Questions tagged [graph-theory]

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

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5
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1answer
87 views

Known property? Maximum radius of connected induced subgraph

I was wondering if the following graph property has a name and has been researched: Consider any connected induced subgraph $H \subseteq G$. Then $r(G)$ denotes the maximum radius of any such $H$. I ...
-2
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1answer
84 views

Minimum vertex cover and odd cycles

Suppose we have a graph G without odd cycles. Consider the minimum vertex cover problem of G formulated as a linear programming problem, that is for each vertex $v_{i}$ we have the variable $x_{i}$, ...
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0answers
72 views

Fast Computation of First k Eigenvectors of Graph Laplacian

I'm looking for references for the following; Given the unnormalized Laplacian matrix of a graph $L = D - A, ~ L \in \mathbb{R}^{n\times n}$ Algorithm(s) to efficiently estimate the first $k$ (...
3
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0answers
81 views

Isomorphic subforest problem

I recently read that the following problem is NP-Complete: Given a tree $T$, and a forest $F$, is there a subgraph of $T$ isomorphic to $F$? The reference provide was to the textbook “Computers and ...
12
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1answer
862 views

Consequences of $\oplus \mathbf{P} \subseteq \mathbf{NP}$

I have part of a proof attempt of $\oplus \mathbf{P} \subseteq \mathbf{NP}$. The proof attempt consists of a Karp reduction from the $\oplus \mathbf{P}$-complete problem $\oplus$3-REGULAR VERTEX COVER ...
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0answers
74 views

How to approach the “traveling salesman problem” with cost changing every time salesman reaches a new city

Let's say instead of finding the shortest path we have to maximize the profit in a year of the salesman under the following constraints. Salesman can go to a different city only on weekends, all ...
4
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1answer
113 views

Hardness of finding if a vertex lies on a simple directed path between two vertices

Given a directed graph $G = (V, E)$ and three vertices $u, v, w \in V$. Is it NP-Hard to find whether there is a simple path from $u$ to $v$ passing through $w$? I found a couple of hardness ...
0
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1answer
54 views

Reduction graph to planar bounded treewidth and bounded diameter graph

We got reduction graph to planar bounded treewidth graph, but this is unlikely to be true. Let $H$, the planarizing gadget, be planar graph with four distinguished vertices $u,u',v,v'$ on the outer ...
5
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1answer
176 views

Is this a known problem, and is it #P-complete?

Let $G=(V,E)$ be an undirected graph. What I call a selection function of $G$ is a partial function $f:V \to E$ such that for every node $v$, if $f(v)$ is defined then it is one of the adjacent edges ...
2
votes
1answer
47 views

Maximum subgraph problem with unknown complexity

Let $Q$ be a polynomial time decidable graph property. In a graph let us call a subgraph $S$ a $Q$-subgraph, if $S$ has the property $Q$. Consider the following optimization problem: Maximum $Q$-...
1
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1answer
46 views

Existence of graphs of every order related to Barnette’s conjecture

Consider the class C3CBP of $3$-connected cubic bipartite planar graphs. They form the class on which the (in)famous Barnette’s conjecture is based. My interest in C3CBP graphs is somewhat orthogonal ...
6
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2answers
143 views

$NP$-Completeness of $\epsilon$-balanced graph partitioning 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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2answers
111 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'...
12
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0answers
119 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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0answers
126 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 ...
6
votes
1answer
188 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 ...
13
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1answer
4k views

What is the correct definition of $k$-tree?

As the title says, what is the correct definition of $k$-tree? There are several papers that talk about $k$-trees and partial $k$-trees as alternative definitions for graphs with bounded treewidth, ...
5
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0answers
133 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
157 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 ...
7
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0answers
175 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 $S = \emptyset$. ...
8
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2answers
472 views

Dijkstra parallelization

I'd like to know what is the best method to parallelize the Dijkstra algorithm. Thanks.
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1answer
100 views
22
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8answers
3k views

NP-hard problems on paths

everybody knows there exist many decision problems which are NP-hard on general graphs, but I'm interested in problems that are even NP-hard when the underlying graph is a path. So, can you help me to ...
1
vote
1answer
119 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 ...
2
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0answers
90 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 ...
22
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5answers
9k views

Vertex Cover applications in the real world

What applications does the Vertex Cover Problem have in the real world? Which industry or research projects use actually implemented software that is based on theoretical results for the Vertex Cover ...
0
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1answer
155 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
99 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?
9
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1answer
121 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 ...
25
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1answer
676 views

Regularity Lemma for Sparse Graphs

Szemeredi's Regularity Lemma says that every dense graph can be approximated as a union of $O(1)$ many bipartite expander graphs. More accurately, there's a partition of most vertices into $O(1)$ sets ...
0
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0answers
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 ...
0
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1answer
73 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 $\...
9
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2answers
274 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
183 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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0answers
129 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
votes
1answer
110 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 ...
0
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0answers
61 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 ...
2
votes
3answers
3k views

Animate a graphviz graph

I'm exploring some graph algorithms, and would like to animate the graph change over time (e.g. when adding a heap node or balancing a tree). Is there a nice way to animate a sequence of ...
7
votes
2answers
194 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”. (...
2
votes
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 ...
0
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0answers
79 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 ...
9
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1answer
432 views

Complexity of k-clique for hypergraphs

Classic Problem: Let a number $k$ be given. The $k$-clique problem is as follows. Given a graph $G$, does there exist a subset $S$ of $k$ vertices so that any two vertices of $S$ are adjacent? ...
1
vote
1answer
1k views

directed or bidirected in relation to mssp (Multiple source shortest path)

Firstly I wanted to ask. If I have a undirected graph and split all the edges into two directed edges is it still called directed or does it become bi-directed? The main question is i have a graph ...
2
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0answers
53 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 ...
8
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1answer
450 views

Approximation algorithms for Directed Minimum Cut with Cardinality Constraints

We would like to know whether there are any known approximation results for the cardinality constrained minimum $s$-$t$-cut on directed graphs. We weren't able to find any such result in literature. ...
3
votes
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 ...
1
vote
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
votes
0answers
74 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
45 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 ...
10
votes
1answer
216 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 ...