# Questions tagged [graph-theory]

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

344 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
136 views

### Sampling small separators

Is there a tractable way to define an approximate distribution over small vertex separators of the graph? I'm looking for something along the lines of [Bayati,2008], a way to turn a single "this is a ...
214 views

### Hardness of approximating chromatic number of triangle-free graphs

The chromatic number of graph, $\chi( G)$ is hard to approximate for general graphs. Are there results of hardness of approximating $\chi(G)$ for triangle-free graphs?
130 views

### Power law for degree distribution of random KNN graphs?

Setup Sample random points from say standard Gaussian (or uniform) distribution in $R^{d}$ for some "d" and consider a KNN (K-nearest neigbour) graph for some K. Look at the degree ...
79 views

### Complexity of Encoding a Matroid Flow Problem in a Matrix

Context: Take a directed graph $G$ with a specified subset of source vertices $S$ and target vertices $T$. We say a subset $I\subseteq T$ of size $r$ is independent if there exist $r$ distinct ...
95 views

### Counting quotient graphs, but not exactly

All graphs considered will be directed graphs $G=(V,E)$, with $E \subseteq V \times V$ (so possibly with self-loops). For $k \in \mathbb{N}_{\geq 1}$, I will write $[k]$ the set $\{1,\ldots,k\}$. A $k$...
144 views

1k views

### Optimizing Maximum Weighted Matching (Edmonds Blossom)

Background: I've ported Edmonds Blossom Algorithm with Maximum Weighted Matching to Java: https://github.com/simlu/EdmondsBlossom/blob/master/src/Blossom.java The original Python implementation ...
142 views

### Graph Isomorphism Algorithm of Vertex Transistive Graphs and other

What are the best known Graph-Isomorphism algorithms for below graph classes- 1.vertex-transitive, 2. edge-transitive, 3.arc-transitive (or symmetric) 4.distance-transitive. Are they GI Complete? ...
193 views

### Graph Isomorphism of Strongly Regular Graph with fixed parameter

$G, H$ are strongly regular graphs with parameter $(n, r, \lambda, \mu)$ where $\lambda$ is constant. Here, $n$ is the number of total vertices. Each graph is $r$ regular. Every two adjacent ...
516 views

### Graph partition with weighted vertices and edges

I am searching for an algorithm to apply to a specific graph partition problem that I am interested in. It feels like a topic that people from CS may have worked on but it is also different from ...
85 views

### size bounds for circuits recognizing graph properties (reachability, cyclicity, …)

I am interested in the following. Let the inputs of the circuit correspond to the arcs of a directed graph. The circuit has to output 1 iff there is a directed path from a given node S to a given node ...
254 views

### Definition of Clique width of graph

The clique width of graph $G$ is defined as minimum number of labels required to construct $G$ by using four operations. I would like to know why the name clique width is given to this definition. ...
400 views

### The largest connected component of a random subgraph

Let a graph on $|V|$ vertices and $|E|$ edges. We randomly sample $s= c \cdot \frac{|V|}{{d_{\text{av}}}}$ vertices, without replacement, where $d_{\text{av}}$ is the average degree of $G$ and $c$ is ...
129 views

### Can we decide Red-blue cut problem in polynomial time?

Given a directed graph whose arcs are coloured red and blue and integers $r$ and $b$, can we decide in polynomial time whether the digraph has a cut with at most $r$ red arcs and at most $b$ blue arcs?...
148 views

### Recognition problem of cycle permutation graphs

A cycle permutation graph is a cubic graph composed from two $n$-cycles each labeled $1, 2,..., n$, with additional edges connecting vertex $i$ in the first cycle to vertex $\pi(i)$ in the second, ...
355 views

### #EXP-Complete problems

Let #EXP be the counting variant of NEXP, in the same way that #P is the counting variant of NP. Are there any known #EXP-complete problems? In particular, has #Succinct Sat (the natural candidate) ...
414 views

### How to go from a DAG model to bounded treewidth?

I am cross posting this question from CS.SE since I believe it is research-related question. Given a Bayesian Network DAG $G$, we can transform it into a junction tree $T_G$ by performing two steps:...
2k views

### DAG partitioning for parallel computing

Consider a set of processes ($P=\{p_1, p_2,\dots, p_n \}$) and their data dependencies. Each process $p_i$ has an execution runtime which is denoted by $d_i$. We are interested to parallelize these ...
120 views

### What is the complexity of decomposing 2k-regular multigraphs into k-stars?

Imagine we have a graph, which is 2k-regular, and we want to decompose it into copies of k-stars. Then we could just at first decompose the graph into 2-factors, orient each of the factors and then we ...
149 views

### Vertex Covers whose vertex induced subgraph has an even number of edges and no isolated vertices

Let $G$ be a graph, and let $C_{E,0}$ be the number of those vertex covers of $G$ satisfying both the following properties: Their corresponding vertex induced subgraph has an even number of edges. ...
453 views

### Dynamic shortest path data structure for DAG

Let $G$ be a dynamic DAG (directed acyclic graph) where new vertices and new edges can be inserted. I am looking for an efficient data structure/algorithm to maintain the shortest path from a fixed ...
2k views

### Can the Hungarian method be used with real edge weights?

I had a problem where I need to apply bipartite weighted matching on a graph where the edge weights are real (positive and negative). I have looked at several implementations of the Hungarian method ...
94 views

### Explicit combinatorial construction minimizing intersection of sets

I'd like to know if anything is known about the following problem: Suppose we choose positive integer $t$ to be constant. Let $S = \{1,2,\dots,n\}$, where $n$ is sufficiently large. Consider a ...
105 views

### Are there any conditions for which ($k$-)apexness is preserved under $Y-\Delta$ transformations?

Lately, something I've been interested in is finding the set of forbidden minors for the apex graphs. One thing I tried to do was to look at the graphs which were known, and try to find a pattern in ...
185 views

### Expansion of constant-size sets

My question refers to the expansion of constant size sets of an expander graph. Suppose we are given an expander graph with Cheeger constant $\alpha$. What can be said about the edge expansion of sets ...
237 views

### Ising Model and Eulerian Subgraphs

Let $\mathcal{X}= \{1,-1\}^n$, $E$ the set of edges and $J$ some real-valued matrix. Van der Waerden's theorem gives constant $c$ and set of edge weights such that \sum_\mathbf{x\in \mathcal{X}} \...
224 views

### Follow-up on Nair/Tetali's Correlation Decay Tree for hypergraphs?

I'm wondering if anybody has followed up on Nair/Tetali's correlation decay tree construction for hypergraphs. I didn't find anything relevant in back-citation on google scholar. Interesting question ...
43 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 ...