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

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Calculate sum of reciprocal rank in arbitrary large graph

For arbitrary graph of n node, can I approximate $\sum_{v\neq u}\frac{1}{Rank_u(v)^a\times Rank_v(u)^b}$ with $\sum_{v\neq u}\frac{1}{Rank_u(v)^{a+b}}$ or not when n is large enough? $a,b>0$, ...
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18 views

Complexity of finding a clique of given size in a directed graph?

Given a directed graph G and 2 vertex V1 and V2 are strongly connected. i.e. there is an edge from V1 to V2 and also an edge from V2 to V1. If we try to find weather they are member of a clique of k ...
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56 views

Edge-partitioning into rainbow triangles

I'm wondering if the following problem is NP-hard. Input: $G = (V,E)$ a simple graph, and a coloring $f : E \to \{1,2,3\}$ of the edges ($f$ does not verify any specific property). ...
3
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1answer
69 views

Partitioning the edges of a complete graph into smaller complete graphs

Consider the (simple, undirected) complete graph $K_n$. Is it possible to partition its edges into less than $n$ cliques, each having less than $n$ vertices? Note: It is easy to see that $n$ cliques ...
4
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1answer
194 views

Counting the number of K4

I was going over this paper and I don't understand a certain proof (section five phase 2). Given a graph G=(V,E) partitioned into the sets of vertices L and H. The vertices in L are at most D where D ...
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32 views

Approximate distance preserving sparse graph representation that are not necessarily subgraphs

I am looking for a type of graph sparsifier that I think I have seen somewhere but now I can't find the paper anymore. I think the paper referred to it as a spanner, but that term is used for so many ...
5
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1answer
169 views

Sparser Bipartite graphs?

Maximal Planar Bipartite graphs are sparser than maximal planar graphs. For which other classes of graphs are maximal Bipartite members sparser than arbitrary maximal members. Let $\mathcal{C}$ be a ...
3
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2answers
135 views

Runtime of Tucker's algorithm for generating a Eulerian circuit

What is the time complexity of Tucker's algorithm for generating a Eulerian circuit? The Tucker's algorithm takes as input a connected graph whose vertices are all of even degree, constructs an ...
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1answer
68 views

TSP heuristics for limited distance information

this is my first question on Theoretical CS. :) I've posted a similiar question on Mathoverflow and a friendly user advised me to post my question on this site. Problem: I'm looking for TSP ...
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32 views

Lower bound on treewidth of co-graph

What is lower bound on tree-width on the connected co-graph with $n$ vertices? The upper bound is $n - 1$, as clique is a co-graph.
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28 views

Characterization of an irreducible matrix

A matrix is irreducible, if it is not similar via some permutation to a block upper triangular matrix that has more than one block of positive size. Equivalently, for a 0-1 matrix, if it is viewed as ...
0
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1answer
54 views

Seeking for a game for modelling a problem using game thoery [closed]

I have a problem which I want to formulate it as a game, using game theory. In this problem there is several agents, we can consider the agents as the employees of different offices, these agents have ...
3
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1answer
106 views

Paths and Probabilities for a Random Walk on a Graph

I'm working on a problem about $N$ nodes that are randomly positioned on a rectangular grid. I want to take a sample of $n\leq N$ nodes by randomly selecting the first node then visiting the nearest ...
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2answers
176 views

Shortest path in DAG with path dependent arc costs

I've got the following problem Consider a DAG $G=(V,E)$ with $V=[v_1,…,v_n]$, and edge-set $E=[e_1,…,e_m]$, with associated costs $c_1,…,c_m$. The problem is to find the shortest paths from an ...
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0answers
45 views

Enumerate all the possible paths in a directed acyclic graph: matrix multiplication or queue based solutions? [closed]

I have the following problem: given a $G$, a source $s$ and a sink $t$ i have to enumerate all the possible paths between $s$ and $t$. Each node has an outdegree always $\leq 3$. Each path has an ...
9
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1answer
124 views

When does a graph admit an orientation in which there is at most one s-t walk?

Consider the following problem: Input: a simple (undirected) graph $G=(V,E)$. Question: Is there an orientation of $G$ satisfying the property that for every $s,t \in V$ there is at most one ...
10
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2answers
202 views

Name the graph class: Disjoint union of a clique and an independent set

Let $G$ be a graph which is the disjoint union of a clique and an independent set, i.e. $$G = K_{n_1} + \overline{K_{n_2}} = K_{n_1} + I_{n_2} .$$ The graph class of all such graphs is characterized ...
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1answer
140 views

Is finding whether k different perfect matchings exist in a bipartite graph co-NP?

Few definitions first. The co-NP problem is a decision problem where the answer "NO" can be verified in polynomial time. The perfect matching in a bipartite graph is a set of pairs of nodes (a pair is ...
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38 views

MRF vs CRF dependencies

If Markov random fields (MRF) model explicit short range dependence between features and implicit long range dependencies (knock-off effect), can we say that a Conditional random field (CRF) models ...
3
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2answers
125 views

Question about an old result of Erdős and Simonovits

My question concerns the following result of Erdős and Simonovits. A graph $G$ is $d$-almost-regular if $d\delta(G)\geq \Delta(G)$. Theorem 1 of the above paper states that a large $n$-vertex ...
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0answers
124 views

Theoretical background of Classes and Objects

I would like to learn about the possible ways of formalizing Classes and Objects (in programing languages like java) using formal languages. Where should I start? This might be related to my previous ...
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71 views

Proper definition of a graph with with loops and parallel edges

When a graph has parallel edges and self loops one cannot identify an edge with the set of two adjacent vertices it connects. I have failed to find a good formal definition of a non-simple graph. ...
3
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1answer
91 views

What is the best way to find an induced cycle basis of a graph?

My question is essentially what comes in the subject line: what is the best way to find an induced cycle basis of a graph (i.e., a cycle basis of the graph in which each cycle is an induced subgraph ...
5
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0answers
128 views

Approximating the clique size of the graph

Let $G=(V,E)$ be a graph. For a given $\rho \leq |V|$ and $\epsilon$ with $(0<\epsilon<1)$, is there any sublinear query algorithm known/possible to decide if the graph has a clique of size ...
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242 views

Is the complexity of this covering problem known?

Let $G=(V,E)$ be a graph. A vertex set $X\subseteq V$ is called critical if $X\neq\emptyset$ and no vertex in $V\setminus X$ is adjacent to exactly one vertex in $X$. The problem is to find a vertex ...
4
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1answer
120 views

Special properties of bipartite expanders

It is well known that expanders, and often the special case of bipartite expanders, have found many uses in derandomization, coding, etc. However, I am curious if there are any special properties of ...
5
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1answer
92 views

When does automaton stay unchanged after string homomorphism?

Suppose we have a string homomorphism $\varphi: \Sigma \rightarrow \Sigma^*$. Consider the languages in $\varphi(\Sigma^*)$ whose letters are elements of $\varphi(\Sigma)$, so here I do not want to ...
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2answers
560 views

Complexity of the recovery of an adjacency matrix from its square

I am interested in the following problem: Given an $n\times n$ matrix, is there an undirected graph on $n$ vertices whose adjacency matrix squared is that matrix? Is the computational complexity of ...
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40 views

Applications of network clustering coefficient

Consider the global clustering coefficient of a graph as defined here . The clustering coefficient describes how likely it is for a random connected triplet of vertices to be closed. My question ...
8
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2answers
362 views

Implications of a problem being in XP when parameterized by diameter

Let $X$ be an NP-complete graph problem. Suppose $X$ is solvable in polynomial time on graphs of bounded diameter. In other words, $X$ parameterized by diameter is in XP. (Recall a problem is in XP if ...
21
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1answer
559 views

Are there subexponential algorithms for PLANAR SAT known?

Some NP-hard problems which are exponential on general graphs are subexponential on planar graphs because the treewidth is at most $4.9 \sqrt{|V(G)|}$ and they are exponential in the treewidth. ...
4
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1answer
195 views

Reduction SAT to a problem on a planar graph with as few vertices as possible

Let $\phi$ be CNF formula with $n$ variables and $m$ clauses. I am looking for a reduction is $\phi$ satisfiable to a problem on a planar graph $G$ with as few vertices as possible. The majority of ...
5
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1answer
251 views

Is it $NP$-complete to decompose bridgeless cubic bipartite graph into edge-disjoint paths of length 3?

Motivated by this post on cubic graphs decompositions, I am interested in decomposing a connected bridgeless graph into edge-disjoint paths of length 3 (P4). My intuition is that it should be ...
7
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1answer
1k views

Several papers appear to imply P=NP via chordal graphs, what is wrong?

Several papers appear to imply P=NP via chordal graphs, which suggests something is wrong. As usual $\gamma(G)$ is the domination number and $i(G)$ and $\gamma^i(G)$ are the independence domination ...
22
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3answers
542 views

Is the Cheeger constant $\mathsf{NP}$-hard?

I have read in uncountably many articles that determining the Cheeger constant of a graph is $\mathsf{NP}$-hard. It seems to be a folk theorem, but I have never found either a quote or a proof for ...
6
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1answer
85 views

Maximum number of geometrically disjoint paths - is the complexity known?

Let $G$ be an undirected graph, given with a planar drawing. We do not assume $G$ is a planar graph, we just fix a planar representation of it, such that each vertex is represented by a distinct ...
4
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1answer
112 views

Does it help for clique if the vertices are partitioned into 3 cliques?

A graph is $(p,q)$-colorable if its vertices can be partitioned into $p$ cliques and $q$ independent sets. For $(2,0)$-colorable graphs clique is polynomial. I am interested how easier (if any) is ...
24
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2answers
509 views

Why is “topological sorting” topological?

Why is "topological sorting" called "topological"? Is it just because it determines an order without altering any vertices or edges -- like a doughnut and coffee cup are topologically equivalent? Why ...
4
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1answer
92 views

Generalized geography on solid grid graphs

A post here For which families of graphs is Generalized Geography in $P$? mentioned that generalized geography on solid grid graphs is open. Is the question still open? A quick search on Google shows ...
2
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1answer
148 views

Efficient recalculation of the maximum flow when edge capacities are reduced

Assume that we have a (directed) graph $G(V \cup \{s, t\}, E)$ and an (integer) capacity function $c : E \mapsto \mathbb{N}$. Let $f : E \mapsto \mathbb{N}$ be a maximum $s-t$ flow on this graph. ...
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1answer
75 views

Minimum order of partite in a bipartite graph

I want to create a bipartite graph where the first partite $U$ contains $L$ vertices with degree $k$ and the second partite $V$ contains $N$ other vertices with degree $a$. I need to find the minimum ...
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1answer
114 views

Many-one reduction from inequality problem to equality problem

Let the k-inequality-MIS problem be the decision problem whether an arbitrary graph $G=(V, E)$ contains a maximal independent set of at least size $k$, that is the corresponding language is: ...
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0answers
97 views

Finding Contextual Nodes in a Knowledge Graph

I'm currently participating in developing a knowledge graph that uses ConceptNet and a few others as its data sources. It uses the same architecture as ConceptNet namely it is stored as a Hypergraph ...
2
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1answer
291 views

Has this formulation of pursuit evasion been researched? Similar to Helicopter Cops and Robbers Game

There are pursuers and evaders in the vertices of a directed graph G with one component. Each vertex must have atleast one outgoing edge (can be a loop). At each time t: The evaders must move ...
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330 views

Example of scalable non-Hamiltonian cubic planar graph

Suppose I have a non-Hamiltonian cubic planar graph $G_1$ = ($V_1$, $E_1$) where no face has fewer than five sides, and that I can partition $V_1$ into two Hamiltonian subgraphs. Suppose further that ...
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0answers
123 views

$CIS_G$ problem deterministic lower bound

In notes, http://www.cs.toronto.edu/~toni/Courses/CommComplexity2/Lectures/clique-notes.pdf, it is mentioned towards end of section $4$ that http://dl.acm.org/citation.cfm?id=237817 shows that ...
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56 views

Transitive reduction not provably minimal

Working on finding minimal equivalent graphs, which unlike transitive reductions only allows for edge removals from the original graph. I was under the impression that if you allow for new edges to be ...
3
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1answer
137 views

Finding a minimum directed cut that splits a weakly connected bipartite graph into two such non-edgeless graphs

Consider a directed, weakly connected bipartite graph $G = (U, V, E)$ where $U$ and $V$ are sets comprising the nodes of $G$, and $E \subseteq U \times V$ is the set of edges. The task is to find a ...
2
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1answer
158 views

Algorithms for computing the minimal vertex separator of a graph

Background: Let $u,v$ be two vertices of an undirected graph $G=(V,E)$. A vertex set $S\subseteq V$ is a $u,v$-separator if $u$ and $v$ belong to different connected components of $G−S$. If no proper ...
3
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1answer
132 views

Comparison between the maximum clique and maximum biclique problem

It seems to be commonly believed that the maximum biclique problem (on a bipartite graph) is more difficult than the original maximum clique problem. Is there a formal proof for this claim? (For ...