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Questions tagged [graph-theory]

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

6
votes
1answer
131 views

Path in a graph with durations [closed]

I have the following problem: given a directed graph $G=(V,E,d)$, where $d:V\to\mathcal{I}(\mathbb{Q}_0^+\cup\{+\infty\})$ (here $\mathbb{Q}_0^+$ denotes the set of non-negative rationals and $\...
3
votes
0answers
78 views

PTIME or NP-Hardness of stochastic objective function

I will begin by linking a previous post where I asked a general question for a stochastic setting which I describe below. It turns out that my "proof" for a restricted case had a mistake and there is ...
4
votes
0answers
83 views

Showing hardness of maximizing stochastic objective function over graph

Consider a graph $G = (V, E)$ with $n$ vertices and $m$ edges. Each vertex $v_i$ can take positive value $a_i$ with probability $p_i$ and value $0$ with probability $1-p_i$. The challenge is to ...
3
votes
1answer
109 views

Results/concepts that also proved useful outside of their “home areas”

There are some results/concepts in TCS which are used in areas other than the "home area" where they emerged. For example, NP-completeness has complexity theory as its home area, but it is also used ...
1
vote
2answers
179 views

Is perfect matching for bipartite graph with no cycles unique?

Given a balanced bipartite graph that satisfies Hall's theorem (is non singular) then it shown that it has at least one perfect matching. My question is if the balanced bipartite graph is also ...
1
vote
0answers
46 views

Generating random labelled trees

I am looking for a simple rejection-free algorithm to uniformly sample random labelled trees (i.e. to generate each of them with the same probability). One possibility is to generate Prüfer sequences ...
1
vote
0answers
75 views

Is NP-complete the existence of paths of a given length in a directed graph? [closed]

Given a directed graph G= (V,E), a pair of vertices s and t, a natural number K encoded in binary, whether the problem to decide there exists a path (not necessarily simple) from s to t of length K is ...
-1
votes
1answer
227 views

Closeness Centrality for Weighted Graphs

In order to determine the Closeness Centrality for a vertex u in a graph, you compute the shortest path between u and all other vertices in the graph. The centrality is then given by: $C(u) = \frac{1}...
0
votes
0answers
90 views

Path Finding: single-source, multi-path, multi-target, and max-depth - approaches and application

Background Definitions (as used here): $\qquad$single-source: for path finding, an algorithm is single-source if it searches from a given node. $\qquad$multi-target: for path finding, an ...
1
vote
0answers
59 views

Finding a Minimal k-Subgraph

Given a complete, positively weighted, bidirectional graph with $n$ nodes without self-loops. Hence the corresponding adjacency matrix $A$ is positive, symmetric, and has zero main diagonal. I am ...
0
votes
1answer
95 views

Is there any better than (2/k)-approximation algorithm for Independent Set in Coloring graph?

I would like to know any results in terms of approximation algorithm about Maximum weight Independent Set problem in coloring graph. Input: Given a graph G with non-negative vertex weights and ...
6
votes
0answers
72 views

Complexity of bounded degree full contraction

This paper defines the problem $\mathrm{B{\scriptsize OUNDED} \ D{\scriptsize EGREE}\ C{\scriptsize ONTRACTION}}$ as follows: Instance: A graph $G$ and two integers $d$ and $k$. Question: Is there a ...
2
votes
0answers
116 views

Hardness of Approximation for minimum path cover in an undirected graph?

Given an undirected graph $G = (V,E)$, a path cover is a set of disjoint paths such that every vertex $v\in V$ belongs to exactly one path. The minimum path cover problem consists of finding a path ...
1
vote
1answer
98 views

How good of an approximate 2-coloring can you get of the halved cube graph?

We say that a 2-coloring $col : V_G \rightarrow \{0, 1\}$ of a graph $G$ is $\epsilon$-approximate if $Pr_{(w, v) \in E_G}(col(w) \neq col(v)) \geq \epsilon$. For every $n$, what is the maximum $\...
2
votes
1answer
96 views

What is the name of a graph with local clustering coefficients equal to zero?

I am struggling to find in literature a name for a kind of graph where all local clustering coefficients are equal to zero (or, at least, bounded). For instance, domino, a subset of cacti and all ...
0
votes
0answers
81 views

Find the maximum induced (weighted) subgraph with edge weights greater than some minimum

I have a (fully connected) weighted undirected graph. I want to find a maximal induced subgraph whose edge weights are all above some minimum value. Or, if not a maximal subgraph, then with some ...
10
votes
2answers
211 views

Common insights into hypothetical complexity of graph problems

I came across two examples of hypothetical hardness of some graph problems. Hypothetical hardness means that refuting some conjecture would imply the NP-completeness of the respective graph problem. ...
2
votes
1answer
87 views

Is Eulerian Path (or Eulerian Cycle) definable in Monadic Second Order Logic?

Does there exist a monadic second order logic formula which is satisfied by a graph if and only if it has an Eulerian path (or Eulerian cycle). I am looking for properties of graphs which are ...
16
votes
1answer
398 views

What is the complexity of this graph problem?

Given a simple undirected graph $G$, find a subset $A\neq \emptyset$ of vertices, such that for any vertex $x\in A$ at least half of the neighbors of $x$ are also in $A$, and the size of $A$ is ...
0
votes
0answers
28 views

Partitions of regular graphs with upper bounds on bipartition width

Are there efficient graph partitioning algorithms with guaranteed upper bounds on the bipartition width in terms of the total number of vertices of the graph, or another non-spectral quantity (...
3
votes
0answers
100 views

Eigenvalues of adjacency matrix of a connected bipartite graph

Let $G=(V,E)$ is a connected d-regular bipartite graph with the same number of vertices on both sides of the bipartition. It's known that that the largest eigenvalue of its adjacency matrix would be d,...
-5
votes
1answer
122 views

Does any DAG can be topologically sorted? [closed]

I am not good enough in computer science. My intention is to solve some programming problem in terms of DAG's. The key point is that before getting them into database, I need run "topological sort" in ...
3
votes
1answer
165 views

Natural (well studied) classes of graphs with treewidth $\Theta(n^\alpha)$ with $1/2 < \alpha < 1$

Which natural (well studied) classes of graphs have treewidth that scales as $\Theta(n^\alpha)$ in the number $n$ of vertices, with $1/2 < \alpha < 1$?
-1
votes
1answer
35 views

Maximize graph with k cut edge operations

I have undirected graph with N nodes each with some weight. There are M edges and in exactly K operations I want to maximize the XOR sum of connected components of the graph. ((n1 XOR n2 XOR n3) + (c1 ...
4
votes
0answers
112 views

Reduce $m$-clause 3SAT to PLANAR-3SAT in $O(m^{2-\varepsilon})$

The classic reduction from 3SAT to PLANAR-3SAT requires a removal of $O(m^2)$ crossings from a rectilinear representation of 3SAT with $m$ clauses. However, the crossing number inequality suggests ...
3
votes
0answers
109 views

Practical polynomial-time implementation of bounded degree graph isomorphishm

There's a well-known article for solving graph isomorphism problem in polynomial time. Many other articles on the subject of isomorphism mention it as a possible "alternative", but note that is not ...
5
votes
2answers
221 views

Efficient way to generate random planar cubic bipartite graphs

3-regular bipartite planar graphs appear in a variety of NP- / #P-complete problems. Suppose one wants to test the complexity of these problems via numerical experiments. Is there an efficient way to ...
1
vote
2answers
85 views

reference for a special modular decomposition

For modular decomposition. X is a module if all members of X have the same set of neighbors among vertices not in X. I need a special modular decomposition. X is my module if all members of X have ...
3
votes
1answer
121 views

Will core decomposition get a maximal clique?

I have read David Eppstein's paper about maximal clique enumeration by using degeneracy order. It has mentioned the core decomposition, which is removing the vertex with the smallest degree ...
1
vote
0answers
25 views

Complexity of recognizing generalized graph join

A join of two graphs is the union of both graphs with additional edges such that every vertex of the first graph is connected to every vertex of the second graph. There is a generalization of this, ...
2
votes
1answer
65 views

Are equally weighted MSTs closely related?

Suppose we have an undirected connected graph $G=(V,E)$ that has several minimum spanning trees. We say two trees $T_1, T_2$ are connected if they share exactly $|V|-2$ edges(*). In other words $T_1$ ...
2
votes
0answers
110 views

Maximum weight non-overlapping paths in a DAG

Suppose we have a weighted DAG $G$. A $m$-path-tuple is defined as $(P_1, ..., P_m)$ in which $P_i$ is a path on the graph, and no $P_i$ and $P_j$ share any edges. In other words each edge of the ...
0
votes
0answers
123 views

Minimum cut on a directed graph with negative term

Suppose we are given a directed Graph $G=(V,E)$ and there is a nonnegative weight $w(u,v)$ is defined for the edge from $u$ to $v$. The task is to partition vertices to $A$ and $B$ (partition means $V ...
1
vote
0answers
75 views

Counting the number of connected components in a dynamic plane graph

I'm working on the following problem: let $G = (V, E)$ be a connected, planar graph. Our goal is to find a $d$-partition of $G$, $P = \{V_1, \ldots, V_d\}$, such that $G[V_i]$ is connected, and $\min_{...
0
votes
0answers
67 views

Why do spectral ultrasparsifiers need to be trees

Given a graph $G$ with $n$ vertices and $m$ edges, a $(\kappa,h)$-ultra-sparsifier $H$ is a graph on the same set of vertices such that: [1] $H$ consists of a spanning tree $T$ along with $\frac{hm}{\...
2
votes
0answers
81 views

Unbalanced connected partition

Let $G = (V, E)$ be a connected graph with (possibly negative) vertex weights $w(v)\in\mathbb{Z}$. We want to partition the vertices into two parts such that the induced graphs $G'$ and $G''$ are ...
6
votes
1answer
314 views

Why isn't the Charikar algorithm for finding the densest subgraph optimal?

I read about the algorithm in Greedy Approximation Algorithms for Finding Dense Components in a Graph by Moses Charikar, and I tried to find an instance/graph where the algorithm returns a different ...
-4
votes
1answer
42 views

Is a k-connected graph also k-1 connected?

So I'm reading about k-connected graphs, and I'm a little confused about them. This is the main definition I've seen: A graph is k connected if and only if for any distinct x, y vertices in the graph,...
0
votes
0answers
143 views

Data structure to search name of files and get its path

I will be inserting names of files in a dynamically way, approximately till 1 billion of names. Besides, I do also want to store the path where the files are located in order to do the following ...
7
votes
1answer
125 views

Random Deterministic Automata

I am familiar with the term of random graphs, such as $G(n,p)$- a distribution over simple undirected graphs with $n$ vertices, where each edge appears in a graph w.p. $p$. That is, each graph $G=(V,E)...
4
votes
1answer
109 views

Properties of toroidal graph

I am interested in work pertaining to graphs that have genus 1 i.e. toroidal graphs. Specifically, i am trying to find answers to the questions below. Since toroidal graphs can be recognized in $P$ , ...
1
vote
1answer
295 views

Common terminology used for lower/upper bounds

Suppose you have developed an upper bound on the number of vertices of a particular graph. This bound is the best possible bound that can be found for any given instance. What do you call such a bound?...
7
votes
3answers
359 views

What are graph grammars?

I have found information on graph grammars and graph rewriting, but the papers that I find on it are a bit thick. Can someone give a quick overview of what graph grammars are, as well as an overview ...
12
votes
1answer
227 views

For any two non-isomorphic graphs $G, H$, does there exist a polysize, polylog quantifier depth first order formula which witnesses this?

I want to be very specific. Does anyone know of a disproof or a proof of the following proposition: $\exists p \in \mathbb{Z}[x], n, k, C \in \mathbb{N},$ $\forall G, H \in STRUC[\Sigma_{graph}] (...
4
votes
0answers
126 views

Girth of graphs that decompose into two disjoint union of spanning trees

Let $G$ be an $n$-node graph that can be decomposed into two disjoint union of spanning trees. In particular, $G$ has $2n-2$ edges. It is not hard to show that the girth of $G$ is at most $O(\log n)...
8
votes
1answer
164 views

Complexity of finding semi-ordered Eulerian tours in a 4-regular graph

I'm trying to figure out the time-complexity of the problem I describe below, which I call the semi-ordered Eulerian tour problem or the SOET problem. Either finding an efficient algorithm for this ...
2
votes
2answers
209 views

One Generalization of Graph Isomorphism Problem

Say I generalize the language which consists of pairs of isomorphic graphs to take the following form: $GI(f) = \{ (G, H) \mid \exists \psi : V_G \rightarrow V_H, Pr_{a, b \in V_G} ((a, b) \in E_G \...
0
votes
0answers
53 views

Preference based group generator

I'm looking for a polynomial time algorithm for the following problem: $n$ number of students must be partitioned into $\frac{n - n\%m}{m}$ sets where $m$ is the ideal cardinality. Each pair of ...
-2
votes
1answer
80 views

Algorithm finding path with maximal ratio of white vertices [closed]

Recently I encountered an interesting graph problem and couldn't find proper solution: given undirected graph G = <V, E>, each vertex is either white or black....
5
votes
1answer
109 views

Minor and subdivision

It is a well known fact that if $H$ is a graph of maximum degree 3, then $H$ is a minor of a graph $G$ if and only if $H$ is a topological minor of $G$. Moreover, a graph $G$ has one of $K_{3,3}$ or $...