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Questions tagged [cut-space]

Use this tag for questions about the family of all cut sets of an undirected graph.

In graph theory, a cut is a partition of the vertices of a graph into two disjoint subsets. Any cut determines a cut set, the set of edges that have one endpoint in each subset of the partition. Those edges are said to cross the cut. In a connected graph, each cut set determines a unique cut, and in some cases cuts are identified with their cut-sets rather than with their vertex partitions.

The family of all cut sets of an undirected graph is known as the cut space of the graph. It forms a vector space over the two-element finite field of arithmetic modulo two with the symmetric difference of two cut sets as the vector addition operation and is the orthogonal complement of the cycle space. If the edges of the graph are given positive weights, the minimum weight basis of the cut space can be described by a tree on the same vertex set as the graph, called the Gomory-Hu tree. Each edge of that tree is associated with a bond in the original graph, and the minimum cut between two nodes s and t is the minimum weight bond among those associated with the path from s to t in the tree.

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Bounds on Cut Sizes on an Erdos-Renyi Graph

Let us have an Erdos-Renyi graph $\mathcal{G} = \mathcal{G}(n,p)$. For a subset of vertices $S$ we define the cut-size $c(S)$ as the number of edges $(u,v)$ such that $u \notin S$ and $v \in S$. Let us assume that $p$ is a constant probability and…
EducationalFerret
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Let $G = (V, E)$ be a graph, and let $e \in E$ be a cut edge. Show that the removal of $e$ increases the number of connected components by exactly $1$

Let $G = (V, E)$ be a graph, and let $e \in E$ be a cut edge. Show that the removal of $e$ increases the number of connected components by exactly $1$, i.e., $$ c(G - e) - c(G) = 1. $$ My Proof: Suppose $e = \{u,v\} \in E$ and that it is a cut…
user20194358
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Show inequality involving capacities of the union and intersection of cuts.

Let $(S,\overline S)$, $(T,\overline T)$ be cuts of a network $G$. I've alredy showed que $(S\cup T,\overline{S\cup T})$ and $(S\cap T,\overline{S\cap T})$ are cuts for $G$. Now, I have left to prove that $$cap(S\cup T,\overline{S\cup T})+cap(S\cap…
Fabrizio G
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If $A$ is a set of edges such that $S \cap T$ is non-empty, where $T$ is the edges of any spanning tree of $G$, then $S$ contains a cutset of $G$

We have a connected, undirected graph $G$, and suppose we have a set of tree edges $T_1,\cdots,T_n$ of $G$ which encompass all possible spanning trees of $G$. Now suppose that we construct a set of edges $A$ such that $A \cap T_i \neq \emptyset$ for…
John Doe
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Strong Relationship Between Abstract and Geometric Dual of a Finite Graph

In what follows, a graph is always a finite multigraph with loops. A geometric dual of a graph is formed by embedding that graph in the plane (taking a set of points for the vertices and curves with non-intersecting interiors between them for the…
Thomas Anton
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K-means clustering of Laplacian matrix eigenvectors or finding which eigenvector has maximum value

I've read this article. In this article. it's explained that $k$-eigenvectors of a Laplacian matrix are an approximation of RatioCut for $k$ clusters. To prove this, a matrix named $H$ is defined as follow: \begin{equation} h_{ij} = \begin{cases} …
user137927
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Size of min-cut in a Rényi–Erdős graph

For a Rényi–Erdős graph $G(n, p)$, what can we say about the size of the min-cut (in the whole graph)? I'm looking for something like this: $$ \Pr(\text{min-cut-size} > x) \geq \cdots $$ or $$ \Pr(\text{min-cut-size} = x) \geq \cdots $$ for an $x…
Daniel
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Definition of cuts in graph theory

Definition of cut: Let $X$ be a set of vertices of a graph $G$. The cut induced by $X$ is the set of all the edges with an endpoint in $X$ and another in $V_G \backslash X$. On https://en.wikipedia.org/wiki/Cut_(graph_theory) there's such a…
k1r1t0
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Minimum Spanning Tree (MST): Cut property direct proof

I'm trying to fully understand the cut property in the concept of minimum spanning trees (MST) in graph theory and graph algorithms. It seems that all the literature out there proves this theorem via contradiction, which is extremely irritating…
Michel H
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In the Max-Cut Problem, Are Graphs with All Edges Cut More Common Than Those Without?

Question: In the context of the Max-Cut problem, I'm trying to understand the distribution of graphs based on their edge-cut properties. Specifically, I'm interested in whether there are generally more graphs where all edges can be cut (i.e., each…
김재희
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Determining whether a set of edges is a cut

I have a connected graph, $G$. For a subset of the edges of that graph, $E$, I would like to know: Do these edges cut the graph, resulting in two subgraphs $A$ and $B$ such that each edge in $E$ has one endpoint in each subgraph? If so, how many…
Matthew Matic
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Do different definitions of cut distances on graphs coincide when the graphs have the same set of weighted nodes?

Please refer to Section 2 in this paper for notations, and specifically Section 2.3 for definitions of $d_\square$ and $\delta_\square$. My question is as follows: If $G$ and $G^\prime$ are graphs with the same set of nodes and all node weights are…
Tien Kha Pham
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Sparse Graphs with a Small Maximum Cut

Let us have a connected graph $\mathcal{G}(V,E)$ where $V$ are the vertices and $E$ the edges. Define the maximum-cut as $M(\mathcal{G})$, which is a division of the vertices into two sets with a maximal number of edges between the sets. In general…
EducationalFerret
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Cut in a connected graph is minimal iff corresponding partition induces connected subgraphs

This was claimed in Reinhard Diestel's Graph Theory textbook. This doesn't seem true at all to me. By cut I mean that if in a graph $G=(V,E)$, we have a vertex partition $\{V_1, V_2\}$, then the set of all edges crossing this partition is called a…
Harma Joma
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Question on cut

I was reading principle of mathematical analysis by Walter Rudin and there i saw this definition of cut: Consider a set $a\subset Q$ to be a cut. If it follows the following property (i)$a$ is non empty. And $a\not=Q$ (ii)if $p\in a $, $q\in Q $…
user426700
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