Questions tagged [graph-theory]
Graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects.
1,404
questions
-3
votes
0answers
81 views
$K$ Disjoint Triangles [closed]
Given an undirected graph $G$ and a parameter $k$, the task is to decide if there exists a collection $C$ of $k$ vertex-disjoint triangles. We need to create a randomized algorithm in O*($2^{3k}$)-...
5
votes
0answers
125 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 ...
6
votes
1answer
93 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
122 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
84 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
158 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
85 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 ...
10
votes
1answer
253 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 ...
1
vote
1answer
56 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
134 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 $p+1$, ...
4
votes
2answers
193 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 ...
0
votes
0answers
32 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
198 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
106 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
43 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]\...
2
votes
0answers
56 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, ...
2
votes
1answer
92 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 \...
0
votes
0answers
47 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
52 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 ...
3
votes
2answers
83 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 $f:V(G)\to\mathbb{N}$ ...
3
votes
0answers
73 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
93 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 ...
5
votes
2answers
330 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_{...
1
vote
1answer
39 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 ...
9
votes
1answer
498 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 ...
2
votes
1answer
34 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 ...
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?
8
votes
3answers
744 views
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 $\sum\limits_{v\in V}|succ(v)|\times|pred(v)|$ ...
4
votes
1answer
222 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
154 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
122 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 ...
2
votes
2answers
161 views
Finding a random regular graph with degree d
I'm trying to find undirected random graphs $G(V,E)$ with $|V|$ = $d^2$ for $d \in \mathbb{N}$ such that $\forall v \in V: deg(v) = d$.
For $d \in 2\mathbb{N} +1$ this trivially is impossible as no ...
1
vote
2answers
90 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 |(...
0
votes
2answers
139 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
115 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 ...
Kristoffer Arnsfelt Hansen
4,0152 gold badges23 silver badges34 bronze badges
3
votes
1answer
50 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, ...
2
votes
1answer
58 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 ...
3
votes
2answers
520 views
Algorithms for graph generation given parameters
I guess there may be a large number of algorithms proposed for generating graphs satisfying some common properties (e.g. clustering coefficient, average path length, degree distribution, etc). I am ...
21
votes
2answers
1k views
Is feedback vertex set problem solvable in polynomial time for 3-degree bounded graphs?
Feedback Vertex Set (FVS) 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 ...
8
votes
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 ...
1
vote
1answer
133 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
25 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
35 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
71 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
42 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 ...
2
votes
1answer
140 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 ...
13
votes
3answers
2k views
What are graph grammars?
I have found information on graph grammars and graph rewriting, but the papers that I find on it are a bit thick. Can someone give a quick overview of what graph grammars are, as well as an overview ...
0
votes
0answers
36 views
Finding the cardinality of classes that divide all possible directed graphs into those that share k-subgraph cardinalities?
Let us have a set of nodes $V$, such that $|V|=N$. Let $G= (V,E)$ be an arbitrary directed graph on $V$. Let $U$ be the set of all possible directed graphs on $V$.Hence, $|U| = 2^{|V|^2}$. Now, for ...
0
votes
1answer
107 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 ...
