Random Clique Instance Generator

Description

Generate a simple graph with a planted clique and independent Bernoulli background edges; the witness is not claimed to be maximum.

A graph uses vertices and edges to represent relationships. Direction, duplicate handling, indexing, and traversal order affect its interpretation; returning a constructed graph together with a witness or calculated answer helps verify an algorithm.

Generate a simple graph with a planted clique and independent Bernoulli background edges; the witness is not claimed to be maximum.

When to use this tool

Use this generator to create reproducible clique 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, cliqueSize, 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, clique. Use the conventions below. Keep answer-bearing fields separate when presenting an exercise.

Sample a planted clique vertex subset and include all edges within it, then independently draw remaining simple undirected edges. Return the graph and sorted planted witness.^[1]

Input field	Meaning and units	Accepted input
nodeCount	Number of vertices, labeled 0 through nodeCount - 1.	Required. Safe integer; minimum 2; maximum 32.
cliqueSize	Clique Size	Required. Safe integer; minimum 2; 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.
clique	Clique	Required. Array of Safe integer; minimum 0.; maximum items 32.
randomness	Replay source and normalized 32-bit seed, or null seed for an advancing context source.	Required. Object; no unknown keys.
randomness.seed	Optional safe-integer 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,
    "cliqueSize": 3,
    "edgeProbability": 0.4,
    "seed": 42
  },
  "output": {
    "nodes": [
      0,
      1,
      2,
      3,
      4
    ],
    "edges": [
      [
        0,
        1
      ],
      [
        0,
        2
      ],
      [
        0,
        4
      ],
      [
        1,
        2
      ],
      [
        2,
        4
      ],
      [
        3,
        4
      ]
    ],
    "clique": [
      0,
      2,
      4
    ],
    "randomness": {
      "seed": 42,
      "source": "seed"
    }
  }
}

Limitations and assumptions

cliqueSize is a witnessed lower bound on maximum clique size, not a solved optimum. Other or larger cliques can occur and graph connectivity is not guaranteed.
A replay requires the same complete input and implementation. The stated sampling convention applies within its eligible space. A finite draw does not exhaust all possible outputs; repeats remain possible unless a distinct-selection rule is explicit.
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 Clique Instance Generator — tool.io

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