OPERACIONES ENTRE GRAFOS Pues Puesto to que que los los graf rafos son defi defini nido doss en tér término minoss de los los conju onjunt ntos os de vérti érticces y arist ristas as,, es natu natura rall que que las las oper operac acio ione ness defi defini nida dass en la teor teoría ía de conj conjun unto toss pued puedan an ser ser aplic plicad adas as a la teorí teoríaa de graf grafos os.. Sea Sean G1 = (V1, A1, f G1 G1) y G2 = (V2, A2, f G2 G2) dos subgrafos de un grafo G = (V,A, ,A, f G)
v1 a
v1 b
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a v3
c
v2
d
G1
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v6
v3
c
f
d
g e
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h
e
v4 G2
v5
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UNION
La unió nión de los subgrafos G, G1 y G2, es otro subgrafo G3 = (V3, A3, f G3 G3) de G tal que V3 = V1 U V2, A3 = A1 U A2 y FG3 asign igna a toda arist ista de A3 un par de vérti rtices deV 3. O P E R A C I O N E S E N T R E
v1
EJEMPLO
v1 a
b
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h
a v2
v3
c
v3 c
d
g
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e G1
v5
v1
v6 h
f
d v4
e G2
v5
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INTERSECCION
O P E R A C I O N E S E N T R E
SeanV1 ∩ V2 ≠ 0 la inter tersección de los subgrafos G1 y G2, G2, G1 ∩ G2, G2, es otr otro subg subgra rafo fo G4 = (V4, A4, f G4 t al que V4 = V1 ∩ V2, A4 = A1 ∩ A2 y FG4 asig asigna na a toda toda aris arista ta G4) de G, ta de A4 un par de vérti érticces deV 4.
v1
EJEMPLO
v1 a
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a v2
v3
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v3 c
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g
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e G1
v5
v1
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f
d v4
e G
v5
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SUMA ANILLO
O P E R A C I O N E S E N T R E
La suma anillo de los subgrafos G 1 y G2, G1 o G2, es otro subgrafo G 5 = (V5, A5, FG5) de G, tal que V5 = V1 U V2, A5 = (A1 U A2) – (A1 ΩU A2) (1) y FG5 asigna a toda arista A5 un par de vértices de V5. (1) Sean M y N dos conjuntos . La diferencia diferencia simétrica de M y N, escrita (M U N) – (M ∩ N), es el conjunto de todos los elementos que pertenecen a M U N, pero que no pertenecen a M ∩ N.
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v1 a
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d v4
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Las tres operaciones mencionadas son conmutativas, es decir: decir : G1 U G2 = G2 U G1, G1 ∩ G2 = G2 ∩ G1 y G1 O P E R A C I O N E S
G1
Si G1 y G2 son arista-disjuntos entonces G1 ∩ G2 es igual al grafo vacio y G 1 G1 U G2. Si G1 y G2 son vértice-disjuntos, entonces G1 ∩ G2 no esta definido
v1 G1
v1
v1 b v2
G2
h
arista-disjuntos E N T R E
G2 = G2
Para todo grafo G se tiene que: G UG = G ∩G = G G G es igual a grafo vacio
G1
v2
b
G2 =
v3 G2
b v4
v2 vértice-disjuntos
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EJEMPLO FUSION DE VERTICES
O P E R A C I O N E S E N T R E
La figura muestra la fusión de los vértices a y b. la arista 2 se convierte en un aro y la arista 4 en una arista paralela a la arista 5. la fusión de los vértices no altera el numero de aristas, pero si reduce el numero de vértices en una unidad.
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(a,b)
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5 9
8
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f
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ADICION DE D E UNA ARISTA ARISTA
O P E R A C I O
Sea G = ( V, A, f ) un grafo y u y v dos vértice vé rticess de G. El grafo G+a, donde f(a) = uv denota denot a el grafo cuyo conjunto de vértices es V(G) y cuyo conjunto de aristas es A(G) U {a} esta operación se llama adición de una arista a. Claramente G es subgrafo de G+a
v2
v2
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CONEXIÓN EN GRAFOS Una RUTA en un grafo G es una sucesión finita no nula. R = v0a1v1a2v2 … ak-1vk-1ak vk Cuy Cuyos térm térmiino noss son son alte altern rnad adam amen ente te vért vértic ices es y ari aristas stas,, tal tal qu quee toda toda aris arista ta
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v2
EJEMPLO
a1 C O N E X
v1
a4
a3 v3
a6
a5
a9
a8 v5
a2
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v6
a10
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v7
a12
v9
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Una ruta R – ( V0 - Vk) de un grafo G en el cual todas sus aristas son diferentes, di ferentes, se llama una CADENA C – ( V0 - Vk) de grafo G. Por ejemplo, en el grafo de la figura figu ra anterior anterio r, la ruta R2 es una cadena.
C O N E X
Si además todos los vértices de una cadena son diferentes, esta se llama CADENA SIMPLE (CS) . Por ejemplo, en el grafo de la figura anterior, la ruta R3 es una cadena simple.
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EJEMPLO
v2
C O N E X
a5
a1
v1
a3
a6
v3
a10 a7
v5
v7
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GRAFOS CONEXOS Y NO CONEXOS
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DISTANCIA DIST ANCIA EN UN GRAFO
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La distancia entre dos vértices de un grafo conexo G, puede ser considerada como una función que asigna a cada par de vértices del grafo G un numero entero no negativo Esta
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EJEMPLO
v1