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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.
2
votes
Accepted
Find edge weights that fit given node weights
You can solve it by using maximum network flow: First you duplicate every vertex $i$, creating a twin $i'$, which inherits the same degree $d_{i'}:= d_i$. Each edge $ij$ becomes two edges $i,j'$ and $ …
2
votes
Accepted
Characterizing Convex Configurations of Quadrupels of Coplanar Points via Linear (In-)equali...
Here is a sketch of an answer that involves the squared Euclidean distances between the four points $A,B,C,D$ (and only the squared ones, not a mixture between squared and non-squared distances.)
Sup …
6
votes
Algorithm to solve Sokoban-like game on graphs - move chips from one set of vertices to another
Here is a polynomial-time algorithm. I assume that the chips are identical, as in Dima's reformulation and in Sokoban. (Another version would be that the chip from Init$_i$ has to go to Final$_i$, for …
1
vote
Algorithm to solve Sokoban-like game on graphs - move chips from one set of vertices to another
The better analogy when the markers are distinct is not Sokoban, but the 15-puzzle. It is even on an undirected graph.
All my remarks below are about the undirected version. ADDITION: At the end ther …
2
votes
Characterizing Convex Configurations of Quadrupels of Coplanar Points via Linear (In-)equali...
No. (This was an answer to a previous, not entirely clear version of the problem.) Take an equilateral triangle ABC of side length 1, plus the midpoint D of the side AC.
By pushing D slightly in or ou …
4
votes
Minimal graphs with a prescribed number of spanning trees
No answer, but a related question:
The number $n$ of spanning trees in a graph with $k+1$ vertices is the determinant of a $k\times k$ matrix with integer entries between $-1$ and $k$.
For given $ …
8
votes
Is the empty graph a tree?
I checked Reinhard Diestel's textbook on Graph Theory. p.2
A graph of order 0 or 1
is called trivial. Sometimes, e.g. to start an induction, trivial graphs can
be useful; at other times they form sil …