Questions tagged [graph-theory]

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

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4
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1answer
131 views

Complexity of finding Exact Size Cut-Sets in Bipartite Graphs

I am interested in the problem of deciding if a cut-set of a given size $k$ (i.e. the number of edges crossing the partitions is $k$) exists in a given bipartite graph (both the graph and $k$ are part ...
1
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1answer
54 views

Problem property name where an optimal solution in a graph can be used as a solution in any subgraph

Suppose one is given a graph optimization problem where the optimal solution $S$ for the problem on graph $G$ can be used as a solution for any subgraph of $G$. In other words, given $S$ is an optimal ...
5
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1answer
131 views

Counting/Enumerating Minimal Edge Covers

A Minimal Edge Cover is an Edge Cover such that no other Edge Cover is a proper subset of it. Questions Which is the complexity of counting Minimal Edge Covers? Do we know any non-trivial ...
12
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1answer
1k views

Number of 4 cycles

Let $C_4$ be a cycle with four vertices. For an arbitrary graph $G$ with $n$ vertices and m edges say $m>n\sqrt n$, how many $C_4$s exist? Is there a lower bound for this?
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0answers
55 views

Reduction of irregular graphs, to regular graphs, while preserving hamiltonicity

I am wondering if this is a topic that has had research done... If I could reduce irregular graphs to regular graphs (including replacing redundant node clusters with dummy nodes), while ensuring ...
8
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1answer
211 views

Could chromatic number be easy to calculate when colouring is hard for some graph class?

Similar question was asked before, but there was an error in it so it was left unanswered Graph class with easy chromatic number, but NP-hard coloring Is there any infinite set of graphs $C$ such as: ...
4
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1answer
243 views

Is graph connectivity definable in existential MSO with vertices and edges?

Can $\exists$MSO$_2$ express graph connectivity? Monadic SO (MSO) is the fragment of second-order logic in which the second-order quantifiers range over relations of arity 1 only. $\exists$MSO is the ...
9
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0answers
102 views

Graphs with minimal-size induced subgraphs

I consider undirected graphs $G = (V, E)$ for which I write $\text{n}(G) := |V|$ the number of vertices and $\text{m}(G) := |E|$ the number of edges. For $d \in \mathbb{N}$, I say that $G$ is $d$-...
4
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0answers
151 views

A variant of hitting set: finding a matching to hit all edge sets

In general, the hitting set problem is given a family $\cal S$ of sub-sets, $\{S_1, \cdots, S_h\}$, and a universal set $U = \bigcup_{i\in [1,h]} S_i$. It asks for a minimum set $H \subseteq U$ ...
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1answer
318 views

Minimum graph cut with constraints

Given an undirected acyclic raph $G = \{V,E\}$, with each edge $e$ having weight $c_e$in the range $[-\infty, +\infty] $, I want to compute a partition of the graph into $N$ disjoint sets $G_i, i=1,......
6
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1answer
211 views

Paths of length $p$ in a Graph, $p$ a prime

I came across the following decision problem, for which I wondered whether anybody of you came across a similar problem and can give me some insight on its nature/complexity. Given as input a ...
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1answer
136 views

Finding Cheapest n-Path [closed]

Given a weighted directed acyclic graph, how can I find the cheapest path from an Origin Vertex to a Destination Vertex which ...
4
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1answer
519 views

Fast algorithm to find a maximum connected subgraph of k vertices

Given an undirected graph $G = (V, E)$ and a function $f: 2^V \to \mathbb{R}^+,$ where $2^V$ is the set of all subsets of $V$. Find a connected subgraph $T = (V_T, E_T)$ of k vertices such that $f(V_T)...
6
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0answers
110 views

Ramsey number for path in a colored tournament

I am looking about the following Ramseyish theorem with additional structure on tournament (see [1]). For all number of colors $k$, there exists $n=T(k)$ such that all tournaments of size $n$ ...
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2answers
174 views

Looking for an algorithm to construct a graph from two subgraphs

I am looking for an algorithm to construct a graph from two subgraphs. The problem is as following: Given two graphs g1(V, E) and g2(V, E), find a graph G(V, E) where V(g1) ⊆ V(G), V(g2) ⊆ V(G), E(g1)...
3
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1answer
148 views

Mutually exclusive replacement paths, an existing problem?

A replacement path is a simple path allocated to an edge $e \in G$ that connects the endpoints of $e$ in $G \setminus \{e\}$. The problem An undirected graph $G$ is given and the task is to allocate ...
-1
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1answer
65 views

How to make any graph 2-degenerate?

I have to show a PPT(polynomial time reduction) from 'Colorful graph Motif' to '2-Degenerate Steiner Tree'. As input graph should be 2-degenerate, but here is normal graph G (that is, basically an ...
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0answers
49 views

Complexity of extending $P_4 $-partition of cubic graphs

This is a question I posted on MathOverflow before but never got an answer. I am cross-posting it here. Surprising phenomena occurs when we want to extend a partial solution of some easy problems. We ...
8
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3answers
329 views

Check if graph stays connected after edge swap

Checking whether a (simple, undirected) graph is connected can be done in linear time in the number of edges. What I am looking for is a more efficient way of checking whether it stays connected after ...
3
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1answer
437 views

What exactly is Lawler's modification to Yen's algorithm and how does it work?

I recently read about Yen's algorithm, I understand the algorithm and it seems correct, however Wikipedia mentions that there exists "Lawler's modification" to the algorithm, which is described as ...
4
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1answer
123 views

Probability of a $k$-path in a random graph

Assume that $G\in G(n,p)$; if $p=\frac{\ln n +\ln \ln n +c(n)}{n}$, the following fact is well known: \begin{eqnarray} Pr [G\mbox{ has a Hamiltonian cycle}]= \begin{cases} 1 & (c(n)\...
4
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1answer
473 views

Minimum Union-Sum Cost Path

I have a minimum cost path selection problem that is different from the usual shortest path in that each type of cost is accounted only once in the total cost of the path if multiple edges on the path ...
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0answers
59 views

Random Multigraph ER-like model?

I was looking into multigraphs recently and I couldn't find a simple "goto" model for generating random multigraphs along the lines of the ER model of simple graphs. Specifically, I was hoping to find ...
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1answer
56 views

NP-Complete graph problems where a special vertex is given as input?

I am currently working on a graph theory problem where the instance includes a graph and a special vertex in the graph. I am trying to prove the NP-completeness of the problem as well as explore ...
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0answers
79 views

k-center 2.0: A stronger k-center condition

Given an unweighted, undirected graph, we can use the classical 2-appx for $k$-center to select a set $S$ of centers such that every vertex is within a distance of 2 of some center in $S$. Note that ...
1
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1answer
115 views

Total flow using minimum number of edges on a bipartite network

If I have a set of sources $S$ with total capacity $C$ and a set of sinks $T$ with the same capacity $C$, but not necessarily the same cardinality, is there an efficient way to find the minimum number ...
5
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1answer
211 views

Lower bound for triangle-free graphs

I was reading a set of notes where it says It can be shown that $\Omega(n^2)$ space is needed for one-pass algorithms to determine if an (unweighted, undirected) graph $G$ with $n$ nodes contains ...
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0answers
81 views

Counting vertex covers on a chain of k nodes that do not contain a sub-chain of length >=3

By a "chain of k nodes", I mean k nodes lined up like a linked-list: o-o-o.....-o . By "do not contain a sub-chain of length >=3", I mean that no cover should contain two edges that shares a node. ...
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0answers
39 views

Planarity testing of directed graph and 3d grid

I wonder if there exist definitions (and known algorithms) of planarity testing for the following case: 1- A directed graph let $G=(V=\{1,\ldots,n\},E)$ be a directed graph. Assume $e_{ij} =(v_i,...
4
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1answer
215 views

How hard is it to determine the chromatic number of a unit distance graph?

For example, is it NP-complete to decide whether a unit distance graph is 3-colorable?
6
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2answers
246 views

The chromatic number of a graph as a functor

I was fooling around with some concept and was wondering if this viewpoint is explored at all. Let INJ-GRAPH be the subcategory of graphs (with morphisms as homomorphisms) whose morphisms consist only ...
5
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0answers
164 views

Can graphons be useful outside of extremal graph theory?

I was recently introduced to the notion of a Graphon ([Quick introduction], [Original definition]) and I have been fascinated by the idea behind them and the way they combine fields such as graph ...
6
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1answer
156 views

Partition edges into edge disjoint walks

Consider an undirected graph $G=(V,E)$ and two sequences of $k$ vertices $S=s_1,\ldots,s_k$ and $T=t_1,\ldots,t_k$. A set of $k$ walks is called a $(S,T)$-walk partition if the walks form a ...
3
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0answers
150 views

Is 3-coloring bounded degree graphs subexponential: $O(\exp{(\sqrt{n}\log^2{n})})$? [closed]

We got an argument that 3-coloring bounded degree graphs is subexponential with complexity $O(\exp{(\sqrt{n}\log^2{n})})$. The treewidth of a planar graphs on $n$ vertices is $O(\sqrt{n})$ and 3-...
4
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1answer
207 views

Can $L=SL$ be shown with the replacement product instead of the zig-zag product?

(This is a bit of follow-up to https://cstheory.stackexchange.com/posts/comments/93266 but is a distinct enough question I though it should be on its own.) In Omer Reingold's logspace USTCON ...
5
votes
1answer
132 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
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0answers
87 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 ...
3
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0answers
93 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
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1answer
119 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 ...
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2answers
502 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 ...
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0answers
72 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
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0answers
87 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
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1answer
1k 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}...
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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
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1answer
182 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 valid ...
5
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0answers
81 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 ...
3
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0answers
353 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
130 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
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1answer
99 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 ...
10
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2answers
244 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. ...

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