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Results tagged with graph-theory
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user 8193
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.
8
votes
Making a graph claw-free by adding as few edges as possible
I don't know whether it is possible to find a small set of edges to add in order to make a graph claw-free efficiently, but I think this is not a good approach for approximating MIS.
Consider the sta …
4
votes
Accepted
Is there a standard term for this graph/set theoretic concept?
You can think in terms of intersection graphs. The transitive adjacency tells you when two vertices are adjacent. The adjacency union is then empty if and only if two of $E_1,\ldots,E_k$ are in distin …
3
votes
Upper-bound for maximal-cliques on perfect graphs
Cographs form a subclass of perfect graphs and there are examples of cographs with an exponential number of cliques. For example, consider the complement of a perfect matching on $n$ vertices. It is e …
3
votes
Bounds on number of simple paths in graph
I would comment if I could, but since I can't, it goes as an answer.
The keyword you are looking for in Property 3 is "Induced Path".
3
votes
Is a simple graph the "sum" of a partial order and its dual?
As pointed out by others, the answer is no. In fact, the graphs that can be obtained this way are exactly the Comparability Graphs. This graph class is very well studied and well understood. It is als …
3
votes
Accepted
Real-world datasets for testing the maximum edge bi-clique problem
Since this problem has many applications in data mining, you probably should take a look at datasets used as benchmarks in papers on the topic. As an example, this recent paper mostly uses graphs from …
2
votes
Can we say a partial order set is 2-dimensional if its comparability graph does not contain ...
It seems that the answer is negative.
A graph $G$ is a co-comparability graph if its complement is a comparability graph. The comparability graphs of posets of dimension 2 are exactly the comparabilit …
1
vote
Partitioning the vertex set of a planar bipartite graph into a tree and an independent set
I don't know the answer for your question, but I know a somewhat related problem.
In this paper it is shown that, given a planar bipartite cubic graph $G$ and a set $W$ of vertices, it is NP-complete …
1
vote
Need a graph theory problem with nontrivial faster approximation algorithm
One example would be maximum matching. A matching $M$ is a set of pairwise nonadjacent edges. It can be solved in polynomial time, but no linear time algorithm is known. On the other hand, the greed a …
1
vote
Minimum number of edges to add in order to have a biclique cover
The kind of problem you are looking is called graph editing problems. There are many variations but, in general, in such kind of problem you are given a graph $G$ and you are asked the minimum number …
1
vote
Have this subclass of split graphs been studied before?
I don't know if this can help you, but it's easy to see that your class is a superclass of connected threshold graphs. I think that the inclusion is strict, since $S_3$(link) is in your class, but is …
1
vote
Enumerating pairs of disconnected cliques in a graph
Another way of thinking about this problem is to consider the complement of your graph $G$. In the complement, the subgraphs you are looking for are complete bipartite subgraphs, also called bicliques …
1
vote
Tree decomposition of graphs with low height
One way to address this question is to think of a tree decomposition as an unrooted tree. Then, it is easy to see that its diameter is related to the depth by a factor of 2. Searching for diameter, in …
0
votes
Are there any non-planar graphs containing only K(3,3) as a subgraph that are not 4-colourable?
If you are looking for a non 4-colorable graph free of a $K_5$ as subgraph, the simpler way would be taking a non 4-colorable graph and adding a disjoint copy of $K_{3,3}$. One such graph is the Mycie …