Random Vertex Cover Instance Generator

Description

Generate a simple graph whose Bernoulli-sampled edges are all incident to a planted vertex cover, without claiming that cover is minimum.

An optimization instance defines choices, constraints, and an objective. A feasible witness demonstrates that its stated constraints can be met; it establishes optimality only when the tool actually computes and verifies an optimum.

Generate a simple graph whose Bernoulli-sampled edges are all incident to a planted vertex cover, without claiming that cover is minimum.

When to use this tool

Use this generator to create reproducible vertex cover instance cases from the supplied model or bank. Inspect the returned evidence before using cases in a test or exercise.

How to use the tool and how it works

Supply nodeCount, coverSize, edgeProbability in the field shapes below. Set seed when you need an explicit replay.
Run the tool and correct any input error. For generated controls, distinguish a successful fixture containing an invalid case from failure to generate the fixture.
Read nodes, edges, cover. Use the conventions below and keep answer-bearing fields separate when presenting an exercise.

Shuffle labeled nodes to plant a cover of the requested size, then independently retain candidate simple edges only when at least one endpoint belongs to that cover. Return the graph and planted cover.^[1]

Input field	Meaning and units	Accepted input
nodeCount	Number of vertices, labeled 0 through nodeCount - 1.	Required. Safe integer; minimum 2; maximum 32.
coverSize	Cover Size	Required. Safe integer; minimum 0; maximum 32.
edgeProbability	Edge Probability	Required. Finite number; minimum 0; maximum 1.
seed	Optional numeric seed; takes precedence over context randomness. Without either, use the fixed tool.io default seed.	Optional. Safe integer.

Output field	Interpretation	Returned shape
nodes	Nodes	Required. Array of Safe integer; minimum 0.; maximum items 32.
edges	Edges	Required. Array of Array of Safe integer; minimum 0.; minimum items 2; maximum items 2.; maximum items 496.
cover	Cover	Required. Array of Safe integer; minimum 0.; maximum items 32.
randomness	Replay metadata: source is seed or context; seed is the normalized 32-bit seed or null for a context source.	Required. Object; no unknown keys.
randomness.seed	Optional numeric replay seed; normalization and precedence are explained below.	Required. Safe integer. or Null.
randomness.source	Source	Required. Text; one of "context", "seed".

An explicit safe-integer seed takes precedence over injected context randomness and is normalized to an unsigned 32-bit integer. With no seed and no context source, replay uses the fixed tool.io default seed. A context source must return finite values from zero inclusive to one exclusive; its advancing state must be retained separately. randomness records the source and normalized seed, or null for context. These generators are not cryptographic random sources.

json

{
  "input": {
    "nodeCount": 5,
    "coverSize": 2,
    "edgeProbability": 0.4,
    "seed": 42
  },
  "output": {
    "nodes": [
      0,
      1,
      2,
      3,
      4
    ],
    "edges": [
      [
        0,
        1
      ],
      [
        0,
        3
      ],
      [
        3,
        4
      ]
    ],
    "cover": [
      0,
      4
    ],
    "randomness": {
      "seed": 42,
      "source": "seed"
    }
  }
}

Limitations and assumptions

The cover is feasible by construction and need not be minimum. coverSize cannot exceed nodeCount, and edgeProbability controls eligible edges rather than all possible graphs uniformly.
A replay requires the same complete input and implementation. Randomized construction does not guarantee uniform coverage of every feasible output; finite samples can repeat unless a distinct-selection rule is explicitly stated.
The field tables state structural bounds; cross-field constraints and bounded search or output budgets also apply. Unknown keys are rejected where the object contract disallows them. Input errors are distinct from null fields and deliberate invalid controls in a successful output.

Alternative or Complementary methods

For a supplied example, inspect the returned construction or delegated check directly. To evaluate a consumer, also test cases chosen for its own boundaries; a generated finite corpus does not establish exhaustive coverage.

References

Tool behavior and example: Random Vertex Cover Instance Generator — tool.io

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