Questions tagged [graph-theory]

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

308 questions with no upvoted or accepted answers
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30
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Combinatorics of Bellman-Ford or how to make cyclic graphs acyclic?

Roughly speaking, my question is: How costly is to make a cyclic graph acyclic while preserving all simple $s$-$t$ paths? Let $K_n$ be a complete undirected graph on vertices $\{0,1,\ldots,n+1\}$. (...
24
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0answers
981 views

Counting Isomorphism Types of Graphs

Polya's counting theorem leads to an algorithm for counting (precisely) the number of isomorphism types of graphs with $n$ vertices in $\exp (\sqrt n )$ steps. From Polya theorem you get a formula ...
21
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0answers
752 views

$\Delta = 57, d=2$ Moore Graph

I am looking into the last open question regarding the existence of Moore Graphs of diameter 2. A problem that has been open in combinatorics for more than 55 years. You may recall that Hoffman and ...
16
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0answers
254 views

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 $...
15
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0answers
215 views

Mixing properties of random walks on graphs

I have a question about this paper (not behind a pay wall) on the Cheeger inequality for graphs. One of the main ideas of the paper is that random walks on graphs can be used to find sets with small ...
14
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0answers
545 views

Is it NP-hard to find (the root of) a small decision tree for a monotone boolean function?

Last year I spent some time trying to prove or disprove the following: Conjecture. Consider a graph $G$ and define a 2-DNF formula $\phi$ that contains a term $x \land y$ iff $x\mathrel{-\!-}y$ is ...
14
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0answers
1k views

What is the fastest deterministic algorithm for incremental DAG reachability?

As the title. The incremental algorithm maintains the reachability information of a DAG when it undergoes a series of edge insertions (but no deletions). And the algorithm supports constant query (if ...
14
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0answers
345 views

Finding all-pairs anti-distance

Thanks for a great forum. This is my first post here. I am working on a signal processing application and the core of one the main algorithms reduces to a graph theoretical problem. Let $G=(V,E)$ ...
14
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0answers
411 views

Question on Products of Graphs

Let $S_{n}(G)$, $C_{n}(G)$, $T_{n}(G)$ be the $n$-fold Strong Product, Cartesian Product and Tensor Product of a graph $G$ on $|V|$ vertices. Let the chromatic number ($\chi(G)$) and the independence ...
14
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0answers
482 views

Bi-partite expander graphs

My question relates to bi-partite expander graphs, defined as bi-partite graphs on $n$ left vertices, $m$ right vertices, constant left-degree $k$, such that For any linear-sized subset $S$ of the ...
14
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0answers
497 views

Approximation algorithm for Minimum Fill-In and/or minimum elimination ordering (for directed graphs)

Recently while working on a problem, I had to go through some of the literature on nested dissection. I happen to have one (maybe two?) questions related to the same. First, I will define a few ...
14
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1answer
322 views

Space-approximation Trade-off

In their paper Approximate Distance Oracles, Thorup and Zwick showed that for any weighted undirected graph, it is possible to construct a data structure of size $O(k n^{1+1/k})$ that can return a $(...
12
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0answers
216 views

Hitting edges in graphs at random and let them die with honor

Let $G=(V,E)$ be a finite simple 2-connected graph. Let $B\subseteq E$ be a set of bad edges of size $k:=|B|$. For each edge $e\in E$ we toss a fair coin ($p=1/2$) and if the outcome is head, we hit ...
12
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0answers
167 views

Minimal rare subgraphs

I am looking for any related work to the following problem. Say you have a large directed graph $G$ and you want to find rare (or unique) subgraphs of minimal size that are not isomorphic to any other ...
12
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0answers
325 views

Directed Sparsest Cut on Planar Graphs?

The (uniform) directed sparsest cut problem asks for a cut $(S,\bar{S})$ in a directed graph $G=(V,E)$ which minimize the ratio $\frac{\delta_{out}(S) }{|S||\bar{S}|}$, where $\delta_{out}$ is the ...
11
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0answers
332 views

A dynamic data structure to list triangles

Consider an undirected graph with $n$ nodes. Is there an efficient data structure that supports the following operations? Insert an edge into the graph Delete an edge from the graph Given a query ...
11
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0answers
157 views

Generating a random graph with constraints on spectrum

Consider two sequences $u_1 \geq u_2 \geq ... \geq u_n$ and $l_1 \geq l_2 \geq ... \geq l_n$ with $u_i \geq l_i$ for every $i$. Let $\mathcal{G}(l_{1:n},u_{1:n})$ be all undirected unweighted simple ...
11
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0answers
370 views

Cheeger's inequality for directed graphs?

Cheeger's inequality can be used to relate the size of the worst cut in the graph to the eigenvalue gap of a simple random walk on that graph. I am wondering if it possible to extend this result to ...
10
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0answers
142 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 $10\log n$. ...
10
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0answers
283 views

A bijection between ordered lambda terms and rooted planar maps?

Consider the following recurrence in two parameters $n$ and $k$: \begin{aligned} NF(0,k) &= 0 \\ NF(n,k) &= Neu(n,k) + NF(n-1,k+1) \\ Neu(n,k) &= [n=1 \wedge k=1] + \sum_{l=1}^{n-1}\sum_{...
10
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0answers
196 views

Complexity of unique coloring of graphs

The Isolation lemma of Mulmuley, Vazirani, and Vazirani can be used to show that certain $\mathsf{NP}$-complete problems can be reduced via randomized polytime reductions to the unique solution ...
10
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0answers
161 views

Number of graphs with prescribed spectrum

I have a question relevant to the number of graphs with prescribed spectral ratio. Let $A$ be the adjacency matrix of a graph on $n$ vertices. Let $\lambda_i$ be its $i$-th largest (signed) eigenvalue....
9
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0answers
78 views

NP-Hardness of 4-cycle packing problem in complete bipartite digraph?

A directed complete bipartite graph is a bipartite graph where there is exactly one directed edge between any two vertices from its two different parts. In other words, it's an orientation of a ...
9
votes
0answers
148 views

Random unbalanced bipartite graphs are good small set expanders

My question is about small set expansion properties of random unbalanced bipartite graphs. Fix a positive $\delta<1/2$, and a positive integers $n,m,d$. Let us call a bipartite graph $\mathcal{G}$...
9
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0answers
99 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 $d$-...
9
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0answers
313 views

Triangle arrangement problem

Suppose you are given an undirected graph $G$, with each vertex representing an equilateral triangle with sides of unit length. Does there exist an arrangement of these triangles in two dimensions (...
9
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0answers
242 views

Advances towards proving the Held-Karp conjecture for TSP

I've only began my research into the Held-Karp conjecture and I was wondering about recent progress in proving the conjecture. The Held-Karp relaxation is conjectured to have an integrality gap of $\...
9
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0answers
461 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 $...
9
votes
0answers
316 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 ...
9
votes
0answers
479 views

Statistical relationship between diameter and density in strongly connected random digraphs

I would like to know if there is any known relation between diameter and density in (connected) simple digraphs. Asymptotic results for n→∞, statistical, conditioned results that would restrict the ...
9
votes
0answers
175 views

Pseudorandom object yielding shrinkage in $\ell_p$ norm?

Extractors have the following property: For a random variable $X$ of min-entropy $k$ and a seed $Y$, denote the output of an $(k,\epsilon)$-extractor by $\mathrm{Ext}(X,Y)$. Then $\|\mathrm{Ext}(X,Y)-...
9
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0answers
267 views

Spectral gap for random bipartite regular graphs

For a graph $G$, let its Laplacian be $\Delta =I − D^{−1/2}AD^{−1/2}$, where $A$ is the adjacency matrix, $I$ is the identity matrix and $D$ is the diagonal matrix with vertex degrees. I'm ...
8
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0answers
269 views

Embedding a graph with specified edge lengths in d-dimensional space

Given any undirected edge-weighted graph (with weights > 0) and some dimension d, is there a way to assign positions in $\mathbb{R}^d$ to those vertices such that all of the edges between them have ...
8
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0answers
339 views

Upper Bound on Number of $n \times n$ Boolean matrices of Boolean rank at most $k$

An $n \times n$ Boolean matrix $B$ has Boolean rank $k$ if there exist matrices $L \in \{0,1\}^{n \times k}$ and $R \in \{0,1\}^{k \times n}$, s.t. $B = L \circ R$. Here $\circ$ denotes the Boolean ...
8
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0answers
100 views

Approximating a max-cut's intersection with other cuts

For the purposes of this question, a cut in a graph $G$ is the edge-set $\delta (S)\subseteq E(G)$ between some vertex-set $S$ and its complement. A max cut is one with at least as many edges as any ...
8
votes
0answers
392 views

Reducing the maximum degree in a graph

Given a weighted undirected graph $G = (V, E)$ with maximum degree $\mu$ and with positive edge weights, is it possible to construct another graph $H = (V \cup V', E')$ with maximum degree $\mu' = o(\...
8
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0answers
134 views

Extension of Cheeger's inequality with distinguished vertices

The standard Cheeger's inequality for graph $G$ states that $\frac{1}{2}$ $\lambda$ < $\phi(G)$ < $\sqrt{2\lambda}$ where $\lambda$ is the second smallest eigenvalue of the normalized ...
8
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0answers
339 views

Combinatorial method for computing the largest eigenvector of the adjacency matrix of a graph

Given a connected and non-bipartite graph $G=(V,E)$ with vertex set $V=\{1,\cdots, n\}$, let $A$ denote its adjacency matrix and let $deg(i)$ denote the degree of vertex $i$. Let $D$ be a diagonal ...
8
votes
0answers
161 views

Is the dominating set problem constant-factor-approximable in undirected path graphs?

I am interested in the complexity of the minimum dominating set problem (MDSP) in some specific graph class. A graph is an undirected path graph if it is the vertex-intersection graph of a family of ...
8
votes
0answers
174 views

Lossless, constant-degree expanders that expand large sets

It is known how to construct "lossless" unbalanced bipartite expanders with the following properties: the bipartite graph has $n$ left vertices, $m$ right vertices, left-degree $D$, and for all left-...
8
votes
0answers
109 views

Efficient Enumeration and Parameterization of Graphs Deriving from Delaunay Tesselations in 3D

This is a repost from a question on Computational Science. No proper answer was found and it was suggested that TCS might be able to answer. So here it goes: Is there an algorithm that enumerates the ...
7
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0answers
108 views

Algebraic methods for testing planarity

Mac Lane's planarity criterion states that a graph is planar if and only if it has a cycle basis such that each edge is contained in at most two cycles, we call such a basis a 2-basis. A 2-basis of a ...
7
votes
0answers
168 views

Lighting up all elements of a poset by toggling upsets

I consider the following game on a finite poset $(P, <)$. At each point of the game, I have a set of elements $S$ of the poset which are "on", and all others are "off". Initially $S = \emptyset$. ...
7
votes
0answers
86 views

Complexity of computing the simplicial width of a graph

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 $\{v_1,v_2\} \in E$, there ...
7
votes
0answers
141 views

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 $G$ ...
7
votes
0answers
167 views

Expansion of the union of two expander graphs

Suppose that $G$ and $H$ are both expander graphs on the same node set $V$ with a second largest eigenvalue of $\lambda_G$ resp. $\lambda_H$. Let $G\cup H$ be the graph on $V$ with the smallest set of ...
7
votes
0answers
323 views

An algorithm to compute the number of paths of length at most k

So I had to answer the following question: Given a graph $G = (V, E)$, and two vertices $v_i, v_j$, compute the number of walks between $v_i$ and $v_j$ of length at most $k$. $G$ is not too large, ...
7
votes
0answers
462 views

Which problems in graph theory can be stated as quadratic programs?

There seem to be many very interesting problems in graph theory that can be written in the form of maximizing/minimizing a quadratic form on either the Adjacency ${\bf A}$ or the Laplacian matrix ${\...
7
votes
0answers
211 views

Minimum weight expander

Expander constructions given an expander which is a sub-graph of a complete graph. Sometimes we don't want to construct an arbitrary expander, want to find an expander inside another given graph. In a ...
7
votes
0answers
111 views

Can one find good distance-2-separators in planar graphs?

It is known that planar graphs admit "good" separators, allowing to design PTASes for specific problems such as MINIMUM INDEPENDENT SET by recursive separation of the graph. However, it seems that ...