Questions tagged [graph-theory]

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

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d-regular graphs and edge expanders

Show that there is no (n, d, ρ)-edge expander for ρ > 0.5 Is this statement even true? My attempt: Let n = 2, then we can have 2 vertices, A and B. Let d = 1, therefore there is an edge between A ...
math-nerd-in-cs's user avatar
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Average case complexity of decision version of NP-hard problem

I am a bit confused regarding the average case complexity of certain graph problems that are NP-hard like graph coloring, clique, dominating set and whose decision version is NP-complete. It is ...
Subhra Mazumdar's user avatar
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Any value in a formula that calculates (not look up) the 'order' of a 'Independent Edge Set' OR a 'I.E.S.' given an 'order' on complete graphs?

Any value or interest in a formula that calculates (not look up) the 'integer order' of a given 'Independent Edge Set' OR given an 'Independent Set' calculates the 'integer order' on Complete Graphs? ...
Tim's user avatar
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Name for a cyclic path in a graph that visits every vertex while minimizing the maximum number of times a given vertex is revisited?

Me and my colleague are interested in whether anyone has looked into a generalization of Hamiltonian cycles where vertices can be revisited, but we want to minimize the maximum number of times a given ...
mich's user avatar
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Representation of binary strings by graphs and hypergraphs

Let $\Sigma$ be the set $\{ 0, 1 \}$, then the set of all finite binary strings of length $n$ is written as $\Sigma^{\star}_{n}$. Question: Which further ways of representing binary strings of length $...
Samdney's user avatar
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3 answers
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Stable/Robust Traveling Salesman Approximation Methods

I was wondering if there are TSP approximations that are "stable". More specifically, consider the set $G = x_1, ..., x_n$ and the set $G^* = G \cup x_{n+1}$, where $x_i$ are points in $R^d$....
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A variation of the longest path problem

What about finding a path of maximum length in a given graph which may contain cycles, with the constraint that a vertex (or an edge) can be visited at most X (say 2 or 3) times ? EDIT: X would be ...
user1454590's user avatar
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Bottom up TSP solution?

I'm not sure if this is something new or if I'm just not getting previous efforts. TSP can be thought of as a list of weighted links and nodes. If one takes the Nearest Neighbor (NN) of every node and ...
Maub Nesor's user avatar
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How to reduce a code down to its configuration

I have built a system where from atomic information of a UI code I could generate a framework specific code. Here is the concept https://github.com/imvetri/ui-editor. For example, the user of this ...
Vetrivel's user avatar
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Tractability of computing generalized hypertreewidth on bounded arity hypergraphs

Generalized hypertreewidth is a generalization of treewidth to hypergraphs. Unlike treewidth, it is not tractable, for a fixed width $k \in \mathbb{N}$, given a hypergraph $H$, to determine if $H$ has ...
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Generalization of the Hamiltonian path problem on Grid Graphs

Fix a cost to each of these actions: move up, move down, move left, move right. I.e. fix some function $f: \{\text{move up, move down, move left, move right}\} \to \mathbb N$. Define the following ...
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5-color graph and minor

We have a 5-color graph G without 5-clique. The question is: is there a minor H of G that is a 5-clique? Here the minor definition. With "5-color graph G" I mean $\chi (G)=5$.
Mario Giambarioli's user avatar
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1 answer
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Efficient Algorithm for Partitioning a Directed Acyclic Graph into Short Paths

I am working on a problem involving partitioning a directed acyclic graph into distinct multiple paths, each with a maximum length constraint. The goal is to minimize the number of paths (this should ...
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Application LCL definition to vertex coloration

I'm reading the article "What can be computed locally?" by Naor & Stockmeyer and I struggle to understand the definition of an LCL they gave. Here is an extract: (page 2) An Locally ...
Qise's user avatar
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What is known about the complexity of Network Diversion?

In the Network Diversion problem, we are given an undirected graph $G$ on $n$ vertices, with specified nodes $s$ and $t$ and specified edge $e$, and a positive integer $k$, and are tasked with ...
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Independent set queries with preprocessing

Suppose we have a sparse undirected graph $G = (V, E)$ with $|E| = O(|V|)$, and we want to process it and then answer queries of the following type: given a set $A$, is it an independent set in the ...
Command Master's user avatar
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What's the exact complexity of a DFS if we revisit nodes?

By "revisit nodes," I mean if we didn't maintain a set of nodes we have visited. So the sum I'm examining is just the number of paths from a root to a node, across all roots and nodes. We'll ...
Adam Jamil's user avatar
3 votes
1 answer
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What is the treewidth of the 3D-grid (mesh or lattice) with sidelength n?

Here, by 3D-grid of sidelength $n$ I mean the graph $G=(V,E)$ with $V= \{1,\ldots,n\}^3$ and $E=\{( (a,b,c) ,(x,y,z) ) \mid |a-x|+|b-y|+|c-z|=1 \}$. I known how to get the treewidth of $n*n$ grid is ...
Jxb's user avatar
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Notion between connected and strongly connected graphs

Let $G$ be a directed graph without loops (or even better an oriented graph). Let us assume that $G$ is finite. The graph $G$ is connected if its underlying graph $G^*$ is connected (i.e., for every ...
Cyriac Antony's user avatar
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What is a combinatorial embedding?

I got a reviewer comment saying that I should consider using combinatorial embeddings rather than idk what I should call what I was doing topological embeddings?. But I'm confused because as far as ...
Hao S's user avatar
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1 answer
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Maximum cardinality matching on DAGs

A question on computational complexity and graph theory. The problem of finding maximum cardinality matchings of undirected graphs (the largest selection of edges such that each vertex is "...
Marco Pegoraro's user avatar
4 votes
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Exploding number of homomorphisms

I'm trying to tackle the following problem: given two graphs $A$ and $B$, if there exists a graph $D$ such that $\hom(A, D) > \hom(B, D)$ (i.e. there is more homomorphisms from $A$ to $D$ than from ...
Franciszek Malinka's user avatar
1 vote
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Bound on the treewidth of a graph from modular contraction

I cannot find a reference for this easy to prove result concerning the treewidth of a graph with respect to the treewidth of a modular contraction of it. Let $G=(V,E)$ be a graph. A module $M \...
holf's user avatar
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Graphs such that every rotation system admits an embedding on a surface of small genus

Let $G$ be a finite, simple, undirected graph. What conditions on $G$ ensure that every rotation system of $G$ corresponds to a cellular embedding of $G$ on an oriented surface of small genus? (e.g. ...
Cyriac Antony's user avatar
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Given several $a_i$-$r$ paths in a planar graph how ``balanced" of a tree rooted at $r$ can I make?

Suppose I am given distinct nodes $a_1,a_2,.., a_l, r$ and several $a_i$-$r$ paths $P_i$ in a planar graph $G$. I wish to construct a tree $T$ connecting $a_1,a_2,.., a_l, r$ that minimizes the ...
Hao S's user avatar
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Asking boolean question on the nodes of a DAG to find the target node

We are given a DAG $G=(V, E)$ and an unknown target node $x \in V$ to find. There is a mechanism to probe a node, $y \in V$, to ask question of the form "Given the node $y$, is the target node $x$...
Azam Ikram's user avatar
3 votes
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Cover all triangles of a graph with n subgraphs as small as possible

What is the smallest number $s(n,\Delta)$ such that for any undirected simple graph $G=(V,E)$ with $n$ vertices and $\Delta$ triangles, there exist $n$ subgraphs of $G$ covering all triangles where ...
walydna's user avatar
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Finding the shortest cycle containing a vertex in a graph

Given a connected undirected graph with edges $E$ and verticies $V$ and a vertex $v \in V$, find the length of the shortest cycle containing $v$. The best I could do is $O(|E|*deg(v))$, by trying to ...
Sharp Edged's user avatar
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(Where) is determining the mixing time of an implicitly defined graph in the polynomial hierarchy?

Consider an implicitly defined graph; for example, let $G$ be a finite group generated with $n$ generators as $\langle g_1,g_2,\ldots g_n\rangle$ and let $\Gamma$ be the Cayley graph of $G$ under ...
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Let $C$ be collection of subsets of $[N]$. Given, $n\in [N]$, what's the terminology for $card\{s\subseteq [N]\mid n\not \in s,\, s \cup\{n\}\in C\}$?

Let $N \ge 1$ be an integer and let $\mathfrak S$ be a nonempty collection of subsets of $[N] := \{1,2,\ldots,N\}$. For any $n \in [N]$, define $\partial_n \mathfrak S := \{S\setminus\{n\} \mid S \in \...
dohmatob's user avatar
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notation in graph theory [closed]

I was reading a paper, and I found a notation that I don't understand: $\mathbb{E}[| \textbf{S} |] $, where $\textbf{S}$ is a set. Are there any differences with the notation $\mathbb{E}[formula]$ (I ...
Palerme's user avatar
6 votes
0 answers
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Cycle packing with degree condition

Given a directed graph where each vertex has the same in-degree as out-degree, I would like to find the maximum number of edge-disjoint cycles. Is this NP-hard? Without the degree condition, the ...
TZM's user avatar
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What's the connection between branchwidth and treewidth

I understand that treewidth and branchwidth are essentially equivalent for a fixed graph, given that $branchwidth(G) = Θ(treewidth(G))$. However, my question pertains to a specific case involving ...
Jxb's user avatar
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Ensuring the connectivity of an undirected graph through linear programming

I am trying to solve a linear programming problem that deals with finding an optimal subgraph as a function of several parameters. The case is I am trying to model a constraint that ensures that the ...
AdCerros's user avatar
3 votes
1 answer
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END OF THE LINE problem finding a node with in-degree $0$ or out-degree $0$ depending on the initial node

The END OF THE LINE problem is stated as Given two circuits, P and N, a node, v, is balanced if $P(N(v)) = N(P(v)) = v$ or $P(N(v)) \neq N(P(v)) \neq v$. Given that $0^n$ is not balanced, find ...
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Embedding degree-3 planar graphs as topological minors in wall graphs in polynomial time

For a proof, I need the fact that we can efficiently embed an input planar graph into a representative of a specific family of high-treewidth graphs. Specifically, I need an embedding as a topological ...
a3nm's user avatar
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0 votes
1 answer
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Entries of the Inverse Laplacian

Let $L$ be a graph Laplacian. What is the meaning of the entries of its (pseudo)inverse $L^{-1}$? In other words, are there any interpretations which might help with understanding the entries of $L^{-...
Zuza's user avatar
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3 votes
1 answer
201 views

Cover a graph with complete graphs

I want to find the smallest possible function $k(n,m)$ such that for any graph $G$ with $n$ vertices and $m$ edges, there exists $n$ vertex sets $S_1,S_2,...,S_n\subseteq V$ each with size $k(n,m)$ ...
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Minimum vertex-separators under edge addition

I am trying to prove the following claim. Let $T$ be a minimum $st$-separator in an undirected graph $G$, and let $x \in T$. Let $S\neq T$ be a minimal $st$-separator (i.e., not necessarily minimum), ...
BBK's user avatar
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6 votes
1 answer
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Complexity of induced Steiner Tree problem

Consider the following problem: we are given an undirected graph $G=(V,E)$ and three terminal vertices $t_1,t_2,t_3\in V$. We are asked whether there exists a set of vertices $S\subseteq V$ such that ...
Michael Lampis's user avatar
6 votes
0 answers
180 views

Two graphs indistinguishable by 4-WL

There are pairs of non-isomorphic graphs that are indistinguishable by the k-WL (but distinguishable by (k+1)-WL) [1]. For example 4x4 rook’s graph and the Shrikhande graph are non-isomorphic but the ...
DeepOak's user avatar
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1 answer
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increasing minimum graph degree by adding edges

My problem: Given a graph $G=(V, E)$ and an integer $\ell$,add a minimum number of edges to $G$ so that in the resulting graph every vertex has degree at least $\ell$. Is there a polynomial-time ...
jpcasti's user avatar
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3 votes
1 answer
159 views

Treewidth for hypergraphs that specify connectedness requirements

This question is about an alternative definition of treewidth, called weak treewidth. It is defined on hypergraphs where hyperedges intuitively require that the connected subtrees of occurrences of ...
a3nm's user avatar
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Maintaining a $K_{3,3}$-minor-free graph

Suppose we are given that an undirected, connected graph $G$ is $K_{3,3}$-minor-free. Let $a,b\in V(G)$ be non-adjacent vertices. Under what conditions is the graph that results by adding the edge $(a,...
BBK's user avatar
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1 answer
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Spectral sparsification of graphs with negative edge weights

I am reading the following well-known paper on spectral sparsification of weighted graphs: https://arxiv.org/pdf/0808.4134.pdf. Page 2 contains most of the definitions relevant to this question. It is ...
K V's user avatar
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3 votes
1 answer
152 views

Hardness of Maximum Independent Set in 3-Colorable Graphs

Let $G = (V,E)$ be an undirected graph such that there is a proper coloring of the vertices of $G$ in three colors. Question: In such graphs, are there known results for the hardness of finding a ...
John's user avatar
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Graph partitioning to minimize sum of intra-partition edge weights

I've seen a lot of graph partitioning algorithms w/ the objective of minimizing the weight of inter-partition edges, (e.g. k-way partitioning) but haven't quite found anything on minimizing the total ...
axizzt's user avatar
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3 votes
2 answers
221 views

What is this graph problem, and how hard is it?

My problem is quite simple to state, so it surely must have a name: Given a graph $G=(V,E)$ with edge weights $w(e) \in \mathbb{Z}$, find a $V' \subseteq V$ that maximizes $\sum_{e \in E' } w(e)$, ...
tobwin's user avatar
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1 vote
0 answers
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On the borderline between natural and artificial problems

While there is no formal definition of what constitutes a natural algorithmic problem, in most cases there is pretty good consensus whether a specific problem is natural or artificial. Natural usually ...
Andras Farago's user avatar
3 votes
1 answer
163 views

Connected dominating set in bipartite graphs

Let $G$ a bipartite graph with two disjoint set of vertices $\mathbf{A}$ and $\mathbf{B}$. Denote $n_a:=|\mathbf{A}|$, $n_b:=|\mathbf{B}|$. Suppose the following conditions hold: $\Theta(1)<n_b<...
Mingzhou Liu's user avatar

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