Questions tagged [graph-theory]

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

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Finding a subgraph in a biparitite graph with neighborhood of the same size

Setup: Let $G$ be a bipartite graph with parts $A$ and $B$ each of size $n$. Assume that $G$ is connected. For $S \subset A$, let $N(S) \subset B$ be the neighbors of $S$ in $B$. Similarly, for $T \...
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Uniquely 4-colorable Maximal Planar Graph Conjecture?

My question is on Uniquely 4-colorable Maximal Planar Graph Conjecture mentioned in On purely tree-colorable planar graphs (and other papers of same(?) team such as "Theory on Structure and ...
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3answers
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Number of connected components of a random nearest neighbor graph?

Let us sample some big number N points randomly uniformly on $[0,1]^d$. Consider 1-nearest neighbor graph based on such data cloud. (Let us look on it as UNdirected graph). Question What would the ...
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Non-rigid isomorphic structures

In many of the problems trying to solve hidden shift over some objects like graphs mainly the rigid classes are considered. For eg. in this and this isomorphism problem restricted over rigid graphs is ...
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Reduce a decision problem to an optimization problem

In this paper, the author shows that finding an edge coloring that minimizes the total sum on a multi-tree is NP-hard. In theorem 3.1, the author does so by reducing from 3SAT. I find his proof quite ...
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1answer
98 views

Who proved that a triangulation is 3-colourable implies its dual is bipartite

Let $G$ be a maximal planar graph (also called a triangulation); i.e, $G$ is a planar graph every face of which is a triangle. It is well known that the following three statements are equivalent: (i) $...
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1answer
85 views

On the paper “Quantum Computing Hamiltonian cycles”

The paper Quantum Computing Hamiltonian cycles claims: An algorithm for quantum computing Hamiltonian cycles of simple, cubic, bipartite graphs is discussed. It is shown that it is possible to evolve ...
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51 views

3-colourability of Eulerian maximal planar graph

The following paragraph is from this answer by David Eppstein (emphasis mine). A maximal planar graph is 3-colorable iff it is Eulerian (if it is not Eulerian, then the odd wheel surrounding a single ...
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4answers
973 views

Trees: complexity of counting the number of vertex covers

Which is the complexity of counting the number of vertex covers of trees? Is it still #P-complete, as for general graphs?
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Quantum error correction and graph codes

I was reading combinatorial approach towards quantum correction. A lot of work in this is on finding diagonal distance of a graph. Let me add definition of diagonal distance so that this remains self-...
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3answers
936 views

Relation between tree-width and clique number

Are there any nice graph classes for which the tree-width $tw(G)$ is upper-bounded by a function of the clique number $\omega(G)$, i.e. $tw(G)\leq f(\omega(G))$? For example, it is a classic fact ...
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2answers
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Is this vertex ordering optimization NP-Hard?

Could you help me to prove that the following problem is NP-hard? Problem. Given an undirected graph $G=(V,E)$, find an ordering of the nodes such that $\sum\limits_{v\in V}|succ(v)|\times|pred(v)|$ ...
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330 views

Is the following graph optimization problem approximable within a constant factor?

Let $G=(V,E)$ be an undirected graph, and let $\pi$ be a permutation of the vertices in $V$. For a node $v\in V$, we denote by $\text{pred}_{\pi}(v)$ (respectively $\text{succ}_{\pi}(v)$) the set of ...
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1answer
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Is there a planar 4-regular graph that is 3-acyclic colourable?

A colouring is said to be an acyclic colouring if there is no bicoloured cycle (i.e each cycle gets at least 3 colours). Burstein proved that 4-regular graphs are 5-acyclic colourable. It seems to me ...
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1answer
146 views

Tree decompositions of optimal width where every vertex is in a bounded number of bags?

Let $G$ be a graph on $n$ vertices whose maximum degree is at most $\Delta$ and whose treewidth is at most $k$. Does there exist a function $f(k, \Delta)$, independent of $n$, such that it is possible ...
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Graph problems in P with unknown lower bounds

I am looking for references to interesting graph problems, which are known to be in P, but their precise big-O lower bounds are elusive. I would split this into 2 classes: problems, where we know of ...
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2answers
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Proof that optimal solutions of LP Relaxation of independent set are half-integral

I saw somewhere that optimal solutions of LP Relaxation of independent set are half-integral, by what I mean the possible values of a solution are ${ \{0,0.5,1 \} }$. I'm looking for proof of that. ...
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Different version of approximation complexity and algorithm for densest-k-subgraph problem

In the densest-k-subgraph problem, we are given a graph $G =(V,E)$ and $k$, and we are asked to find a set $S \in V$ of vertices to maximize the number of edges in the induced graph of $S$, i.e. $|Ind[...
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1answer
88 views

Intuition behind the Charikar's LP formulation for densest subgraph problem

I understand why the LP gives the optimal solution for the densest subgraph problem. But don't understand the intuition behind the LP in this paper. Just mentioning the LP for maximum density of a ...
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1answer
696 views

Algorithm for finding a 3-cycle cover

Given: An undirected, unweighted graph Looking for: A disjoint vertex cycle cover where every cycle has at least 3 edges Is there any algorithm that solves this problem, possibly with some ...
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2answers
1k views

Why is “topological sorting” topological?

Why is "topological sorting" called "topological"? Is it just because it determines an order without altering any vertices or edges -- like a doughnut and coffee cup are topologically equivalent? Why ...
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1answer
505 views

Ordering of a DAG minimizing some definition of cost

Consider a DAG $(V,A)$ with a topological ordering $(v_1,v_2,\ldots,v_n)$. I define the cost of this ordering as the maximum over all $1\leq i\leq n$ of $|\{j\leq i \mid \exists k>i: (v_j,v_k)\in A\...
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2answers
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Is feedback vertex set problem is solvable in polynomial time for 3-degree bounded graphs?

Feedback Vertex Set is NP-complete for general graphs. It is known to be NP-complete for degree-8 bounded graphs due to a reduction from vertex cover. The Wikipedia article says that it is poly-time ...
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0answers
101 views

Network design with reachability pattern

We are given two sets of terminals $A$ and $B$. For each $a\in A$, we are also given $R_a\subseteq B$. Let $|A|+|B|=n$. We want to find a directed acyclic graph $G$ where $A$ and $B$ are subsets of ...
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1answer
136 views

How to efficiently find a loop between two nodes in a directed graph?

Given two nodes in a directed graph, how can I find a loop (if exists) that pass these two nodes? The loop cannot pass a node more than once. And if there isn't such a loop, how to efficiently ...
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1answer
257 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 $S = \emptyset$. ...
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1answer
94 views

Reference request: Depth- (or Breadth-) first search with hints?

Consider the standard s-t reachability problem of finding a path between nodes $s$ and $t$ in a directed graph $G$. A DFS or BFS could solve it. Would it be possible to pre-process the graph and ...
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2answers
220 views

Is the maximum independent set in cubic planar graphs NP-complete?

In their famous book, Garey and Johnson, write a comment that the maximum independent set problem, in cubic planar graphs is NP-complete(page 194 of the book). They say this is by a transformation ...
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1answer
625 views

Is there a problem that is easy for cubic graphs but hard for graphs with maximum degree 3?

Cubic graphs are graphs where every vertex has degree 3. They have been extensively studied and I'm aware that several NP-hard problems remain NP-hard even restricted to subclasses of cubic graphs, ...
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1answer
687 views

Rapidly mixing Markov chains on 3-colorings of a cycle

The Glauber dynamics is a Markov chain on the colorings of a graph in which at each step one attempts to recolor a randomly chosen vertex with a random color. It does not mix for the 3-colorings of a ...
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1answer
110 views

What is the complexity of this weighted b-edge matching problem?

I'm wondering about the complexity of the following variant of the Generalized Weighted b-edge Matching problem: Input: An undirected multigraph $G = (V, E)$ without loops, an edge partition $(E_1,...
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5answers
2k views

Finding outer face in plane graph (embedded planar graph)

I have a planar graph, for which I have computed a combinatorial embedding and coordinates in the plane. So all my arcs are now oriented in the plane, respective to their end vertices. Computing a ...
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1answer
47 views

Distinguish Graph from Tree using Adjacency Matrix

Given an adjacency matrix, is there a way to determine if the graph will be a tree or a graph (whether or not there is a cycle). For example, given the adjacency matrix: ...
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0answers
28 views

Best known bounds on feedback arcset in high-girth directed graphs?

I asked this question over at MathOverflow, but thinking about it a little more I think it is a more natural fit here. Let $G$ be a directed graph with $n$ vertices and $m$ edges such that every ...
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0answers
76 views

Variation of edge-disjoint spanning trees

In a directed graph, I want to find 2 edge-disjoint spanning trees (arborescence), with the extra restrictions that edges in the 1st tree are not forward arcs in the 2nd tree. Are there existing ...
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1answer
147 views

Fast Computation of First k Eigenvectors of Graph Laplacian

I'm looking for references for the following; Given the unnormalized Laplacian matrix of a graph $L = D - A, ~ L \in \mathbb{R}^{n\times n}$ Algorithm(s) to efficiently estimate the first $k$ (...
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164 views

Sublinear time path existence

Consider a graph $G = (V,E)$ with $N$ edges. Consider two vertices $u_1, v_1 \in V$. We wish to find whether there exists a path of length $4$ between these two vertices or not. This is easy to do in $...
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1answer
49 views

When can partial spectral sparsifiers be combined?

A few important papers about spectral sparsifiers and friends contain a technical idea that involves building many different sparsifiers that each "partially" solve the problem, and then combining ...
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2answers
113 views

List of NP-Complete graph problems/ properties?

Is there a good source to find various decision problems on graph and networks? For a project I'm doing it'd be useful to be able to look at lots of different problems. Is there a good source for ...
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1answer
56 views

Reachability Query for Tree

What is the best complexity for reachability queries on trees so far please? There is no constraint on the directions of the edges in the tree. According to Mikkel Thorup, there is an oracle of size $...
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1answer
136 views

Problems that are NP-hard to approximate even when the input graph is regular

Are there graph problems which are NP-hard to approximate even when the input graph is regular? For instance, are there optimization problems that are NP-hard to approximate within $O(n^\epsilon)$ ...
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1answer
99 views

Is Hamiltonian cycle fixed parameter tractable with parameter clique cover?

We conjecture that Hamiltonian cycle is fixed parameter tractable with parameter clique cover, given $k$-clique cover. Let $G$ be connected simple graph. $k$-clique cover of graph $G$ is partition ...
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1answer
129 views

Known property? Maximum radius of connected induced subgraph

I was wondering if the following graph property has a name and has been researched: Consider any connected induced subgraph $H \subseteq G$. Then $r(G)$ denotes the maximum radius of any such $H$. I ...
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1answer
158 views

Minimum vertex cover and odd cycles

Suppose we have a graph G without odd cycles. Consider the minimum vertex cover problem of G formulated as a linear programming problem, that is for each vertex $v_{i}$ we have the variable $x_{i}$, ...
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0answers
88 views

Isomorphic subforest problem

I recently read that the following problem is NP-Complete: Given a tree $T$, and a forest $F$, is there a subgraph of $T$ isomorphic to $F$? The reference provide was to the textbook “Computers and ...
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1answer
888 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 ...
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94 views

How to approach the “traveling salesman problem” with cost changing every time salesman reaches a new city

Let's say instead of finding the shortest path we have to maximize the profit in a year of the salesman under the following constraints. Salesman can go to a different city only on weekends, all ...
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1answer
124 views

Hardness of finding if a vertex lies on a simple directed path between two vertices

Given a directed graph $G = (V, E)$ and three vertices $u, v, w \in V$. Is it NP-Hard to find whether there is a simple path from $u$ to $v$ passing through $w$? I found a couple of hardness ...
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1answer
61 views

Reduction graph to planar bounded treewidth and bounded diameter graph

We got reduction graph to planar bounded treewidth graph, but this is unlikely to be true. Let $H$, the planarizing gadget, be planar graph with four distinguished vertices $u,u',v,v'$ on the outer ...
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1answer
185 views

Is this a known problem, and is it #P-complete?

Let $G=(V,E)$ be an undirected graph. What I call a selection function of $G$ is a partial function $f:V \to E$ such that for every node $v$, if $f(v)$ is defined then it is one of the adjacent edges ...

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