# Questions tagged [graph-theory]

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

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### On the paper “Quantum Computing Hamiltonian cycles”

The paper Quantum Computing Hamiltonian cycles claims: An algorithm for quantum computing Hamiltonian cycles of simple, cubic, bipartite graphs is discussed. It is shown that it is possible to evolve ...
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### 3-colourability of Eulerian maximal planar graph

The following paragraph is from this answer by David Eppstein (emphasis mine). A maximal planar graph is 3-colorable iff it is Eulerian (if it is not Eulerian, then the odd wheel surrounding a single ...
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### Trees: complexity of counting the number of vertex covers

Which is the complexity of counting the number of vertex covers of trees? Is it still #P-complete, as for general graphs?
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### Quantum error correction and graph codes

I was reading combinatorial approach towards quantum correction. A lot of work in this is on finding diagonal distance of a graph. Let me add definition of diagonal distance so that this remains self-...
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### Relation between tree-width and clique number

Are there any nice graph classes for which the tree-width $tw(G)$ is upper-bounded by a function of the clique number $\omega(G)$, i.e. $tw(G)\leq f(\omega(G))$? For example, it is a classic fact ...
585 views

### Is this vertex ordering optimization NP-Hard?

Could you help me to prove that the following problem is NP-hard? Problem. Given an undirected graph $G=(V,E)$, find an ordering of the nodes such that $\sum\limits_{v\in V}|succ(v)|\times|pred(v)|$ ...
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### Is the following graph optimization problem approximable within a constant factor?

Let $G=(V,E)$ be an undirected graph, and let $\pi$ be a permutation of the vertices in $V$. For a node $v\in V$, we denote by $\text{pred}_{\pi}(v)$ (respectively $\text{succ}_{\pi}(v)$) the set of ...
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### Is there a planar 4-regular graph that is 3-acyclic colourable?

A colouring is said to be an acyclic colouring if there is no bicoloured cycle (i.e each cycle gets at least 3 colours). Burstein proved that 4-regular graphs are 5-acyclic colourable. It seems to me ...
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### Tree decompositions of optimal width where every vertex is in a bounded number of bags?

Let $G$ be a graph on $n$ vertices whose maximum degree is at most $\Delta$ and whose treewidth is at most $k$. Does there exist a function $f(k, \Delta)$, independent of $n$, such that it is possible ...
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### Graph problems in P with unknown lower bounds

I am looking for references to interesting graph problems, which are known to be in P, but their precise big-O lower bounds are elusive. I would split this into 2 classes: problems, where we know of ...
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### Proof that optimal solutions of LP Relaxation of independent set are half-integral

I saw somewhere that optimal solutions of LP Relaxation of independent set are half-integral, by what I mean the possible values of a solution are ${ \{0,0.5,1 \} }$. I'm looking for proof of that. ...
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### Is feedback vertex set problem is solvable in polynomial time for 3-degree bounded graphs?

Feedback Vertex Set is NP-complete for general graphs. It is known to be NP-complete for degree-8 bounded graphs due to a reduction from vertex cover. The Wikipedia article says that it is poly-time ...
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### Network design with reachability pattern

We are given two sets of terminals $A$ and $B$. For each $a\in A$, we are also given $R_a\subseteq B$. Let $|A|+|B|=n$. We want to find a directed acyclic graph $G$ where $A$ and $B$ are subsets of ...
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### How to efficiently find a loop between two nodes in a directed graph?

Given two nodes in a directed graph, how can I find a loop (if exists) that pass these two nodes? The loop cannot pass a node more than once. And if there isn't such a loop, how to efficiently ...
257 views

### Lighting up all elements of a poset by toggling upsets

I consider the following game on a finite poset $(P, <)$. At each point of the game, I have a set of elements $S$ of the poset which are "on", and all others are "off". Initially $S = \emptyset$. ...
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### Reference request: Depth- (or Breadth-) first search with hints?

Consider the standard s-t reachability problem of finding a path between nodes $s$ and $t$ in a directed graph $G$. A DFS or BFS could solve it. Would it be possible to pre-process the graph and ...
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### Is the maximum independent set in cubic planar graphs NP-complete?

In their famous book, Garey and Johnson, write a comment that the maximum independent set problem, in cubic planar graphs is NP-complete(page 194 of the book). They say this is by a transformation ...
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### Is there a problem that is easy for cubic graphs but hard for graphs with maximum degree 3?

Cubic graphs are graphs where every vertex has degree 3. They have been extensively studied and I'm aware that several NP-hard problems remain NP-hard even restricted to subclasses of cubic graphs, ...
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### Rapidly mixing Markov chains on 3-colorings of a cycle

The Glauber dynamics is a Markov chain on the colorings of a graph in which at each step one attempts to recolor a randomly chosen vertex with a random color. It does not mix for the 3-colorings of a ...
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### When can partial spectral sparsifiers be combined?

A few important papers about spectral sparsifiers and friends contain a technical idea that involves building many different sparsifiers that each "partially" solve the problem, and then combining ...