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ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-01
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  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Complete, anticomplete and mixed vertex-set pairs in P4

Statement

In the path P4=1,2,3,4:

  • ({2},{1,3}) is complete;
  • ({1},{3,4}) is anticomplete; and
  • ({2},{3,4}) is mixed.

Facts & Assumptions

Given: P4 with edge set {12,23,34}.

[F1]

Complete means all cross pairs are edges, anticomplete means none are edges, and mixed means some but not all are edges (Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs).

[F2]

The only edges of the path are consecutive pairs (Empty and complete graphs, complete bipartite graphs, and the convention that Pn and Cn have n vertices).

Verification

technique · direct
1.1

Both 21 and 23 are edges, so ({2},{1,3}) is complete.

F1F2
1.2

Neither 13 nor 14 is an edge, so ({1},{3,4}) is anticomplete.

F1F2
1.3

The pair 23 is an edge but 24 is not, so ({2},{3,4}) is mixed.

F1F2
2.1

These verify the three claimed pair types.

step 1.1step 1.2step 1.3∎

Remarks


% Anticomplete pair. \node[aset] (a1) at (4.25,0) {$1$}; \node[vertex] (a2) at (5.25,0) {$2$}; \node[bset] (a3) at (6.25,0) {$3$}; \node[bset] (a4) at (7.25,0) {$4$}; \draw[edge] (a1)--(a2)--(a3)--(a4); \node[caption] at (5.75,-.65) {$A=\{1\}$, $B=\{3,4\}$\\anticomplete};

% Mixed pair. \node[vertex] (m1) at (8.5,0) {$1$}; \node[aset] (m2) at (9.5,0) {$2$}; \node[bset] (m3) at (10.5,0) {$3$}; \node[bset] (m4) at (11.5,0) {$4$}; \draw[edge] (m1)--(m2)--(m3)--(m4); \node[caption] at (10,-.65) {$A=\{2\}$, $B=\{3,4\}$\\mixed};

\node[font=\scriptsize,blue!70!black] at (4.55,1.0) {blue: $A$}; \node[font=\scriptsize,orange!80!black] at (6.95,1.0) {orange: $B$}; \end{tikzpicture} ```

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources