in \mathbb{R}^d$ for each vertex $v \in V$ such that $||x_v - x_u|| = 1$ whenever $(u,v) \in E$ and $ |...
11
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0answers
332 views

A dynamic data structure to list triangles

Consider an undirected graph with $n$ nodes. Is there an efficient data structure that supports the following operations? Insert an edge into the graph Delete an edge from the graph Given a query ...
2
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0answers
84 views

Which computational framework lies behind the Chinese “Social Credit System”?

BACKGROUND The Social Credit System is a data-driven reputation system which draws on several sources to label various entities, namely businesses and individual citizens, with a trustworthiness ...
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68 views

Unknown gaps in computation models

I'm looking for computatuon models where it is known that there are problems that we can solve in time T1 and T2. where T1 is smaller then T2 and it is unknown if there are problems where their ...
0
votes
1answer
87 views

finding maximum weight subgraph

My graph is as follows: I need to find a maximum weight subgraph. The problem is as follows: There are n Vectex clusters, and in every Vextex cluster, there are some vertexes. For two vertexes in ...
3
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0answers
48 views

Problem of determining if a $4$ connected graph has $k$ Hamiltonian cycles

Definition: Define the $k$-HamiltonianCycles problem as the decision problem that asks if a given graph has at least $k$ distinct Hamiltonian cycles. Question: Is there some constant $k$ so that the $...
-1
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1answer
51 views

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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0answers
56 views

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 $w_{\sigma(i),\sigma(j)}=w_{ij}$ ...
2
votes
1answer
164 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 ...
0
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0answers
19 views

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 ...
0
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1answer
33 views

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 ...
0
votes
1answer
69 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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1answer
109 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 ...