I'm looking for any idea or algorithm that solves the problem of finding bipartite subgraphs in any graph. In general, I'd like to divide graph into the smallest number of bipartite subgraphs. My first thought was to find the largest bipartite subgraph and repeat it until they set of the remaining vertices is empty, but I don't know how to solve this problem in polynomial (or pseudo-polynomial) time.
2026-03-30 18:09:47.1774894187
The (largest) bipartite subgraph
1.1k Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in GRAPH-THEORY
- characterisation of $2$-connected graphs with no even cycles
- Explanation for the static degree sort algorithm of Deo et al.
- A certain partition of 28
- decomposing a graph in connected components
- Is it true that if a graph is bipartite iff it is class 1 (edge-coloring)?
- Fake induction, can't find flaw, every graph with zero edges is connected
- Triangle-free graph where every pair of nonadjacent vertices has exactly two common neighbors
- Inequality on degrees implies perfect matching
- Proving that no two teams in a tournament win same number of games
- Proving that we can divide a graph to two graphs which induced subgraph is connected on vertices of each one
Related Questions in BIPARTITE-GRAPHS
- Is it true that if a graph is bipartite iff it is class 1 (edge-coloring)?
- Perfect Matching
- Complete bipartite planar graphs
- Is the graph described below bipartite?
- Prove that an even order ($n=2k$) graph without cycle of order 3, has a size $m \le k^2$
- min cost flow in offline bipartite graph problem
- Rearrangeable matrix visualization
- Is there a name for Chain of complete bipartite graphs?
- Determine if G is bipartite. Find a maximal path and Eulerian circuit in G.
- Does this graph have a Hamiltonian cycle?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
I assume here that you're looking for induced subgraphs, and I address your "general" problem. It's NP-complete.
A bipartite graph is a 2-colorable graph ; so an induced subgraph that is bipartite is an incomplete (not going through all the vertices) 2-coloration of the graph. Looking for the smallest decomposition in bipartite graphs, it means looking for a complete coloration of the graph with $2k$ colors, where $k$ is your number of bipartite graphs. Say you have decomposition in $k$ bipartite graphs, then you can easily deduce a $2k$ coloration of your graph, and conversely when you have a $2k$ coloration you can group your colors two by two, and the induced subgraphs will be bipartite.
But deciding whether a graph is $2k$-colorable is NP-complete. So either your subproblem itself is NP-complete, or it is unoptimal for your global problem. Probably both.
Concerning your "local" subproblem, it can be reformulated as finding the biggest set of two colors in the graph. I'd guess that it must be NP-hard ; finding the biggest set of one color (which is the largest anticlique, or independent set) is already strongly NP-hard.