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Question 19527: Exact Extremal Sizes Need the Complete Vertex-Subset Search

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0038-19527 · q-20261014-advance-20261106-r02-clique-and-independent-set-0052

Original question and givens

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

  • G001 — Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source operation=clique-exact-extrema;original snapshot and mathematics retained

After

explicit ordered source output fields=[maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate]

Conditions: none

Keep the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not mathematical evidence,and No retains the operation's claim scope.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

complete ORIGINAL finite carrier Fin(6);ALL canonical simple undirected original edges=[[0, 1], [0, 2], [0, 3], [1, 2], [3, 4]]

After

all vertices and edge indices retained;extrema slice n<=9 includes empty carrier;complement reverses ONLY distinct-vertex adjacency

Conditions: none

Keep empty carriers and isolates.Complementation reverses adjacency only between distinct vertices,without adding loops.This is a bounded complete-graph certificate,not an extremal inference from a sampled subgraph.

Step identifiers and dependencies

Rule: clique_complete_graph_carrier_not_sample_or_loop_complement · Uses: G001, S001

S003 · implies

Before

next ORIGINAL vertex bitmask=0;decoded increasing vertices=[];ALL pair rows=[];previous best bitmasks=[0, 0]

After

clique=Yes;independent=Yes;size=0;strict-larger ONLY update gives best clique=[] at mask0,best independent=[] at mask0

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S002

S004 · implies

Before

next ORIGINAL vertex bitmask=1;decoded increasing vertices=[0];ALL pair rows=[];previous best bitmasks=[0, 0]

After

clique=Yes;independent=Yes;size=1;strict-larger ONLY update gives best clique=[0] at mask1,best independent=[0] at mask1

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S003

S005 · implies

Before

next ORIGINAL vertex bitmask=2;decoded increasing vertices=[1];ALL pair rows=[];previous best bitmasks=[1, 1]

After

clique=Yes;independent=Yes;size=1;strict-larger ONLY update gives best clique=[0] at mask1,best independent=[0] at mask1

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S004

S006 · implies

Before

next ORIGINAL vertex bitmask=3;decoded increasing vertices=[0, 1];ALL pair rows=[[0, 1, Yes]];previous best bitmasks=[1, 1]

After

clique=Yes;independent=No;size=2;strict-larger ONLY update gives best clique=[0, 1] at mask3,best independent=[0] at mask1

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S005

S007 · implies

Before

next ORIGINAL vertex bitmask=4;decoded increasing vertices=[2];ALL pair rows=[];previous best bitmasks=[3, 1]

After

clique=Yes;independent=Yes;size=1;strict-larger ONLY update gives best clique=[0, 1] at mask3,best independent=[0] at mask1

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S006

S008 · implies

Before

next ORIGINAL vertex bitmask=5;decoded increasing vertices=[0, 2];ALL pair rows=[[0, 2, Yes]];previous best bitmasks=[3, 1]

After

clique=Yes;independent=No;size=2;strict-larger ONLY update gives best clique=[0, 1] at mask3,best independent=[0] at mask1

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S007

S009 · implies

Before

next ORIGINAL vertex bitmask=6;decoded increasing vertices=[1, 2];ALL pair rows=[[1, 2, Yes]];previous best bitmasks=[3, 1]

After

clique=Yes;independent=No;size=2;strict-larger ONLY update gives best clique=[0, 1] at mask3,best independent=[0] at mask1

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S008

S010 · implies

Before

next ORIGINAL vertex bitmask=7;decoded increasing vertices=[0, 1, 2];ALL pair rows=[[0, 1, Yes], [0, 2, Yes], [1, 2, Yes]];previous best bitmasks=[3, 1]

After

clique=Yes;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0] at mask1

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S009

S011 · implies

Before

next ORIGINAL vertex bitmask=8;decoded increasing vertices=[3];ALL pair rows=[];previous best bitmasks=[7, 1]

After

clique=Yes;independent=Yes;size=1;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0] at mask1

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S010

S012 · implies

Before

next ORIGINAL vertex bitmask=9;decoded increasing vertices=[0, 3];ALL pair rows=[[0, 3, Yes]];previous best bitmasks=[7, 1]

After

clique=Yes;independent=No;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[0] at mask1

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S011

S013 · implies

Before

next ORIGINAL vertex bitmask=10;decoded increasing vertices=[1, 3];ALL pair rows=[[1, 3, No]];previous best bitmasks=[7, 1]

After

clique=No;independent=Yes;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S012

S014 · implies

Before

next ORIGINAL vertex bitmask=11;decoded increasing vertices=[0, 1, 3];ALL pair rows=[[0, 1, Yes], [0, 3, Yes], [1, 3, No]];previous best bitmasks=[7, 10]

After

clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S013

S015 · implies

Before

next ORIGINAL vertex bitmask=12;decoded increasing vertices=[2, 3];ALL pair rows=[[2, 3, No]];previous best bitmasks=[7, 10]

After

clique=No;independent=Yes;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S014

S016 · implies

Before

next ORIGINAL vertex bitmask=13;decoded increasing vertices=[0, 2, 3];ALL pair rows=[[0, 2, Yes], [0, 3, Yes], [2, 3, No]];previous best bitmasks=[7, 10]

After

clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S015

S017 · implies

Before

next ORIGINAL vertex bitmask=14;decoded increasing vertices=[1, 2, 3];ALL pair rows=[[1, 2, Yes], [1, 3, No], [2, 3, No]];previous best bitmasks=[7, 10]

After

clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S016

S018 · implies

Before

next ORIGINAL vertex bitmask=15;decoded increasing vertices=[0, 1, 2, 3];ALL pair rows=[[0, 1, Yes], [0, 2, Yes], [0, 3, Yes], [1, 2, Yes], [1, 3, No], [2, 3, No]];previous best bitmasks=[7, 10]

After

clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S017

S019 · implies

Before

next ORIGINAL vertex bitmask=16;decoded increasing vertices=[4];ALL pair rows=[];previous best bitmasks=[7, 10]

After

clique=Yes;independent=Yes;size=1;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S018

S020 · implies

Before

next ORIGINAL vertex bitmask=17;decoded increasing vertices=[0, 4];ALL pair rows=[[0, 4, No]];previous best bitmasks=[7, 10]

After

clique=No;independent=Yes;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S019

S021 · implies

Before

next ORIGINAL vertex bitmask=18;decoded increasing vertices=[1, 4];ALL pair rows=[[1, 4, No]];previous best bitmasks=[7, 10]

After

clique=No;independent=Yes;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S020

S022 · implies

Before

next ORIGINAL vertex bitmask=19;decoded increasing vertices=[0, 1, 4];ALL pair rows=[[0, 1, Yes], [0, 4, No], [1, 4, No]];previous best bitmasks=[7, 10]

After

clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S021

S023 · implies

Before

next ORIGINAL vertex bitmask=20;decoded increasing vertices=[2, 4];ALL pair rows=[[2, 4, No]];previous best bitmasks=[7, 10]

After

clique=No;independent=Yes;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S022

S024 · implies

Before

next ORIGINAL vertex bitmask=21;decoded increasing vertices=[0, 2, 4];ALL pair rows=[[0, 2, Yes], [0, 4, No], [2, 4, No]];previous best bitmasks=[7, 10]

After

clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S023

S025 · implies

Before

next ORIGINAL vertex bitmask=22;decoded increasing vertices=[1, 2, 4];ALL pair rows=[[1, 2, Yes], [1, 4, No], [2, 4, No]];previous best bitmasks=[7, 10]

After

clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S024

S026 · implies

Before

next ORIGINAL vertex bitmask=23;decoded increasing vertices=[0, 1, 2, 4];ALL pair rows=[[0, 1, Yes], [0, 2, Yes], [0, 4, No], [1, 2, Yes], [1, 4, No], [2, 4, No]];previous best bitmasks=[7, 10]

After

clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S025

S027 · implies

Before

next ORIGINAL vertex bitmask=24;decoded increasing vertices=[3, 4];ALL pair rows=[[3, 4, Yes]];previous best bitmasks=[7, 10]

After

clique=Yes;independent=No;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S026

S028 · implies

Before

next ORIGINAL vertex bitmask=25;decoded increasing vertices=[0, 3, 4];ALL pair rows=[[0, 3, Yes], [0, 4, No], [3, 4, Yes]];previous best bitmasks=[7, 10]

After

clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S027

S029 · implies

Before

next ORIGINAL vertex bitmask=26;decoded increasing vertices=[1, 3, 4];ALL pair rows=[[1, 3, No], [1, 4, No], [3, 4, Yes]];previous best bitmasks=[7, 10]

After

clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S028

S030 · implies

Before

next ORIGINAL vertex bitmask=27;decoded increasing vertices=[0, 1, 3, 4];ALL pair rows=[[0, 1, Yes], [0, 3, Yes], [0, 4, No], [1, 3, No], [1, 4, No], [3, 4, Yes]];previous best bitmasks=[7, 10]

After

clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S029

S031 · implies

Before

next ORIGINAL vertex bitmask=28;decoded increasing vertices=[2, 3, 4];ALL pair rows=[[2, 3, No], [2, 4, No], [3, 4, Yes]];previous best bitmasks=[7, 10]

After

clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S030

S032 · implies

Before

next ORIGINAL vertex bitmask=29;decoded increasing vertices=[0, 2, 3, 4];ALL pair rows=[[0, 2, Yes], [0, 3, Yes], [0, 4, No], [2, 3, No], [2, 4, No], [3, 4, Yes]];previous best bitmasks=[7, 10]

After

clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S031

S033 · implies

Before

next ORIGINAL vertex bitmask=30;decoded increasing vertices=[1, 2, 3, 4];ALL pair rows=[[1, 2, Yes], [1, 3, No], [1, 4, No], [2, 3, No], [2, 4, No], [3, 4, Yes]];previous best bitmasks=[7, 10]

After

clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S032

S034 · implies

Before

next ORIGINAL vertex bitmask=31;decoded increasing vertices=[0, 1, 2, 3, 4];ALL pair rows=[[0, 1, Yes], [0, 2, Yes], [0, 3, Yes], [0, 4, No], [1, 2, Yes], [1, 3, No], [1, 4, No], [2, 3, No], [2, 4, No], [3, 4, Yes]];previous best bitmasks=[7, 10]

After

clique=No;independent=No;size=5;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S033

S035 · implies

Before

next ORIGINAL vertex bitmask=32;decoded increasing vertices=[5];ALL pair rows=[];previous best bitmasks=[7, 10]

After

clique=Yes;independent=Yes;size=1;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S034

S036 · implies

Before

next ORIGINAL vertex bitmask=33;decoded increasing vertices=[0, 5];ALL pair rows=[[0, 5, No]];previous best bitmasks=[7, 10]

After

clique=No;independent=Yes;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S035

S037 · implies

Before

next ORIGINAL vertex bitmask=34;decoded increasing vertices=[1, 5];ALL pair rows=[[1, 5, No]];previous best bitmasks=[7, 10]

After

clique=No;independent=Yes;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S036

S038 · implies

Before

next ORIGINAL vertex bitmask=35;decoded increasing vertices=[0, 1, 5];ALL pair rows=[[0, 1, Yes], [0, 5, No], [1, 5, No]];previous best bitmasks=[7, 10]

After

clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S037

S039 · implies

Before

next ORIGINAL vertex bitmask=36;decoded increasing vertices=[2, 5];ALL pair rows=[[2, 5, No]];previous best bitmasks=[7, 10]

After

clique=No;independent=Yes;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S038

S040 · implies

Before

next ORIGINAL vertex bitmask=37;decoded increasing vertices=[0, 2, 5];ALL pair rows=[[0, 2, Yes], [0, 5, No], [2, 5, No]];previous best bitmasks=[7, 10]

After

clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S039

S041 · implies

Before

next ORIGINAL vertex bitmask=38;decoded increasing vertices=[1, 2, 5];ALL pair rows=[[1, 2, Yes], [1, 5, No], [2, 5, No]];previous best bitmasks=[7, 10]

After

clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S040

S042 · implies

Before

next ORIGINAL vertex bitmask=39;decoded increasing vertices=[0, 1, 2, 5];ALL pair rows=[[0, 1, Yes], [0, 2, Yes], [0, 5, No], [1, 2, Yes], [1, 5, No], [2, 5, No]];previous best bitmasks=[7, 10]

After

clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S041

S043 · implies

Before

next ORIGINAL vertex bitmask=40;decoded increasing vertices=[3, 5];ALL pair rows=[[3, 5, No]];previous best bitmasks=[7, 10]

After

clique=No;independent=Yes;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S042

S044 · implies

Before

next ORIGINAL vertex bitmask=41;decoded increasing vertices=[0, 3, 5];ALL pair rows=[[0, 3, Yes], [0, 5, No], [3, 5, No]];previous best bitmasks=[7, 10]

After

clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3] at mask10

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S043

S045 · implies

Before

next ORIGINAL vertex bitmask=42;decoded increasing vertices=[1, 3, 5];ALL pair rows=[[1, 3, No], [1, 5, No], [3, 5, No]];previous best bitmasks=[7, 10]

After

clique=No;independent=Yes;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S044

S046 · implies

Before

next ORIGINAL vertex bitmask=43;decoded increasing vertices=[0, 1, 3, 5];ALL pair rows=[[0, 1, Yes], [0, 3, Yes], [0, 5, No], [1, 3, No], [1, 5, No], [3, 5, No]];previous best bitmasks=[7, 42]

After

clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S045

S047 · implies

Before

next ORIGINAL vertex bitmask=44;decoded increasing vertices=[2, 3, 5];ALL pair rows=[[2, 3, No], [2, 5, No], [3, 5, No]];previous best bitmasks=[7, 42]

After

clique=No;independent=Yes;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S046

S048 · implies

Before

next ORIGINAL vertex bitmask=45;decoded increasing vertices=[0, 2, 3, 5];ALL pair rows=[[0, 2, Yes], [0, 3, Yes], [0, 5, No], [2, 3, No], [2, 5, No], [3, 5, No]];previous best bitmasks=[7, 42]

After

clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S047

S049 · implies

Before

next ORIGINAL vertex bitmask=46;decoded increasing vertices=[1, 2, 3, 5];ALL pair rows=[[1, 2, Yes], [1, 3, No], [1, 5, No], [2, 3, No], [2, 5, No], [3, 5, No]];previous best bitmasks=[7, 42]

After

clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S048

S050 · implies

Before

next ORIGINAL vertex bitmask=47;decoded increasing vertices=[0, 1, 2, 3, 5];ALL pair rows=[[0, 1, Yes], [0, 2, Yes], [0, 3, Yes], [0, 5, No], [1, 2, Yes], [1, 3, No], [1, 5, No], [2, 3, No], [2, 5, No], [3, 5, No]];previous best bitmasks=[7, 42]

After

clique=No;independent=No;size=5;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S049

S051 · implies

Before

next ORIGINAL vertex bitmask=48;decoded increasing vertices=[4, 5];ALL pair rows=[[4, 5, No]];previous best bitmasks=[7, 42]

After

clique=No;independent=Yes;size=2;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S050

S052 · implies

Before

next ORIGINAL vertex bitmask=49;decoded increasing vertices=[0, 4, 5];ALL pair rows=[[0, 4, No], [0, 5, No], [4, 5, No]];previous best bitmasks=[7, 42]

After

clique=No;independent=Yes;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S051

S053 · implies

Before

next ORIGINAL vertex bitmask=50;decoded increasing vertices=[1, 4, 5];ALL pair rows=[[1, 4, No], [1, 5, No], [4, 5, No]];previous best bitmasks=[7, 42]

After

clique=No;independent=Yes;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S052

S054 · implies

Before

next ORIGINAL vertex bitmask=51;decoded increasing vertices=[0, 1, 4, 5];ALL pair rows=[[0, 1, Yes], [0, 4, No], [0, 5, No], [1, 4, No], [1, 5, No], [4, 5, No]];previous best bitmasks=[7, 42]

After

clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S053

S055 · implies

Before

next ORIGINAL vertex bitmask=52;decoded increasing vertices=[2, 4, 5];ALL pair rows=[[2, 4, No], [2, 5, No], [4, 5, No]];previous best bitmasks=[7, 42]

After

clique=No;independent=Yes;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S054

S056 · implies

Before

next ORIGINAL vertex bitmask=53;decoded increasing vertices=[0, 2, 4, 5];ALL pair rows=[[0, 2, Yes], [0, 4, No], [0, 5, No], [2, 4, No], [2, 5, No], [4, 5, No]];previous best bitmasks=[7, 42]

After

clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S055

S057 · implies

Before

next ORIGINAL vertex bitmask=54;decoded increasing vertices=[1, 2, 4, 5];ALL pair rows=[[1, 2, Yes], [1, 4, No], [1, 5, No], [2, 4, No], [2, 5, No], [4, 5, No]];previous best bitmasks=[7, 42]

After

clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S056

S058 · implies

Before

next ORIGINAL vertex bitmask=55;decoded increasing vertices=[0, 1, 2, 4, 5];ALL pair rows=[[0, 1, Yes], [0, 2, Yes], [0, 4, No], [0, 5, No], [1, 2, Yes], [1, 4, No], [1, 5, No], [2, 4, No], [2, 5, No], [4, 5, No]];previous best bitmasks=[7, 42]

After

clique=No;independent=No;size=5;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S057

S059 · implies

Before

next ORIGINAL vertex bitmask=56;decoded increasing vertices=[3, 4, 5];ALL pair rows=[[3, 4, Yes], [3, 5, No], [4, 5, No]];previous best bitmasks=[7, 42]

After

clique=No;independent=No;size=3;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S058

S060 · implies

Before

next ORIGINAL vertex bitmask=57;decoded increasing vertices=[0, 3, 4, 5];ALL pair rows=[[0, 3, Yes], [0, 4, No], [0, 5, No], [3, 4, Yes], [3, 5, No], [4, 5, No]];previous best bitmasks=[7, 42]

After

clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S059

S061 · implies

Before

next ORIGINAL vertex bitmask=58;decoded increasing vertices=[1, 3, 4, 5];ALL pair rows=[[1, 3, No], [1, 4, No], [1, 5, No], [3, 4, Yes], [3, 5, No], [4, 5, No]];previous best bitmasks=[7, 42]

After

clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S060

S062 · implies

Before

next ORIGINAL vertex bitmask=59;decoded increasing vertices=[0, 1, 3, 4, 5];ALL pair rows=[[0, 1, Yes], [0, 3, Yes], [0, 4, No], [0, 5, No], [1, 3, No], [1, 4, No], [1, 5, No], [3, 4, Yes], [3, 5, No], [4, 5, No]];previous best bitmasks=[7, 42]

After

clique=No;independent=No;size=5;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S061

S063 · implies

Before

next ORIGINAL vertex bitmask=60;decoded increasing vertices=[2, 3, 4, 5];ALL pair rows=[[2, 3, No], [2, 4, No], [2, 5, No], [3, 4, Yes], [3, 5, No], [4, 5, No]];previous best bitmasks=[7, 42]

After

clique=No;independent=No;size=4;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S062

S064 · implies

Before

next ORIGINAL vertex bitmask=61;decoded increasing vertices=[0, 2, 3, 4, 5];ALL pair rows=[[0, 2, Yes], [0, 3, Yes], [0, 4, No], [0, 5, No], [2, 3, No], [2, 4, No], [2, 5, No], [3, 4, Yes], [3, 5, No], [4, 5, No]];previous best bitmasks=[7, 42]

After

clique=No;independent=No;size=5;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S063

S065 · implies

Before

next ORIGINAL vertex bitmask=62;decoded increasing vertices=[1, 2, 3, 4, 5];ALL pair rows=[[1, 2, Yes], [1, 3, No], [1, 4, No], [1, 5, No], [2, 3, No], [2, 4, No], [2, 5, No], [3, 4, Yes], [3, 5, No], [4, 5, No]];previous best bitmasks=[7, 42]

After

clique=No;independent=No;size=5;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S064

S066 · implies

Before

next ORIGINAL vertex bitmask=63;decoded increasing vertices=[0, 1, 2, 3, 4, 5];ALL pair rows=[[0, 1, Yes], [0, 2, Yes], [0, 3, Yes], [0, 4, No], [0, 5, No], [1, 2, Yes], [1, 3, No], [1, 4, No], [1, 5, No], [2, 3, No], [2, 4, No], [2, 5, No], [3, 4, Yes], [3, 5, No], [4, 5, No]];previous best bitmasks=[7, 42]

After

clique=No;independent=No;size=6;strict-larger ONLY update gives best clique=[0, 1, 2] at mask7,best independent=[1, 3, 5] at mask42

Conditions: none

Each mask denotes exactly one subset of the original carrier.Audit all pairs and update only for a strictly larger valid size.Ties retain the first numeric bitmask rather than lexicographically sorting vertex lists;sampling cannot replace a row.

Step identifiers and dependencies

Rule: clique_every_bitmask_exact_extrema_strict_tie_order · Uses: G001, S065

S067 · implies

Before

ALL masks0..63 processed once;best clique=[0, 1, 2] at mask7;best independent=[1, 3, 5] at mask42

After

every possible subset covered;both actual witnesses valid;no valid subset larger;omega=3,alpha=3;FIRST numeric-mask ties retained

Conditions: none

Complete subset coverage gives global upper bounds,and the valid witnesses attain them.Exact extrema require both bounds;inclusion-maximality alone is insufficient.On an empty carrier both extrema are0.

Step identifiers and dependencies

Rule: clique_global_upper_and_attained_lower_not_maximal_only · Uses: G001, S066

Reverse checks

  • S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 9dd803d78e772a921fc3d13e5cc07e21e161a09d56dd658bcb597e08d44aa4d4
  • candidate record SHA-256: ab4ae763271d042ba4d5a860150b49b4c6ca98743b30a46383f3e57a909a127e
  • semantic review receipt SHA-256: d165cad30849a3aba67390a59c572c0d965269d4b405e43df36da19e28d85612
  • question-first witness SHA-256: c222ca47a07939c019865a1e4d9ce633205cdee68db92bf96a96f5683a62ebc7
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON