# Questions tagged [vertex-cover]

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### Proving that the number of leaves in a tree >= number of unmatched vertices

Consider a rooted tree $T$. A matching in $T$ is said to be proper if for every unmatched vertex $v$ it holds that the parent of $v$ is matched to one of the siblings of $v$. It is known that every ...
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### How to prove that if Eternal Vertex Cover is Polynomial it's possible to detect its vertices and edges

EVG is defined as EVC = { <G,m,k>| G is an undirected graph and there is as et of m edges in G that are covered by at most k nodes} If EVG was decidable in polynomial time how could we find the ...
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### Reduction from vertex-cover to system of quadratic equations

Define $$\text{SQE}=\{S\ |\ S\ \text{is a system of quadratic equations with real solutions}\}$$ and $$\text{VC}=\{G\ |\ G\ \text{is a simple undirected graph with a vertex cover}\ \leq k\}$$ I am ...
1 vote
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### Trivial vertex cover in regular graph is 2-approximation Proof

I need to show that in any regular graph, taking all nodes gives a 2-approximation vertex cover. My attempt: I am proving that every $k$-regular graph can be reduced to a 2-regular fully connected ...
1 vote
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### Prove that a dominating set has minimum cardinality in a "unit interval graph"

I am given the definition of a unit interval graph, e.g. $G = (V, E)$ such that $\forall v \in V$ there is a weight $x_v \in \mathbb{R}$ and nodes $u, w$ has an edge iff $|x_u - x_w| < 1$. I am ...
1 vote
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### Vertex cover of minimal graph

I'm looking for algorithm that, for given undirected graph $G=(V,E)$, find graph $G'=(V,E')$ with minimal amount of edges that have same vertex cover as G. I mean, vertices $U$ are vertex cover of $G$ ...
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### Is this exponential-sized vertex cover problem in P?

Suppose P $\neq$ NP. Prove or disprove if language is in P using a reduction or an algorithm: $$\left\{ \left(G = (V,E), k, 0^{2^{|V|}} \right) \mid (G,k) \in VC \right\}$$ Suppose I have the this ...
1 vote
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### What is the complexity class of finding vertex cover number of a simple graph?

Suppose we have a simple graph $G$. We know that finding the minimum vertex covering set for $G$ is in the NP-hard class. But, what about the complexity class of finding the size of the set, i.e., the ...
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If we know the vertex cover number for a simple graph $G$ denoted by $\tau(G)$, is it possible to find the minimum vertex cover set for $G$‌ in quasi-polynomial time? As I found, we cannot find any ...