Original question and givens
The finite simple undirected graph has V={-1, 0, 1, 2, 3, 4} and E={{-1, 0}, {-1, 1}, {-1, 2}, {-1, 3}, {-1, 4}, {0, 1}, {0, 2}, {0, 3}, {0, 4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-1 belongs to V. Find deg(v).
G001 — The finite simple undirected graph has V={-1, 0, 1, 2, 3, 4} and E={{-1, 0}, {-1, 1}, {-1, 2}, {-1, 3}, {-1, 4}, {0, 1}, {0, 2}, {0, 3}, {0, 4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-1 belongs to V. Find deg(v).
Step-by-step derivation
S001 · implies
Before
source roles = [qf-simple-graph-degree, [E, V, operation, v]]
After
derived = [degree, no-invented-domain]
Conditions: none
Keep EACH bank’s keys/operation/domain distinct:LDL/PULSE/function legacy table underscores,TREE full tree_* with its declared domain,and permissive BASE signed without invented op/domain.All arithmetic respects the binary-primitive contract.
Step identifiers and dependencies
Rule: R69_SOURCE_specific_keys_operation_domain_AND_binary_contract · Uses: G001
S002 · implies
Before
source roles = [LDL graph vertices ORIGINAL, 100, [-1, 0, 1, 2, 3, 4]]
After
derived = [[-1, 0, 1, 2, 3, 4]]
Conditions: none
Type EVERY RAW SAFE label,distinctness and THIS bank’s cardinality bound;empty is allowed.Sort ONLY the derived representation,not original JSON.Do not exchange TREE and LDL limits.
Step identifiers and dependencies
Rule: Finite_RAW_ALL_SAFE_distinct_labels_then_derived_sort · Uses: G001, S001
S003 · implies
Before
source roles = [LDL raw edges ORIGINAL, 0, [-1, 0]]
After
derived = [[-1, 0]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S002
S004 · implies
Before
source roles = [LDL raw edges ORIGINAL, 1, [-1, 1]]
After
derived = [[-1, 1]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S003
S005 · implies
Before
source roles = [LDL raw edges ORIGINAL, 2, [-1, 2]]
After
derived = [[-1, 2]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S004
S006 · implies
Before
source roles = [LDL raw edges ORIGINAL, 3, [-1, 3]]
After
derived = [[-1, 3]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S005
S007 · implies
Before
source roles = [LDL raw edges ORIGINAL, 4, [-1, 4]]
After
derived = [[-1, 4]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S006
S008 · implies
Before
source roles = [LDL raw edges ORIGINAL, 5, [0, 1]]
After
derived = [[0, 1]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S007
S009 · implies
Before
source roles = [LDL raw edges ORIGINAL, 6, [0, 2]]
After
derived = [[0, 2]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S008
S010 · implies
Before
source roles = [LDL raw edges ORIGINAL, 7, [0, 3]]
After
derived = [[0, 3]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S009
S011 · implies
Before
source roles = [LDL raw edges ORIGINAL, 8, [0, 4]]
After
derived = [[0, 4]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S010
S012 · implies
Before
source roles = [LDL raw edges ORIGINAL, 9, [1, 2]]
After
derived = [[1, 2]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S011
S013 · implies
Before
source roles = [LDL raw edges ORIGINAL, 10, [1, 3]]
After
derived = [[1, 3]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S012
S014 · implies
Before
source roles = [LDL raw edges ORIGINAL, 11, [1, 4]]
After
derived = [[1, 4]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S013
S015 · implies
Before
source roles = [LDL raw edges ORIGINAL, 12, [2, 3]]
After
derived = [[2, 3]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S014
S016 · implies
Before
source roles = [LDL raw edges ORIGINAL, 13, [2, 4]]
After
derived = [[2, 4]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S015
S017 · implies
Before
source roles = [LDL raw edges ORIGINAL, 14, [3, 4]]
After
derived = [[3, 4]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S016
S018 · implies
Before
source roles = [[-1, 0, 1, 2, 3, 4], [[-1, 0], [-1, 1], [-1, 2], [-1, 3], [-1, 4], [0, 1], [0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], No]
After
derived = [[[-1, 0], [-1, 1], [-1, 2], [-1, 3], [-1, 4], [0, 1], [0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], [[-1, [0, 1, 2, 3, 4]], [0, [-1, 1, 2, 3, 4]], [1, [-1, 0, 2, 3, 4]], [2, [-1, 0, 1, 3, 4]], [3, [-1, 0, 1, 2, 4]], [4, [-1, 0, 1, 2, 3]]]]
Conditions: none
Type the WHOLE raw edge list,then normalize undirected endpoints/deduplicate;reversed duplicates are allowed.INPUT edges have no loops/outside endpoints,while isolated vertices stay in V.TREE raw E does NOT acquire LDL’s200-entry limit.
Step identifiers and dependencies
Rule: Graph_WHOLE_simple_undirected_edges_normalized_NO_loops_or_outside · Uses: G001, S017
S019 · implies
Before
source roles = [-1, [-1, 0, 1, 2, 3, 4]]
After
derived = [[0, 1, 2, 3, 4], 5]
Conditions: none
A degree/neighbor QUERY first requires v∈V.Find distinct neighbors through ALL edges,not sketch crossings or raw duplicate rows.
Step identifiers and dependencies
Rule: Graph_QUERY_vertex_must_belong_V_complete_distinct_neighbors · Uses: G001, S018