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

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30 views

Connecting partial paths to form a hamiltonian cycle

For an undirected graph that consists of partial paths such that each vertex is a part of one of those paths and that there are edges between all the paths, is there an efficient algorithm to connect ...
0
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0answers
54 views

Best Hamiltonian Cycle Problem solver

What is the best Hamiltonian Cycle Problem (HCP) solvers available in the market? Googling so far shows that there is one created by Flinders University that can solve at most 5000 node instances. I ...
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0answers
48 views

how to create general star from spanning tree [closed]

i had read a paper "Approximation algorithms for the shortest total path length spanning tree problem" .I am not getting what's a star and general star.can you explain with an example ...
-2
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0answers
41 views

Determine if subgraph is spanning tree [closed]

I am looking for algorithm that allow me to determine whether given subgraph is spanning tree or not.
3
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0answers
107 views

Can we decide Red-blue cut problem in polynomial time?

Given a directed graph whose arcs are coloured red and blue and integers $r$ and $b$, can we decide in polynomial time whether the digraph has a cut with at most $r$ red arcs and at most $b$ blue ...
-1
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0answers
24 views

Find all the Cycle Bases in a Undirected Graph

How to find all the Cycle Bases in a Undirected Graph For example, given the graph: 0 --- 1 | | \ | | \ 4 --- 3 - 2 the algorithm should return 1-2-3 ...
-1
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0answers
41 views

Edge-based NP-hard problems for reduction

I have a problem formulation where the input is a undirected graph $G=(V,E)$, and the task is to add a set of new edges $F \subseteq V\times V \setminus E$ to $E$, but no node $v\in V$ receives more ...
2
votes
2answers
91 views

Graph-theoretic properties of the Wiener index

The Wiener index of a graph is the sum of the lengths of the shortest paths between all pairs of its vertices. Are there useful graph-theoretic properties of this index?
-2
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0answers
51 views

shortest path algorithm cost calculation [closed]

so the question goes like this: You are given a directed graph G=(V,E) and a weight function wt: E->R+. You are also given two distinguished nodes s,t ∈ V Write an algorithm that marks every node v ...
5
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0answers
109 views

The minimum entropy of a proper coloring of a graph

The chromatic number $\chi(G)$ of an undirected graph $G$ is the minimum number of colors in a proper coloring of the vertices of $G$ (where a proper coloring uses different colors for two vertices ...
4
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0answers
108 views

Recognition problem of cycle permutation graphs

A cycle permutation graph is a cubic graph composed from two $n$-cycles each labeled $1, 2,..., n$, with additional edges connecting vertex $i$ in the first cycle to vertex $\pi(i)$ in the second, ...
8
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0answers
268 views

Is it possible to solve perfect matching in linear time

As we know matching can be solve in polynomial time. One classical and famous algorithm is designed by Karp and Hopcroft. Is it possible to solve perfect matching problem in linear time for given ...
-1
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0answers
46 views

Find correspondences of nodes between two graphs

I have two graphs (actually two street maps, one very fine grained and exact, the other coarser and maybe a bit faulty w.r.t. the nodes coordinates and the topology) and I want to find corresponding ...
3
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1answer
336 views

Is DAG isomorphism NP-C

Given two directed acyclic graphs $G_1$ and $G_2$, is it NP-Complete to find a one-to-one mapping $f:V(G_1) \rightarrow V(G_2)$ such that $(v_i,v_j) \in D(G_1)$ if and only if $(f(v_i),f(v_j)) \in ...
5
votes
0answers
102 views

lower bound for difference between max cut and min cut

Let $G=(V,E)$ be a graph on $n$ vertices with edge weights $w=(w_e)_{e\in E}$. Let $M^+$ and $M^-$ be the maximum and the minimum cut values, i.e. ...
4
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0answers
152 views

Is this graph polynomial known? Can it be efficiently computed?

Consider a connected simple graph $G$ with $n$ vertices and $m$ edges. View each edge $\ell$ as a transposition $t_{\ell}$ acting on the set of vertices. [To be more explicit, given an edge $\ell$ ...
-2
votes
1answer
122 views

Finding cliques in weighted graph

We have given a weighted graph $G=\{V,E\}$, where $V=\{v_1, v_2,...,v_n\}$, and for all $i,j$, the weight of edges $w(v_i, v_j)\in (0,W)$. And we have also given a weight threshold s $w$ (where ...
2
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1answer
124 views

Planar separator theorem and tree decomposition

The Wikipedia article about the Planar Separator Theorem states that it is possible to use a hierarchy of separators to construct a tree decomposition for a planar graph and moreover provides an ...
6
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2answers
320 views

Complexity of finding even cuts for a graph

Given a graph $G=(V,E)$, what is known about the classical computational complexity of finding a non-trivial cut which partitions the vertices into two sets $V_a$ and $V_b$ such that every vertex in ...
5
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0answers
75 views

General Results for Complicated Constraint Satisfaction Problem

Consider the following problem: on a finite two-dimensional grid (say the grid points are vertices of a graph), I need to color the vertices in such a way to satisfy the existence and nonexistence of ...
2
votes
1answer
63 views

When polynomial GI implies polynomial (edge) colored GI?

Crossposted from MO. (edge) colored graph isomorphism is GI which preserves the colors (of edges if it is edge colored). There are several reductions using transformations/gadgets of (edge) colored ...
5
votes
1answer
61 views

Is there an extension to the stable roommates problem with multiple roommates per room?

The stable roommates problem presents a set of N two-person rooms and 2N would-be roommates with preferences over each other, and asks for a stable allocation of roommates to rooms (and, really, to ...
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1answer
72 views

Minimum vertex cover for bipartite graphs

I know that it is possible to calculate the minimum vertex cover of a bipartite graph, However, i want that the minimum vertex cover which contains vertices from only one partite set, which will be ...
8
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1answer
93 views

Is there a gadget that reduces generalized geography to undirected graphs?

The directed Generalized Geography game is well-known to be PSPACE-complete, however, I could not find anything for the undirected version. I saw that in Hans L. Bodlaender, Complexity of path-forming ...
1
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1answer
128 views

Rings and the set of all minimum s-t-cuts

Let $N$ be a flow network with nodes $V$ and edges $E$. For technical reasons, the source side of a minimum $s$-$t$ cut $(A,B)$ with $s \in A$ and $t \in B$ is defined as $A - \{s\}$. Now, let ...
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0answers
18 views

introducing correlations in the adjacency (or connectivity ) matrix

I am trying to build an adjacency matrix with correlation. i.e The probability of connection from node A to node B is set to a constant factor of order 1 (say $\alpha$ ), if node B is connected to ...
3
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0answers
63 views

Online bridge and nonbridge counting (identification)

I was wondering if there is any efficient (possibly armortized poly-logarithmic) online algorithm which supports counting (identification) of bridges- and non-bridges online, i.e. during a sequence of ...
14
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3answers
301 views

What separates easy global problems from hard global problems on graphs of bounded treewidth?

Plenty of hard graph problems are solvable in polynomial time on graphs of bounded treewidth. Indeed, textbooks typically use e.g. independet set as an example, which is a local problem. Roughly, a ...
2
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0answers
115 views

Regular graph coloring conjecture

Almost all regular graphs with degree $k^2$ are $k+1$ colorable. The near absence (finitude) of adjacent small cycles in random regular graphs as $N$ (the size of the graph) grows may help imply the ...
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1answer
88 views

Finding an exactly weighted st-path in a digraph [closed]

I have a weighted digraph graph $G = (V,E)$ where the weights are positive and negative integers. The graph $G$ is not necessarily acyclic. The question is: given 2 nodes $v_1$ and $v_2$, is there a ...
0
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1answer
67 views

How to map random cardesian points in a 2d array

I was wondering if there is any algorithm, theoretical or already implemented, or if its even possible at all, where, given N random ...
-3
votes
1answer
64 views

tagging and graph “compression”

I have a question on stack-overflow about "compressing" a graph. Suppose I have tags from a finite set $T$ and objects from a finite set $O$. Moreover there are (uni-directional) links from elements ...
4
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0answers
71 views

Indexing structure for all-pairs min-cuts in a huge DAG

I have a huge DAG - e.g., the dependency graph of all packages in a linux distribution. Suppose I'd like to make a user-friendly tool that makes it very easy to understand how to break the transitive ...
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2answers
77 views

Arrangements of Objects

Suppose there are $n$ bins each having $k$ objects. Assume that capacity of each bin is also $k$. Now we want to rearrange the objects such that each bin contains $k$ objects but this time if $x,y$ ...
1
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1answer
73 views

Connecting vertices after struction operation in J.Chen, I.Kanj, G.Xia vertex cover algorithm

EDIT: I'm sorry if this question belongs more to cs.SE, I've had a dilemma about where to put it. Please let me know if it's inappropriate. I'm currently implementing the Vertex Cover problem solving ...
3
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2answers
148 views

Decompose a complete graph into smaller cliques

The following exercise problem is from the book of D.B.West which i could solve: If a complete graph can be decomposed into triangles then $n-1$ or $n-3$ is divisible by 6. So my questions are ...
6
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0answers
467 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 ...
3
votes
3answers
378 views

Sub-exponential algorithm for Hamiltonian cycle problem on cubic planar graphs?

There are several graph $NP$-complete problems that have sub-exponential time algorithm on planar graph instances. What is the fastest algorithm for HC problem on cubic planar graphs? Is there a ...
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0answers
115 views

Graph sparsification

I would like to ask if any one is aware of any results related to graph sparsification with bounded degrees? What I mean is anyone aware of graph sparsification results such that the resultant graph ...
4
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2answers
196 views

Algorithms for online clique detection

Are there any algorithms which let you detect cliques when adding/deleting edges based on previously detected cliques? What would be the time/memory complexity of this approach?
6
votes
2answers
184 views

Partition planar graph into connected subgraphs of equal size

Work Jünger, Michael, Gerhard Reinelt, and William R. Pulleyblank. "On partitioning the edges of graphs into connected subgraphs." Journal of graph theory 9.4 (1985): 539-549. states that for ...
11
votes
1answer
301 views

Existence of long induced paths in expander graphs

Let's say that a graph family $\mathcal{F}$ has long induced paths if there is a constant $\epsilon > 0$ such that every graph $G$ in $\mathcal{F}$ contains an induced path on $|V(G)|^{\epsilon}$ ...
9
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0answers
121 views

Maximum local edge connectivity

For a simple graph, the local edge connectivity of vertices $x,y$ where $x\neq y$ is $\lambda(x,y)$ and defined as the maximum number of edge disjoint paths from $x$ to $y$. One can find this by a ...
3
votes
1answer
92 views

Is there a known extension of Dirac's / Ghoulia-Houri's theorems for $k$-path existence?

In the well studied problem of Hamiltonicity, several papers/theorems gave sufficient "degree conditions" for the existence of Hamiltonian path in a graph. These include: Dirac's theorem , 1952, ...
9
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0answers
127 views

Is RAMSEY COLORING in $NC$?

Say that a function $f$ is a RAMSEY COLORING if on unary input $n$, it returns a complete graph on $n$ vertices with its edges colored red and blue without a monochromatic clique of size $10\log n$. ...
3
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0answers
83 views

Finding modular decomposition of graph

I am trying to learn how to find modular decomposition of graph using the method given in the paper Simpler Linear-Time Modular Decomposition via Recursive Factorizing Permutations. I am unable to ...
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0answers
62 views

Pruning a graph by removing vertices not part of any minimal Steiner tree

Given a sizable graph $G = (V, E)$, with $|V| \approx 1400$ vertices and $|E| \approx 1600$ edges, I have an optimization problem to find a connected subgraph of $G$ of size $k$ such that a given ...
6
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0answers
190 views

NP-hardness of tasks graph assignment to two heterogenous servers

I have a problem with determinig if the following assignment problem is NP-hard. Any comments and suggestions would be appreciated. Problem definition Given is a directed acyclic graph $G=(V,E)$ ...
1
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1answer
206 views

Is there simple reduction Dominating Set to Vertex Cover?

Is there simple reduction Dominating Set to Vertex Cover? In the other direction the reduction is simple. Searching the web returned blog. It warns This is not finished yet and experiments ...
0
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0answers
35 views

Estimating Graph/ Network Accuracy

If I'm creating a (social) network using some automatic system, which I know is not 100% accurate but for which I can estimate the rate of error, what, if anything, can I say about the accuracy of the ...