# Questions tagged [graph-theory]

Graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects.

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### The origin of the notion of treewidth

My question today is (as usual) a bit silly; but I would request you to kindly consider it. I wanted to know about the genesis and/or motivation behind the treewidth concept. I sure understand that ...
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### Consequences of a quasi-polynomial time algorithm for the graph isomorphism problem

The Graph Isomorphism problem (GI) is arguably the best known candidate for an NP-intermediate problem. The best known algorithm is sub-exponential algorithm with run-time $2^{O(\sqrt{n \log n})}$. ...
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### Data for testing graph algorithms

I am looking for a source of huge data sets to test some graph algorithm implemention. Please also provide some information about the type/distribution (e.g. directed/undirected, simple/not simple, ...
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### Are there subexponential algorithms for PLANAR SAT known?

Some NP-hard problems which are exponential on general graphs are subexponential on planar graphs because the treewidth is at most $4.9 \sqrt{|V(G)|}$ and they are exponential in the treewidth. ...
896 views

### Coloring complexity of graphs

Suppose $G$ is a graph with coloring number $d = \chi(G)$. Consider the following game between Alice and Bob. At each round, Alice picks a vertex, and Bob answers with a color in $\{1,\ldots,d-1\}$ ...
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### When is relaxed counting hard?

Suppose we relax the problem of counting proper colorings by counting weighted colorings as follows: every proper coloring gets weight 1 and every improper coloring gets weight $c^v$ where $c$ is some ...
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### Hamiltonicity of k-regular graphs

It is known that it is NP-complete to test whether a Hamiltonian cycle exists in a 3-regular graph, even if it is planar (Garey, Johnson, and Tarjan, SIAM J. Comput. 1976) or bipartite (Akiyama, ...
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### Why Ramanujan graphs are named after Ramanujan?

I recently taught expanders, and introduced the notion of Ramanujan graphs. Michael Forbes asked why they are called this way, and I had to admit I don't know. Anyone?
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### Reverse Graph Spectra Problem?

Usually one constructs a graph and then asks questions about the adjacency matrix's (or some close relative like the Laplacian) eigenvalue decomposition (also called the spectra of a graph). But what ...
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### The complexity of determining if a fixed graph is a minor of another

The result by Robertson and Seymour demonstrates an $O(n^3)$ algorithm for testing whether a fixed graph $G$ is a minor of $H$. I have two and a half questions on this topic: 1) It appears that there ...
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### Natural CLIQUE to k-Color reduction

There is clearly a reduction from CLIQUE to k-Color because they're both NP-Complete. In fact, I can construct one by composing a reduction from CLIQUE to 3-SAT with a reduction from 3-SAT to k-Color. ...
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### Regularity Lemma for Sparse Graphs

Szemeredi's Regularity Lemma says that every dense graph can be approximated as a union of $O(1)$ many bipartite expander graphs. More accurately, there's a partition of most vertices into $O(1)$ sets ...
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### Cliquewidth of Almost Cographs

(I posted this question to MathOverflow two weeks ago, but so far without a rigorous answer) I have a question about graph width measures of undirected simple graphs. It is well-known that cographs (...
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### NP-hard problems on paths

everybody knows there exist many decision problems which are NP-hard on general graphs, but I'm interested in problems that are even NP-hard when the underlying graph is a path. So, can you help me to ...
I'm looking for undirected, unweighted, connected graphs $G=(V,E)$, in which for every pair $u,v \in V$, there is a unique $u \rightarrow v$ path that realizes the distance $d(u,v)$. Is this class of ...