# Tagged Questions

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

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### Problem of graph bi-partition (related to graph isomorphism)

I am considering the following problem: Input: 3 graphs $G=(V,E)$, $H_1$, $H_2$ Question: Is there some $V_1\subseteq V$ such that $G[V_1]$ (the subgraph induced by $V_1$) is isomorphic to ...
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### non-Hamiltonian graphs [on hold]

What is the relation between the set of non-Hamiltonian graphs, all of whose members are presumably not known, and the dynamic linear programming and factorial solutions to the TSP in general graphs? ...
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### Is there any digraph data set that gives all directed graphs satisfying certain requirements? [closed]

I'm looking for a digraph dataset that can return all directed graphs satisfying certain requirements. Following are some examples: All tournament with 12 vertices; All connected digraphs with 10 ...
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### Max-sum graph-partition for maximizing intra-edge weights?

I would like to know if the following problem has already been studied, and if so how is it called. In particular I'm interested in approximability results. Input: A graph G with negative or ...
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### How similar are two DFAs? -not just binary equivalence- [on hold]

I'm looking for a metric, a measure, to compute similarity (or distance) between two DFAs. I think it should considers alphabet, transitions, states and global (or local) structure. Have you any ...
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### Is the proceedings of WG 2015 published?

Within theoretical computer science, the "Workshop on Graph-Theoretic Concepts in Computer Science (WG)" is one of the main specialized venues for publication of papers dealing with graph theory. It's ...
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### About complexity of recovering or learning Bayesian networks

Are there complexity theoretic results about recoverability or learnability of the marginals (of the source vertices) and the conditionals (along each of the edges) of a Bayesian network from having ...
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### Enumerating all (super)orientations of an undirected graph

Given an undirected graph $G$, an orientation of $G$ is a directed graph obtained by assigning every edge a direction, a superorientation of $G$ is a directed graph obtained by orienting every edge in ...
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### Graph partition with weighted vertices and edges

I am searching for an algorithm to apply to a specific graph partition problem that I am interested in. It feels like a topic that people from CS may have worked on but it is also different from ...
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### Kleinberg Rubinfeld Short Paths in Expander Graphs for Hypergraphs

In the 1996 paper "Short Paths in Expander Graphs" by Kleinberg and Rubinfeld, the authors show a randomized polynomial-time algorithm for finding an embedding of a graph $H$ into a graph $G$, if $G$ ...
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### Machine learning algorithms on hypergrap models

Graphical models are a very useful tool with many applications, whereby a joint distribution of a set of random variables is modeled using only pairwise dependencies between the variables, and two ...
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### Tree-decomposition with clique interfaces

Let $G=(V,E)$ be a finite undirected graph. A tree decomposition $(T,\lambda)$ of $G$ is a tree $T$ with labeling function $\lambda : T \to 2^{V}$ such that: For every edge $\{v_1,v_2\} \in E$, ...
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### Classes of graphs with superconstant treewidth

There are several interesting classes of graphs with bounded treewidth. For instance, trees (treewidth 1), series parallel graphs (treewidth 2), outerplanar graphs (treewidth 2), $k$-outerplanar ...
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### Graph class with easy chromatic number, but NP-hard coloring

Is there a graph class for which the chromatic number can be computed in polynomial time, but finding an actual $k$-coloring with $k=\chi(G)$ is NP-hard? Without any further restriction the answer ...
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### Is there a special name for the following type of graphs?

Weighted graphs such that if $x$ and $y$ are nodes in the graph, and $p$ and $p'$ are two paths in the graph between $x$ and $y$, then the weight of $p$ equals the weight of $p'$.
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### The relationship between degree of vertex and size of dominating set

I was wondering is there any relationship between degree of vertex and size of dominating set. For example, if I know the number of vertices is $n$, and I could know each vertex in the graph has ...
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### How to efficiently generate a random 0-1 matrix of a given rank

How to efficiently generate a random $n\!\times\!n$ $0$-$1$ matrix of rank $k<n$?
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### Complexity of a graph-rewriting problem

I recently came across the following problem which seems to fall in the context of graph rewriting problems: Input: A graph $G=(V,E)$ with maximum degree 3, an edge $e_0 \in E$ and pairs of graphs ...
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### Canonical way of coloring graphs (individualization) for isomorphism purpose

Stefan Kratsch and Pascal Schweitzer in their paper Graph Isomorphism for Graph Classes Characterized by two Forbidden Induced Subgraphs give a characterization of graphs on which Graph Isomorphism ...
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### Minor closed properties that are explicitly MSO expressible

Below, MSO denotes the monadic second order logic of graphs with vertex-set and edge-set quantifications. Let $\mathcal{F}$ be a minor closed family of graphs. It follows from Robertson and ...
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### Optimal distribution of integer edge weights

I am not sure whether the following problem has been studied. Any help would be greatly appreciated. I have $L$ sets, $S_1, S_2,...,S_L$, each of $n$ elements, taken from a universe of $N$ elements. ...
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### Does such model exists?

I have a problem on distributed graph, with the following model: 1. There is a Global Graph $G=(V,E)$ 2. There are $k$ computers. 3. Each computer $1 \leq i \leq k$ knows ALL the nodes of the ...
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### Generate connected induced subgraphs as the satisfying assignments to a SAT instance

I want a SAT instance (in CNF) whose set of satisfying assignments are the connected induced subgraphs of a given input graph. A general solution would be helpful, but I really only need this when the ...
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### Fractional but not integer multi-commodity minimum cost flow

I'm searching for an example digraph for the multi-commodity minimum cost flow problem with integer demand. There shouldn't be an integer but fractional optimal solution. I found here a similar ...
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### Find max weight induced graph in a multipartite graph with one vertex from each part

Consider the follow problem: Input: $G=(V,E)$, a weighted $k$-partite graph with $n$ vertices. Output: $U \subseteq V$, one vertex from each part, maximizing the total weight of the induced graph ...
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### Is graph isomorphism still open for bounded clique width or bounded rank width? 2015 paper claims it is polynomial [closed]

To my knowledge, graph isomorphism for graphs with bounded clique width or bounded rank width is open. 2015 arxiv paper claims it is polynomial: Isomorphism Testing for Graphs of Bounded Rank Width ...
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### Partition graph into 2 or more claw-free subgraphs

Is it NP-hard to partition the vertex set of graph G into k subsets so that they induce k claw-free subgraphs of G?
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### Probability of two vertices being connected by some path in a random directed graph

Define $G(n, p)$ as a random directed graph ($n$ vertices; we put edge between two vertices with probability $p$). What are the known results for the following problem: Fix two vertices $v$ and ...
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### Embedding distortion under group quotient

The high level question is as follows: Suppose some group (here assumed to be a vector space of $\mathbf{F}_2^n$) has a low-distortion embedding into $l_1$. Under what condition does the quotient of ...
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### How to constrain a finite automaton (NFA and DFA) to a tree?

I have a finite automaton by the standard model Hopcroft & Ullman define: $$M = (Q, \Sigma, \delta, q_0, F)$$ Where $\delta$ is the transition function mapping $Q \times \Sigma \mapsto Q$, such ...
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### Is the topsort from “Structuring Depth-First Search Algorithms” guaranteed to be (reverse) stable?

In "Structuring Depth-First Search Algorithms in Haskell", implemented in Data.Graph in the Haskell standard library, an algorithm for topologically sorting graphs is given: ...
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### Vertices adjacent to Exterior region of a Planar Graph(Algorithm)

Problem: I am looking for an algorithm which finds all vertices that are adjacent to exterior region of a planar graph(For a planar graph, any region=face can be considered as the exterior region ...
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### Submatrix of small rank

Let $G=(V,E)$ be a graph with adjacency matrix $M=(m_{ij};i,j \in V )$ over $\mathbb{F}_2$ and $k \in \mathbb{Z^+}$. How can we find in polynomial time a subset $A \subseteq V$ such that The rank ...
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### Graph that maximizes minimum hitting time?

Let $G$ be some connected bidirectional (or undirected) graph. We define a random walk as a walk that begins at a vertex chosen uniformly at random, and at each step proceeds to one of its current ...
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### When does adding edges decrease the cover time of a graph?

When first learning about random walks on a graph $G$, one may have an intuitive feeling that adding edges to $G$ will decrease its cover time $C(G)$. However, this is not the case. The path graph ...
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### Reconstructing a 2D lattice graph from an unordered adjacency list

Is there any kind of algorithm that can map a set of points and their unordered adjacent neighbours to a 2D lattice graph that would then be addressable using X, Y coordinates? For example, given the ...
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### Multicuts (or multiway cuts) for 3 Terminals of Minimum Capacity

For each fixed $k \geq 3$, the following well-known problem is NP-complete. k-Multicut (aka "Multiway Cut", "k-Way Cut", "Multicut") Input: $(N,l)$ where $N=(V,E,T,c)$ is an undirected $k$-terminal ...
### Approximation algorithms for the maximum $2$-independence set problem
I am interested in approximating the maximum $2$-independent set problem in arbitrary graphs. In a graph $G$ a set $I$ of vertices is called $2$-independent if the distance between any two distinct ...
### Given a 4-cycle free graph $G$, can we determine if it has a 3-cycle in quadratic time?
The $k$-cycle problem is as follows: Instance: An undirected graph $G$ with $n$ vertices and up to $n \choose 2$ edges. Question: Does there exist a (proper) $k$-cycle in $G$? Background: For any ...