Questions tagged [graph-theory]
Graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects.
1,451
questions
0
votes
1answer
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 ...
3
votes
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 ...
8
votes
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 ...
-1
votes
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.
-2
votes
1answer
96 views
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
votes
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 ...
1
vote
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 ...
5
votes
0answers
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 $p+1$, ...
0
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0answers
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 $v\in V(G)$)....
4
votes
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 ...
7
votes
0answers
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 ...
1
vote
0answers
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]\...
3
votes
0answers
64 views
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, ...
0
votes
0answers
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 $\delta(S)$...
1
vote
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 ...
7
votes
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 ...
2
votes
0answers
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 ...
2
votes
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 \...
1
vote
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 ...
5
votes
0answers
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 ...
1
vote
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 ...
3
votes
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 $f:V(G)\to\mathbb{N}$ ...
0
votes
0answers
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?
2
votes
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 ...
6
votes
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_{...
5
votes
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)\...
2
votes
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
votes
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 ...
1
vote
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 |(...
3
votes
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, ...
0
votes
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
votes
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
votes
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 ...
1
vote
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 ...
0
votes
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 ...
-1
votes
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
votes
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 ...
0
votes
0answers
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
votes
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 ...
0
votes
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 ...
1
vote
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
votes
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 = \...
1
vote
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 ...
1
vote
0answers
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
votes
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
votes
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
votes
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
votes
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 ...
3
votes
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
votes
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 ...
