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Results tagged with graph-theory
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user 6094
Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.
4
votes
What is the cycle structure of a graph?
Another possible answer is the cycle space of a graph, which is a vector space and so supports the application of many tools from linear algebra.
- 151k
3
votes
four color proof
Just as background, definitely not an answer to your question: You are probably aware of
the paper,
"A new proof of the four colour theorem,"
by N. Robertson, D. P. Sanders, P. D. Seymour and R. Thom …
- 151k
3
votes
Recognition of graph families.
Permit me to direct you to the Wikipedia page on "Forbidden graph characterization,"
which contains a long table of graph classes that have a forbidden subgraph characterizations.
For example, chordal …
- 151k
4
votes
Minimum dimension for sphere packing a graph in Euclidean space
This is a longish comment rather than a complete answer.
Long ago Fred Roberts introduced the notion of the "boxicity" of a graph,
and later the related notion of "sphericity" of a graph was studied. …
- 151k
5
votes
determining k-edge-connectivity of a graph
Generally $k$-connectivity is computed using max-flow min-cut algorithms. I cannot quote you complexities off the top of my head, but you should be able to find the number of edges that disconnect an …
- 151k
8
votes
Accepted
5
votes
How to find central vertex in a graph?
Mathematica has a function GraphCenter[] that computes the center of a graph (the set of vertices with minimum eccentricity--exactly your definition).
You can find a description in the documentation h …
- 151k
4
votes
Degree of faces in a regular graph
This will likely only serve to sharpen your question, but I will just observe that there
is no upper bound on the number of edges of a face of a 4-regular planar graph:
The octagon vertices …
- 151k
5
votes
Accepted
Convex representation of (planar) graphs
Although what constitutes a "practical justification" is in the eye of the beholder,
I would think this might qualify.
There is a notion called geometric routing that is used to solve network routing …
- 151k
2
votes
Two graph constructions: new, old?
As a public service, so to speak, here is $C'_4$, if I've followed the construction correctly:
- 151k
7
votes
Is there anything similar to the four color theorem for 3-dimensional objects?
This is an easy result, not at all comparable to the $4$-color theorem, but it perhaps has the flavor you are seeking:
A collection of tetrahedra forming a pure simplicial complex may be "solid 4- …
- 151k
14
votes
Always a planar-drawn cycle through $n$ points
Here is a quote from the first paper cited below: Steinhaus posed a version of your question, which has become known as simple polygonization of a set of points:
1Agarwal, Pankaj K., Ferran …
- 151k
2
votes
Shortest path in a weighted graph with coloured edges
There is quite a bit of work on finding shortest paths with turn penalties.
In a typical model, such as that used in the paper cited below, "Turn
costs are stored in tables that are assigned to nodes. …
- 151k
2
votes
Bound on the number of unlabeled cographs on n vertices
The paper "Enumeration and limit laws of series-parallel graphs"
by Manuel Bodirsky, Omer Gimenez, Mihyun Kang, and Marc Noy,
establishes that the number of labeled series-parallel graphs on $n$ verti …
- 151k
1
vote
In search for isotropic graphs: Straight lines and parallels
Perhaps it will help to explore the world of
pseudoline arrangements.
A pseudoline is a simple curve in the projective plane that is topologically
a line. Each pair of pseudoines in an arrangment meet …
- 151k