# A spider is a graph with an even number of vertices

A spider is a graph with an even number of vertices, say 2g, such that g vertices form a clique, and each of the other g vertices is adjacent to exactly one distinct vertex in the clique. The picture shows spiders with 6 and 8 vertices. The spiderVerse problem takes as input a graph G=(V,E) and a natural number g >= 1 and outputs a set of 2g vertices such that the induced subgraph is a spider, or outputs NO if such set of vertices does not exist. Show that the SpiderVerse is NP-complete

### Best Answer

Since SpiderVerse is both in NP and NP-hard, it is NP-complete. This completes the proof that the SpiderVerse Answer is NP-complete.

#### Step-by-step explanation

To show that the SpiderVerse Answer is NP-complete, we need to demonstrate two things:

1. SpiderVerse is in NP: This means that given a potential solution (a set of vertices), we can verify in polynomial time whether this set forms a valid spider as defined.

2. SpiderVerse is NP-hard: This requires showing that any Answer in NP can be reduced to SpiderVerse in polynomial time.

# 1. SpiderVerse is in NP

To verify if a set of vertices forms a spider:

- Check if the induced subgraph on these vertices is connected.

- Count the vertices: ensure there are exactly

vertices.- Check the structure: verify that

vertices form a clique, and each of the other vertices is adjacent to exactly one vertex in the clique.

Each of these verifications can be done in polynomial time relative to the size of the input graph

. Thus, SpiderVerse is in NP.

# 2. SpiderVerse is NP-hard

To prove NP-hardness, we will reduce a known NP-complete Answer to SpiderVerse. Let's reduce the Vertex Cover Answer to SpiderVerse.

Reduction from Vertex Cover to SpiderVerse

Vertex Cover Answer: Given a graph

and a number , determine if there exists a set such that and every edge in is incident to at least one vertex in .

Reduction Strategy:

- Given an instance of the Vertex Cover Answer

:- Construct a spider graph

as follows:- Create a clique

with vertices.- Add

additional vertices, each connected to exactly one vertex in .

Construction Details:

- Let

be the input graph for Vertex Cover.- Compute

.- Construct

:-

will have vertices:-

will be the clique with vertices.- Add

vertices, each connected to one distinct vertex in .

Verification:

- If there exists a Vertex Cover of size

in , then select the corresponding vertices from that are connected to . This set will form a valid spider in .- If there exists a set of

vertices in forming a spider, then the vertices forming the clique and the vertices adjacent to form a Vertex Cover of size in .

Since Vertex Cover is known to be NP-complete, and we have shown a polynomial-time reduction from Vertex Cover to SpiderVerse, SpiderVerse is also NP-hard.

To show that the SpiderVerse problem is NP-complete, we need to demonstrate two things:

SpiderVerse is in NP: We need to show that given a solution (a set of 2g vertices that form a spider), we can verify it in polynomial time.

SpiderVerse is NP-hard: We need to show that any problem in NP can be reduced to SpiderVerse in polynomial time.

### Step 1: SpiderVerse is in NP

Given a set of 2g vertices, we can verify whether they form a spider in polynomial time by performing the following checks:

Clique Check: Verify that g of these vertices form a clique. This involves checking that there is an edge between every pair of these g vertices. This can be done in (O(g^2)) time.

Adjacency Check: Verify that each of the remaining g vertices is adjacent to exactly one distinct vertex in the clique. This involves checking the adjacency of each of these g vertices to the g vertices in the clique, which can be done in (O(g^2)) time.

Since both checks can be done in polynomial time, verifying a solution is polynomial, and thus SpiderVerse is in NP.

### Step 2: SpiderVerse is NP-hard

To show that SpiderVerse is NP-hard, we can reduce a known NP-complete problem to SpiderVerse. We will use the Clique problem, which is known to be NP-complete.

Clique Problem: Given a graph (G = (V, E)) and a number (k), determine if there is a clique of size (k) in (G).

Reduction from Clique to SpiderVerse:

Input Transformation: Given an instance of the Clique problem, (G = (V, E)) and (k), we construct an instance of the SpiderVerse problem as follows:

Set (g = k).

The graph (G' = (V', E')) is the same as (G).

Output Transformation: We need to show that (G) has a clique of size (k) if and only if (G') has a spider with (2g = 2k) vertices.

If (G) has a clique of size (k): Let (C) be the set of (k) vertices that form a

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