Grafo bipartito completo

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Nella teoria dei grafi, si definisce grafo bipartito completo un grafo bipartito $\,G=\langle (V_{1};V_{2}),E\rangle$ , con $V_{1}$ e $V_{2}$ ad indicare i sottoinsiemi dei nodi, tale che:

$\forall \;(v_{i},v_{j}):[(v_{i}\in V_{1})\land (v_{j}\in V_{2})]\;\exists \;\{v_{i};v_{j}\}$

È quindi un grafo bipartito in cui esistono tutti gli archi che connettono gli elementi di un insieme a quelli dell'altro, o, come dice la definizione, per ogni coppia di vertici di cui il primo nell'insieme $V_{1}$ e il secondo nell'insieme $V_{2}$ esiste un arco che abbia inizio nel primo e termine nel secondo.

Questo genere di grafi è utilizzato in alcuni algoritmi, in particolare nella soluzione di problemi di assegnamento.

Esempi

Per ogni k, K_1,k è chiamato stella. Tutti i grafi bipartiti completi che sono alberi sono stelle.
Il grafo K_1,3 è chiamato artiglio.
Il grafo K_3,3 è chiamato grafo dei servizi.

Altri progetti

Wikimedia Commons contiene immagini o altri file su grafo bipartito completo

Collegamenti esterni

(EN) Eric W. Weisstein, Grafo bipartito completo, su MathWorld, Wolfram Research.

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