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

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9
votes
2answers
120 views

Can a hereditary graph class contain almost all, but not all, n-vertex graphs?

Let $Q$ be a hereditary class of graphs. (Hereditary = closed with respect to taking induced subgraphs.) Let $Q_n$ denote the set of $n$-vertex graphs in $Q$. Let us say that $Q$ contains almost all ...
2
votes
1answer
100 views

Perfect Matching with ``set-over-like" constraints?

Problem Description: Let k and n be some natural numbers. We are given a complete bipartite graph G where each side of G has n vertices. G is edge-labeled with labels being subsets of {1,...,k}. We ...
7
votes
1answer
174 views

Combinatorial discrepancy of the system of all cuts

This is a variant of this previous question. In the meantime I have learned that what I am really interested in is the discrepancy of the system of all cuts of the complete graph on $n$ vertices. More ...
4
votes
1answer
83 views

Minimum weight matching in general graphs with additional input specifying the number of matched edges

We know of the minimum weight perfect matching problem in general graphs which can be solved using a primal-dual algorithm. Assume, we have an additional constraint specifying the exact number of ...
3
votes
2answers
646 views

Finding the two shortest paths while minimizing the number of nearby/common edges

The shortest path problem between 2 arbitrary nodes is one that has been covered extensively and the solution is well-known. Consider the edge costs to be arbitrary. Consider the following variant: ...
4
votes
0answers
67 views

Random Sampling Threshold to Get a Connected Induced Subgraph

Working on network design this summer I have come across certain applications that have inspired me to ask the following question: Given an undirected connected graph $G=(V,E)$ what is the minimum ...
5
votes
1answer
82 views

Existence of certain graph gadget related to coloring odd hole free graph

Crossposted from MO. Wondering about the existence of graph gadget related to coloring (or 3-coloring) odd hole free graphs. Let $G$ be simple $k$-chromatic connected graph with two vertices $u,v$. ...
0
votes
0answers
62 views

Find the number of vertices that belong to all the maximum matchings of a general connected graph [duplicate]

The given graph is connected but not necessarily bipartite. Please describe the complete approach with useful links , I read stuff related to augmenting paths but could not comprehend well. An O(VE) ...
9
votes
2answers
123 views

Exact formula for the number of spanning trees of a rectangle

This blog talks about generating "twisty little mazes" using a computer an enumerating them. The enumeration can be done using Wilson's algorithm to get the UST, but I don't remember the formula for ...
1
vote
1answer
162 views

Two paper appear to imply collapse via coloring $P_5$-free graphs

Found this from graphclasses.org. Two papers give conflicting results for coloring $P_5$-free graphs which appear to imply $P=NP$. From Polynomial-time algorithm for vertex k-colorability of ...
2
votes
0answers
17 views

centralized deterministic Spanner construction with low degree and low stretch

Does there exist a centralized deterministic spanner construction with low degree and low stretch both independent of the graph diameter (no log D factor), but can be dependent on the number of nodes. ...
-2
votes
2answers
76 views

Is there a typo in this definition of Minimal Maximal Matching [closed]

in the following paper http://journal.frontiersin.org/article/10.3389/fphy.2014.00005/full the author describes the minimax problem as follows: Given a graph G = (V, E), we are looking for a ...
4
votes
0answers
70 views

Graph factors of maximum weight

I am trying to find references to a weighted version of the graph factor problem for the case when the "target degree" is a set of integers with "gaps" of size at most one. The unweighted version of ...
2
votes
1answer
140 views

Complexity of simple undirected graph isomorphism problem

We define a simple undirected graph as a graph where no vertex has a loop and there is only zero or one undirected, unweighted edge between any pair of vertices. My question: What is the complexity ...
11
votes
2answers
330 views

Number of vertices present in all maximum matchings

Given a graph $G$, we need to find the cardinality of the largest set of vertices so that each of them are present in every maximum matching possible. Is there a solution beside the obvious remove ...
0
votes
0answers
37 views

How to prove that the probability that a random graph has a stable set of size $2\lceil \log n\rceil$ is sub-constant? [migrated]

Given a random graph on $n$ vertices where each edge is included with probability $1/2$. Lets call it $G=(n,1/2)$. How can we show that the probability that this graph has a stable set of size at ...
2
votes
1answer
91 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 ...
0
votes
1answer
71 views

How many edges are cut in a balanced partition of a graph?

Consider a graph on n nodes and e edges that is partitioned into k "balanced" subgraphs in the sense that each block has an equal number of nodes and the number of cut edges is minimized. Is there a ...
1
vote
1answer
72 views

Finding all possible simple cyclic paths in a digraph

I have a strongly connected component with over 200 vertices and more than 600 edges. I need to iterate through each simple cycle in the graph exhaustively, without specifying a particular node. Is ...
6
votes
1answer
90 views

Lexicographic perturbation for euclidean shortest path instances?

Assume we have an undirected graph $G=(V,E)$ and vertex locations $\pi: V \rightarrow \mathbb{R}^2$. I am looking for a procedure to perturb the vertex positions to obtain new positions $\pi'$ such ...
16
votes
0answers
227 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 ...
14
votes
1answer
274 views

How expensive may it be to destroy all long s-t paths in a DAG?

We consider DAGs (directed acyclic graphs) with one source node $s$ and one target node $t$; parallel edges joining the same pair of vertices are allowed. A $k$-cut is a set of edges whose removal ...
3
votes
0answers
92 views

How many neighbors does a vertex has which are closest to a source vertex in random regular graphs?

Let $G=(V,E)$ be an undirected, random $r$-regular graph. Let $s,t\in V$, and denote by $N(v)=\{u\in V\mid (u,v)\in E\}$ the neighborhood of $v$. I'm looking for the distribution of the number of ...
5
votes
1answer
177 views

$d$-regular bipartite expander graph

I have seen that there exists $d$-left regular bipartite graphs. My question is do there exists $d$-regular bipartite expander graphs in which both the degree of the left and the right vertices is ...
12
votes
1answer
223 views

Does treewidth $k$ imply the existence of a $K_{1,k}$ minor?

Let $k$ be fixed, and let $G$ be a (connected) graph. If I'm not mistaken, it follows from the work of Bodlaender [1, Theorem 3.11] that if the treewidth of $G$ is roughly at least $2k^3$, then $G$ ...
1
vote
0answers
65 views

Matrix of ranking

In any graph of n nodes in any dimension, define matrix $M_r$ of ranking as $\forall r_{ij}\in M_r$, $r_{ij}$ is the ranking of j to i. That is, j is the $r_{ij}$th nearest node to i. Therefore, any ...
12
votes
1answer
264 views

Structure of graphs that exclude a perfect matching on four vertices as an induced graph

I am interested in understanding the structure of the class of graphs $G$ such that there is no vertex induced subgraph on four vertices that is a perfect matching. Stated differently for any four ...
0
votes
1answer
69 views

Calculate sum of reciprocal rank in arbitrary large graph

For arbitrary graph of n node, can I approximate $\sum_{v\neq u}\frac{1}{Rank_u(v)^a\times Rank_v(u)^b}$ with $\sum_{v\neq u}\frac{1}{Rank_u(v)^{a+b}}$ or not when n is large enough? $a,b>0$, ...
6
votes
0answers
88 views

Edge-partitioning into rainbow triangles

I'm wondering if the following problem is NP-hard. Input: $G = (V,E)$ a simple graph, and a coloring $f : E \to \{1,2,3\}$ of the edges ($f$ does not verify any specific property). ...
3
votes
1answer
90 views

Partitioning the edges of a complete graph into smaller complete graphs

Consider the (simple, undirected) complete graph $K_n$. Is it possible to partition its edges into less than $n$ cliques, each having less than $n$ vertices? Note: It is easy to see that $n$ cliques ...
4
votes
1answer
249 views

Counting the number of K4

I was going over this paper and I don't understand a certain proof (section five phase 2). Given a graph G=(V,E) partitioned into the sets of vertices L and H. The vertices in L are at most D where D ...
0
votes
0answers
39 views

Approximate distance preserving sparse graph representation that are not necessarily subgraphs

I am looking for a type of graph sparsifier that I think I have seen somewhere but now I can't find the paper anymore. I think the paper referred to it as a spanner, but that term is used for so many ...
5
votes
1answer
250 views

Sparser Bipartite graphs?

Maximal Planar Bipartite graphs are sparser than maximal planar graphs. For which other classes of graphs are maximal Bipartite members sparser than arbitrary maximal members. Let $\mathcal{C}$ be a ...
3
votes
2answers
166 views

Runtime of Tucker's algorithm for generating a Eulerian circuit

What is the time complexity of Tucker's algorithm for generating a Eulerian circuit? The Tucker's algorithm takes as input a connected graph whose vertices are all of even degree, constructs an ...
0
votes
1answer
79 views

TSP heuristics for limited distance information

this is my first question on Theoretical CS. :) I've posted a similiar question on Mathoverflow and a friendly user advised me to post my question on this site. Problem: I'm looking for TSP ...
0
votes
0answers
35 views

Characterization of an irreducible matrix

A matrix is irreducible, if it is not similar via some permutation to a block upper triangular matrix that has more than one block of positive size. Equivalently, for a 0-1 matrix, if it is viewed as ...
0
votes
1answer
60 views

Seeking for a game for modelling a problem using game thoery [closed]

I have a problem which I want to formulate it as a game, using game theory. In this problem there is several agents, we can consider the agents as the employees of different offices, these agents have ...
3
votes
1answer
217 views

Paths and Probabilities for a Random Walk on a Graph

I'm working on a problem about $N$ nodes that are randomly positioned on a rectangular grid. I want to take a sample of $n\leq N$ nodes by randomly selecting the first node then visiting the nearest ...
1
vote
2answers
236 views

Shortest path in DAG with path dependent arc costs

I've got the following problem Consider a DAG $G=(V,E)$ with $V=[v_1,…,v_n]$, and edge-set $E=[e_1,…,e_m]$, with associated costs $c_1,…,c_m$. The problem is to find the shortest paths from an ...
9
votes
1answer
135 views

When does a graph admit an orientation in which there is at most one s-t walk?

Consider the following problem: Input: a simple (undirected) graph $G=(V,E)$. Question: Is there an orientation of $G$ satisfying the property that for every $s,t \in V$ there is at most one ...
10
votes
2answers
212 views

Name the graph class: Disjoint union of a clique and an independent set

Let $G$ be a graph which is the disjoint union of a clique and an independent set, i.e. $$G = K_{n_1} + \overline{K_{n_2}} = K_{n_1} + I_{n_2} .$$ The graph class of all such graphs is characterized ...
0
votes
1answer
148 views

Is finding whether k different perfect matchings exist in a bipartite graph co-NP?

Few definitions first. The co-NP problem is a decision problem where the answer "NO" can be verified in polynomial time. The perfect matching in a bipartite graph is a set of pairs of nodes (a pair is ...
0
votes
0answers
47 views

MRF vs CRF dependencies

If Markov random fields (MRF) model explicit short range dependence between features and implicit long range dependencies (knock-off effect), can we say that a Conditional random field (CRF) models ...
3
votes
2answers
128 views

Question about an old result of Erdős and Simonovits

My question concerns the following result of Erdős and Simonovits. A graph $G$ is $d$-almost-regular if $d\delta(G)\geq \Delta(G)$. Theorem 1 of the above paper states that a large $n$-vertex ...
0
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0answers
133 views

Theoretical background of Classes and Objects

I would like to learn about the possible ways of formalizing Classes and Objects (in programing languages like java) using formal languages. Where should I start? This might be related to my previous ...
0
votes
0answers
76 views

Proper definition of a graph with with loops and parallel edges

When a graph has parallel edges and self loops one cannot identify an edge with the set of two adjacent vertices it connects. I have failed to find a good formal definition of a non-simple graph. ...
3
votes
1answer
103 views

What is the best way to find an induced cycle basis of a graph?

My question is essentially what comes in the subject line: what is the best way to find an induced cycle basis of a graph (i.e., a cycle basis of the graph in which each cycle is an induced subgraph ...
5
votes
0answers
137 views

Approximating the clique size of the graph

Let $G=(V,E)$ be a graph. For a given $\rho \leq |V|$ and $\epsilon$ with $(0<\epsilon<1)$, is there any sublinear query algorithm known/possible to decide if the graph has a clique of size ...
9
votes
1answer
321 views

Is the complexity of this covering problem known?

Let $G=(V,E)$ be a graph. A vertex set $X\subseteq V$ is called critical if $X\neq\emptyset$ and no vertex in $V\setminus X$ is adjacent to exactly one vertex in $X$. The problem is to find a vertex ...
4
votes
1answer
129 views

Special properties of bipartite expanders

It is well known that expanders, and often the special case of bipartite expanders, have found many uses in derandomization, coding, etc. However, I am curious if there are any special properties of ...