Questions tagged [graph-theory]

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

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51 views

Do such instances always admit a 3D matching?

I want to know whether the following kinds of special instances of the 3D Matching problem are ``yes" instances, i.e., admit a 3D matching. We are given 3 sets $A,B,C$ containing $m$ elements ...
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1answer
150 views

TSP with "enemy" nodes

I am curious if the following variation of the traveling salesman problem (TSP) (or a vehicle routing problem (VRP) version) occurs in the literature and has a name I could search for. The story/idea ...
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1answer
152 views

Finding vertex separator such that the induced subgraph has minimal number of edges

My problem is related to edge and vertex cuts with a little twist. Given a graph $G$ and two vertexes $u$ and $v$. I want to find a set of vertexes $S \subset V$ that disconnects $u$ and $v$ such that ...
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1answer
131 views

What are the known classes of undirected graphs such that every graph belonging to that class is guaranteed to have a Hamiltonian Path?

One trivial class of graphs is the class consisting of complete graphs or complete bipartite graphs with equal sized partitions. I would love to know if more such classes exist.
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1answer
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upper bound on the total number of fixed-length paths in an acyclic graph [closed]

I was wondering if there is an upper bound on the total number of fixed-length paths (path length from 1 to $n-1$ given $n$ nodes) in an acyclic graph (not directed) of $n$ nodes? If so, can you point ...
4
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1answer
179 views

Does such a bipartite graph exist?

In the course of my studies on graphs I sometimes use gadgets. I recently came upon a need for a certain bipartite graph with the following properties, and I am wondering if anyone knows if such a ...
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0answers
94 views

Complexity of counting 3-colourings of planar bounded degree graphs

The following are known: It is #P-complete to count the number of 3-colourings of a planar graph (with respect to randomized reductions) [1]. For all $d\geq 3$, it is #P-complete to count the number ...
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187 views

How to find the second smallest cut in a graph?

For an undirected graph, how do we find the second smallest $s,t-$cut(s) for some $s,t\in V$? What's the time complexity of this computation? What if we only cared about finding a cut of size $p+1$, ...
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34 views

Listing Eulerian orienations with special properties

An orientation of a simple undirected graph is said to be Eulerian if every vertex has the same number of in-coming edges and out-going edges (i.e., in-degree($v$)=out-degree($v$) for all $v\in V(G)$)....
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1answer
219 views

Complexity of relaxed edge colouring

A (proper) $k$-edge colouring of a graph $G(V,E)$ is a function $f:E\to\{1,2,\dots,k\}$ such that adjacent vertices are mapped to different colours; that is, $f(e)\neq f(e')$ if $e$ and $e'$ are ...
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112 views

Decomposing graph homomorphisms

A homomorphism $h: G\to H$ from a graph $G$ to a graph $H$ is a function from the vertices of $G$ to those of $H$ which preserves edges, that is, if $(x,y)$ is an edge of $G$ then $(h(x),h(y))$ is an ...
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49 views

Generalizing PageRank for tripartite graphs

Problem I have the following directed tripartite graph $G(E\cup V\cup P, A)$, where there is a many-to-one symmetric relationship between the subsets V and E - $e\in E,v\in V,[e, v]\in A \iff [v, e]\...
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Number of connected partitions (or labelings) in a grid graph

Let $G$ be a 2D lattice graph (undirected) of size $W\times H$. Each "inner" vertex has $4$ adjacent vertices, whereas "boundary" vertices have $2$ or $3$ adjacent vertices, ...
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64 views

Prove that this linear relaxation has half-integral extreme points

Given a graph $G=(V,E)$, here is a Linear Relaxation of the edge cover polytope: (1) For each $v \in V, \sum_{e \in \delta(v)} x_e \geq 1.$ (2) For each $e \in E$, $0 \leq x_e \leq 1.$ Here $\delta(S)$...
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1answer
58 views

Weak incidence colouring

Let $G(V,E)$ be a simple undirected graph, let $k\in \mathbb{N}$, and let $I(G)=\{(v,e)\ :\ e\in E, v\in e \}$ (here, $v\in e$ means $v$ is incident on $e$). A $k$-incidence colouring of $G$ is a ...
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1answer
133 views

Monotone circuit representations of paths in a graph?

Consider a directed graph $G = (V, E)$ with a source $s \in V$ and sink $t \in V$. From $G$, I can define a monotone Boolean function $\phi_G$ on the set of variables $E$, in the following way: every ...
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98 views

Graph recovery from pairwise-common neighborhoods

Define the common neighborhood of two vertices $u$ and $v$ of a simple undirected graph as the set $N(u,v)=N(u)\cap N(v)$. For a simple bipartite graph $G=(U,V,E)$, define the pairwise-common ...
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1answer
100 views

How long does it take at most for $k$ boolean variables to map back to themselves with a positive disjunctive update rule?

I have a vector of boolean variables $v=(x_1,\dots,x_k)$. In each step each variable is updated according to a positive disjunction like so: $x_1 \leftarrow x_i \vee \dots \vee x_j$ $\dots$ $x_k \...
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2answers
138 views

Maximum cliques of the transitive closure of a chordal DAG

Let $G=(V,A)$ be a directed acyclic graph, for which the underlying undirected graph is chordal (so that every induced cycle in the underlying undirected graph is a triangle). It is known that in a ...
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163 views

Minimum spanning tree, but with an unusual objective function

This is a problem that came up in my study of rumour networks. I was wondering if anyone had thoughts or references on this problem. If we have a rooted tree $T = (V,E)$ with root $r$, I first label ...
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1answer
55 views

Bipartite graph projections, with threshold

Let $G=(\top,\bot,E)$ be a bipartite graph: $E\subseteq \top\times\bot$. The projections $G_\bot = (\bot,E_\bot)$ and $G_\top = (\top,E_\top)$ of $G$ are defined as follows: two vertices are linked ...
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2answers
95 views

Graph labelling where vertices with a common neighbour get different labels

Is the following notion of graph labeling (let me call it special labeling) ever studied in the literature? A special labelling of a (simple undirected) graph $G$ is a function $f:V(G)\to\mathbb{N}$ ...
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27 views

Unique naming/labeling of $40$-node strongly regular graphs

Brendan McKay's webpage lists all possible $40$-node strongly regular graphs. Is there a standard way to name them uniquely?
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1answer
36 views

Maximum weight matching with classes of edges in a multi-edge bipartite graph

Posted a similar question in mathoverflow, have tried to reduce this to Ford Fulkerson, but been stuck. Thought I'd ask TCS community to see if there are any ideas from individuals, here. Consider a ...
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2answers
390 views

Is that edge orientation optimization problem NP-hard?

Is the following optimization problem NP-hard? Problem. For a given undirected graph $G=(V,E)$, find an orientation of the edges that minimizes the objective value $\sum_\limits{u\in V} ~\left( d_{...
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1answer
256 views

Is this edge orientation optimization problem NP-hard?

Is the following optimization problem NP-hard? Problem. For a given undirected graph $G=(V,E)$, find an orientation of the edges that minimizes the objective value $\sum_\limits{v\in V} ~d_{out}(v)\...
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0answers
177 views

Is this node permutation optimization NP-Hard?

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 $\text{succ}_{\pi}(v)$ the set of neighbors of $v$ that occur after $v$ in ...
6
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1answer
146 views

Random Cerny Conjecture

For simplicity, all DFAs will be using the binary alphabet $\{0,1\}$. Let $M$ be a synchronizable DFA. We let $p(M,n)$ be the probability that a random $x\in \{0,1\}^n$ will synchronize $M$. We define ...
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2answers
108 views

Name of this graph partitioning problem? (related to coloring)

Given a graph $G=(V,E)$ and an integer $k$, find a partition $P_1, P_2, \dots, P_k$ of $V$ into $k$ parts that minimize the total number of edges between two vertices in the same part, i.e. $\sum_i |(...
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1answer
57 views

Problem conditions to use Laplacian solvers

I am trying to use Laplacian Solvers to solve a linear equation. I am just learning it (form here), so my question is very basic and it might not even make sense. Suppose that we want to solve Ax=b, ...
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2answers
208 views

Minimum number of triangles required to cover a complete graph?

Let $K_n$ be a complete graph, I am interested in knowing the minimum number of triangles required to get a edge cover of $K_n$. In case there is no closed-form solution to this problem, then I would ...
3
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1answer
178 views

A stronger Flow Decomposition Theorem?

In the classic Network Flows: Theory, Algorithms, and Applications book (pages 80/81) the flow decomposition theorem is stated as follows: Every nonnegative arc flow x can be represented as a path ...
2
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1answer
67 views

random sampling DAGs via nilpotent matrix sampling

The adjacency matrix of an acyclic graph is known to be a nilpotent matrix (all eigenvalues are zero). I am interested in sampling DAG adjacency matrices or equivalently sample random nilpotent ...
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1answer
155 views

Effect of self loops on mixing time?

Consider 2 graphs G1 and G2. G1: Any non-regular graph. G2: Same graph but with added self-loops such that degree of each node is the same (either some $\Delta$, or maximum '$n$', where $n$ is the ...
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0answers
28 views

Optimum first stage solution of two stage stochastic shortest path induces tree

I struggle with the proof of Lemma 1 in the Paper "Hedging Uncertainty: Approximation Algorithms for Stochastic Optimization Problems" by Ravi and Sinha and hope this is the right community ...
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1answer
43 views

Multi agent path following with collision avoidance with pre-determined path

I am working on a multi-agent pathfinding algorithm. I am aware of other techniques, but planned on the folowing strategy only. The problem: There is 12x12 grid, with a few solid blockades within them....
3
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0answers
78 views

A class name for series-parallel graphs of same length

I'm currently working on graphs classes where the distance between two specific vertices is the same in every connected spanning subgraphs, and I am looking for a name for this class. Given a ...
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29 views

Optimum partitioning of vertices into mutually disjoint subsets in a weighted graph

tl;dr I'm trying to partition my students into groups with respect to their preferences, i.e. they can declare if they want to be with someone in a group or if they do not want to be with someone in a ...
2
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0answers
48 views

Internal as well as external partition of (regular) graphs

Let $G$ be a simple finite undirected graph. Let $\{V_1,V_2\}$ be a partition of its vertex set; that is, $V_1\cup V_2=V(G)$ and $V_1\cap V_2=\emptyset$. The partition $\{V_1,V_2\}$ is said to be an ...
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1answer
123 views

When does a bipartite graph have bounded treewidth?

As the title says, I want to know when the treewidth of a bipartite graph is bounded by a constant. What families of graphs are both bipartite and bounded treewidth? More generally, I would like to ...
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0answers
50 views

Are there classes where all Eulerian orientations can be listed in polynomial time?

Is there is a subclass of regular graphs (say 4-regular graphs) for which there is a polynomial time algorithm to list all Eulerian orienations? An Eulerian orientaiton of an (undirected simple) graph ...
3
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1answer
115 views

Minimal clique edge cover vs minimalist (assignment-minimum) ones

Given a graph $G=(V,E)$, a clique edge cover is a collection $C$ of subsets of $V$ such that each element $c$ of $C$ is a clique ($c \times c \subseteq E$) and $G$ is the union of these cliques ($E = \...
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0answers
87 views

Can someone recommend a reference on graph minors structure theorem and sublinear treewidth?

Can someone recommend a reference on graph minors structure theorem and sublinear treewidth? Doesn't have to be the newest/strongest results as long as it's easier than tracking down all the papers ...
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41 views

Directed tree decompositions on subtrees of DAGs

Given a DAG, is the arboreal decomposition of the DAG with the guarantee that given a node $x$, $v$ such that $x$ is reachable from $v$ are in the subtree of $x$? If not, is there a similar ...
4
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1answer
116 views

Standard Name for a vertex removal like operation

I have an operation that looks a lot like vertex removal, and I'm wondering if there's a standard name for it. Given a graph $G$ we remove a vertex $v$, but instead of removing the edges that were ...
4
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2answers
223 views

Does replacing each vertex of $G$ by $H$ increase treewidth of $G$ by at most $\Delta(G)$?

I am asking this question from the context of parameter preserving reductions which has implications for kernelization (See Theorem 18 of [1] for an example). For simplicity, here I am assuming that ...
2
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1answer
73 views

Has this bipartite graph problem been studied?

I have a directed bipartite graph with vertex sets $U$ and $V$, directed edge sets $E(U,V)$ and $E(V,U)$, and a demand function $d \colon U \rightarrow \mathbb{Z}$. I want to find a function $f \colon ...
2
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1answer
182 views

Obtaining Sets of Ancestors Quickly in a Directed Acyclic Graphs

Suppose I have a DAG, $G = (V, E)$ and we know that all nodes in the DAG have at most $A$ ancestors. Let $V' \subseteq V$ be a subset of vertices of $V$. Is there a way to obtain the set of all ...
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0answers
50 views

Counting subsets of bipartite graph part which admit an induced perfect matching

Given a bipartite graph $G=(U \sqcup V, E)$, count $U^\prime \subseteq U$ for which $\exists V^\prime \subseteq V$ such that the induced subgraph $G[U^\prime \sqcup V^\prime]$ contains a perfect ...
2
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1answer
125 views

Complexity of MAX-ONEs Monotone 2-SAT with $n^{3/2}$ or $C n^2$ clauses?

Let $\phi$ be negative monotone 2-CNF on $n$ variables and $n^{3/2}$ clauses. What is the complexity of finding satisfying assignment with maximum number of ones $k$? Alternatively let $G$ be a graph ...

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