Prime Factorization in the Generalized Hierarchical Product

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1 Prime Factorization in the Generalized Hierarchical Product Sarah Anderson and Kirsti Wash Clemson University January 18, 2014

2 Background Definition A graph G = (V, E) is a set of vertices V (G) together with a set of edges E(G) which are unordered pairs {u, v} of vertices of G [commonly written uv]. If uv E(G) we say that u is adjacent to v. Additionally, if we require u and v to be distinct for the edge uv, then we say G is a simple graph, i.e. G contains no loops. Figure 1: C 5

3 Background Definition Given two graphs G and H, the Cartesian product of G and H, denoted G H, is the graph whose vertex set is V (G) V (H) whereby two vertices (u 1, u 2 ) and (v 1, v 2 ) are adjacent if u 1 v 1 E(G) and u 2 = v 2, or u 1 = v 1 and u 2 v 2 E(H).

4 Background w 2 v 2 u 2 u 1 v 1 w 1 Figure 2: P 3 K 3

5 Background Definition An isomorphism between graphs G and H is a bijection between the vertex sets of G and H f : V (G) V (H) such that uv E(G) if and only if f (u)f (v) E(H).

6 Background Figure 4: Isomorphic copies of K,3 2

7 Definition A graph is prime with respect to a given product if it is nontrivial and cannot be represented as the product of two nontrivial graphs.

8 Definition A graph is prime with respect to a given product if it is nontrivial and cannot be represented as the product of two nontrivial graphs. So if G = G 1 G 2 is prime, then either G 1 or G 2 is K 1.

9 Theorem (Sabidussi 1960, Vizing 1963) Prime factorization with respect to the Cartesian product is not unique in the class of possibly disconnected simple graphs.

10 Theorem (Sabidussi 1960, Vizing 1963) Prime factorization with respect to the Cartesian product is not unique in the class of possibly disconnected simple graphs. Theorem (Sabidussi 1960, Vizing 1963) Every connected graph has a unique prime factor decomposition with respect to the Cartesian product.

11 Theorem (Sabidussi 1960, Vizing 1963) Prime factorization with respect to the Cartesian product is not unique in the class of possibly disconnected simple graphs. Theorem (Sabidussi 1960, Vizing 1963) Every connected graph has a unique prime factor decomposition with respect to the Cartesian product. Cartesian?

12 We wish to show that every connected graph has a unique prime factor decomposition with respect to the Cartesian product.

13 We wish to show that every connected graph has a unique prime factor decomposition with respect to the Cartesian product. We follow the proof found in Handbook of Product Graphs.

14 Definition Consider the Cartesian product G 1 G n. For any index 1 i n, there is a projection map defined as p i (x 1,..., x n ) = x i. p i : G 1 G n G i

15 Definition Consider the Cartesian product G 1 G n. For any index 1 i n, there is a projection map defined as p i (x 1,..., x n ) = x i. p i : G 1 G n G i Definition Given a vertex a = (a 1,..., a n ) of the product G = G 1 G n, the G i -layer through a is the induced subgraph Gi a = {x V (G) p j (x) = a j for j i} = {a 1 } G i {a n }.

16 P 3 K 3 w 2 v 2 u 2 u 1 v 1 w 1

17 P 3 K 3 w 2 v 2 u 2 u 1 v 1 w 1

18 P 3 K 3 w 2 v 2 u 2 u 1 v 1 w 1

19 P 3 K 3 w 2 v 2 u 2 u 1 v 1 w 1

20 Theorem (Sabidussi (1960) and Vizing (1963)) Let φ be an isomorphism between the connected graphs G and H that are representable as Cartesian products G = G 1 G k H = H 1 H l of prime graphs. Then k = l and to any a V (G), there is a permutation π of {1,..., k} such that φ (Gi a) = Hφ(a) π(i) for 1 i k.

21 Theorem (Sabidussi (1960) and Vizing (1963)) Let φ be an isomorphism between the connected graphs G and H that are representable as Cartesian products G = G 1 G k H = H 1 H l of prime graphs. Then k = l and to any a V (G), there is a permutation π of {1,..., k} such that φ (Gi a) = Hφ(a) π(i) for 1 i k. We need to show that φ (Gi a ) is a layer in H.

22 Theorem (Sabidussi (1960) and Vizing (1963)) Let φ be an isomorphism between the connected graphs G and H that are representable as Cartesian products G = G 1 G k H = H 1 H l of prime graphs. Then k = l and to any a V (G), there is a permutation π of {1,..., k} such that φ (Gi a) = Hφ(a) π(i) for 1 i k. We need to show that φ (Gi a ) is a layer in H. How do we do this???

23 Definition A subgraph W G is convex in G if every shortest G-path between vertices of W lies entirely in W.

24 P 4 P 3

25 P 4 P 3 W is the red subgraph of P 4 P 3.

26 P 4 P 3 W is the red subgraph of P 4 P 3. W is not convex.

27 Notice that by definition, every Gi a convex. layer in G = G 1 G k is

28 Notice that by definition, every Gi a convex. layer in G = G 1 G k is G a 1 is the red subgraph.

29 Lemma (HPG) A subgraph W of G = G 1 G k is convex if and only if W = U 1 U k, where each U i is convex in G i.

30 Lemma (HPG) A subgraph W of G = G 1 G k is convex if and only if W = U 1 U k, where each U i is convex in G i. Really, we only need one direction of this lemma: convexity of W = W is a subproduct

31 Assume W is convex and V (W ) {green vertices}.

32 Assume W is convex and V (W ) {green vertices}.

33 Assume W is convex and V (W ) {green vertices}.

34 Assume W is convex and V (W ) {green vertices}.

35 Assume W is convex and V (W ) {green vertices}.

36 Assume W is convex and V (W ) {green vertices}.

37 Assume W is convex and V (W ) {green vertices}. W is a subproduct.

38 Theorem (Sabidussi (1960) and Vizing (1963)) Let φ be an isomorphism between the connected graphs G and H that are representable as Cartesian products G = G 1 G k H = H 1 H l of prime graphs. Then k = l and to any a V (G), there is a permutation π of {1,..., k} such that φ (Gi a) = Hφ(a) π(i) for 1 i k.

39 Proof Sketch Take layer G a i in G, which is a convex subgraph.

40 Proof Sketch Take layer Gi a in G, which is a convex subgraph. Then φ (Gi a ) is a convex subgraph of H.

41 Proof Sketch Take layer G a i Then φ (G a i in G, which is a convex subgraph. ) is a convex subgraph of H. So φ (G a i ) = U 1 U l.

42 Proof Sketch Take layer Gi a in G, which is a convex subgraph. Then φ (Gi a ) is a convex subgraph of H. So φ (G a i ) = U 1 U l. We know G i = G a i = φ (G a i ) are prime.

43 Proof Sketch Take layer Gi a in G, which is a convex subgraph. Then φ (Gi a ) is a convex subgraph of H. So φ (Gi a) = U 1 U l. We know G i = G a i = φ (G a i ) are prime. So U j = {b j } for all but one index, call it π(i).

44 Proof Sketch Take layer Gi a in G, which is a convex subgraph. Then φ (Gi a ) is a convex subgraph of H. So φ (Gi a) = U 1 U l. We know G i = G a i = φ (G a i ) are prime. So U j = {b j } for all but one index, call it π(i). φ (G a i ) = {b 1 } U π(i) {b l }

45 Proof Sketch Take layer Gi a in G, which is a convex subgraph. Then φ (Gi a ) is a convex subgraph of H. So φ (Gi a) = U 1 U l. We know G i = G a i = φ (G a i ) are prime. So U j = {b j } for all but one index, call it π(i). φ (G a i ) = {b 1 } U π(i) {b l } So φ (G a i ) Hφ(a) π(i).

46 Proof Sketch Take layer Gi a in G, which is a convex subgraph. Then φ (Gi a ) is a convex subgraph of H. So φ (Gi a) = U 1 U l. We know G i = G a i = φ (G a i ) are prime. So U j = {b j } for all but one index, call it π(i). φ (G a i ) = {b 1 } U π(i) {b l } So φ (G a i ) Hφ(a) π(i). The reverse argument shows φ (G a i ) = Hφ(a) π(i).

47 Hierarchical Product (2008)- Barrière, Dalfó, Fiol, and Mitjana introduced a new graph product, referred to as the generalized hierarchical product of graphs.

48 Hierarchical Product (2008)- Barrière, Dalfó, Fiol, and Mitjana introduced a new graph product, referred to as the generalized hierarchical product of graphs. Definition Given graphs G 1,..., G n and nonempty vertex subsets U i V (G i ) for 1 i n 1, the generalized hierarchical product, denoted G = G 1 (U 1 ) G n 1 (U n 1 ) G n, is the graph with vertex set V (G 1 ) V (G n ) and adjacencies (y 1, x 2,..., x n ) if x 1 y 1 E(G 1 ) (x 1,..., x n 1, x n ) (x 1, y 2, x 3,..., x n ) if x 2 y 2 E(G 2 ), x 1 U 1.. (x 1,..., x n 1, y n ) if x n y n E(G n ), x i U i, 1 i n 1.

49 Hierarchical Product Example w 2 v 2 u 2 u 1 v 1 w 1 x 1 Figure 5: P 4 (U 1 ) K 3 where U 1 = {v 1, w 1 }

50 Hierarchical Product We now wish to show that every connected graph has a unique prime factor decomposition with respect to the hierarchical product.

51 Hierarchical Product We now wish to show that every connected graph has a unique prime factor decomposition with respect to the hierarchical product. What part of the proof for the Cartesian product fails in the context of the hierarchical product?

52 Hierarchical Product We now wish to show that every connected graph has a unique prime factor decomposition with respect to the hierarchical product. What part of the proof for the Cartesian product fails in the context of the hierarchical product? convexity = subproduct

53 Hierarchical Product Example

54 Hierarchical Product Example W is the green subgraph of G = P 4 (U 1 ) P 3.

55 Hierarchical Product Example W is the green subgraph of G = P 4 (U 1 ) P 3. W is convex, but not a subproduct.

56 Hierarchical Product Solution

57 Hierarchical Product Solution Just change the definition of convexity...kind of.

58 Hierarchical Product Solution Just change the definition of convexity...kind of. Definition Let H be a subgraph of G = G 1 (U 1 ) G n 1 (U n 1 ) G n, and let G = G 1 G n. We say that H is hierarchically convex if given any two distinct vertices a and b of H and any shortest G -path P between a and b, each edge of P contained in G is also contained in H.

59 Hierarchical Product Example a: H 1 is not hierarchically convex b: H 2 is hierarchically convex Figure 7: P 3 (U 1 ) P 3 where V (P 3 ) = {0, 1, 2} and U 1 = {1, 2}

60 Hierarchical Product Lemma (Anderson and W.) If H is a subgraph of a hierarchical product G that is hierarchically convex, then H is a subproduct.

61 Hierarchical Product Theorem (Anderson and W.) Let φ be an isomorphism between the connected graphs G and H that are representable as hierarchical products G = G 1 (U 1 ) G n 1 (U n 1 ) G n, H = H 1 (W 1 ) H m 1 (W m 1 ) H m of prime graphs, where U i V (G i ) for i {1,..., n 1} and W j V (H j ) for j {1,..., m 1}. For any a = (a 1,..., a n ) V (G) where a i U i for 1 i n 1, there exists a permutation π of {1,..., n} such that φ(gi a) = Hφ(a) π(i) for 1 i n and m = n.

62 Hierarchical Product Proof Sketch Choose a = (a 1, a 2,..., a n ) V (G) such that a i U i for 1 i n 1.

63 Hierarchical Product Proof Sketch Choose a = (a 1, a 2,..., a n ) V (G) such that a i U i for 1 i n 1. For any j {1,..., n}, Gj a is a connected layer in G, which is hierarchically convex in G.

64 Hierarchical Product Proof Sketch Choose a = (a 1, a 2,..., a n ) V (G) such that a i U i for 1 i n 1. For any j {1,..., n}, Gj a is a connected layer in G, which is hierarchically ( ) convex in G. Then φ is hierarchically convex in H. G a j

65 Hierarchical Product Proof Sketch Choose a = (a 1, a 2,..., a n ) V (G) such that a i U i for 1 i n 1. For any j {1,..., n}, Gj a is a connected layer in G, which is hierarchically ( ) convex in G. Then φ Gj a is hierarchically convex in H. ( ) So φ = F 1 (V 1 ) F m 1 (V m 1 ) F m, where G a j F i H i for 1 i m and V i W i for 1 i m 1.

66 Hierarchical Product Proof Sketch Choose a = (a 1, a 2,..., a n ) V (G) such that a i U i for 1 i n 1. For any j {1,..., n}, Gj a is a connected layer in G, which is hierarchically ( ) convex in G. Then φ Gj a is hierarchically convex in H. ( ) So φ = F 1 (V 1 ) F m 1 (V m 1 ) F m, where G a j F i H i for 1 i m and ( V) i W i for 1 i m 1. We know G j = G a j = φ Gj a are prime.

67 Hierarchical Product Proof Sketch Choose a = (a 1, a 2,..., a n ) V (G) such that a i U i for 1 i n 1. For any j {1,..., n}, Gj a is a connected layer in G, which is hierarchically ( ) convex in G. Then φ Gj a is hierarchically convex in H. ( ) So φ = F 1 (V 1 ) F m 1 (V m 1 ) F m, where G a j F i H i for 1 i m and ( V) i W i for 1 i m 1. We know G j = G a j = φ Gj a are prime. So F k = {b k } for all but one index, call it π(j).

68 Hierarchical Product Proof Sketch Choose a = (a 1, a 2,..., a n ) V (G) such that a i U i for 1 i n 1. For any j {1,..., n}, Gj a is a connected layer in G, which is hierarchically ( ) convex in G. Then φ Gj a is hierarchically convex in H. ( ) So φ = F 1 (V 1 ) F m 1 (V m 1 ) F m, where G a j F i H i for 1 i m and ( V) i W i for 1 i m 1. We know G j = G a j = φ Gj a are prime. So( F k ) = {b k } for all but one index, call it π(j). φ = {b 1 } F π(j) (V π(j) ) {b m } G a j

69 Hierarchical Product Proof Sketch Choose a = (a 1, a 2,..., a n ) V (G) such that a i U i for 1 i n 1. For any j {1,..., n}, Gj a is a connected layer in G, which is hierarchically ( ) convex in G. Then φ Gj a is hierarchically convex in H. ( ) So φ = F 1 (V 1 ) F m 1 (V m 1 ) F m, where G a j F i H i for 1 i m and ( V) i W i for 1 i m 1. We know G j = G a j = φ Gj a are prime. So( F k ) = {b k } for all but one index, call it π(j). φ Gj a = {b 1 } F π(j) (V π(j) ) {b m } ( ) So φ Gj a H φ(a) π(j).

70 Hierarchical Product Proof Sketch Choose a = (a 1, a 2,..., a n ) V (G) such that a i U i for 1 i n 1. For any j {1,..., n}, Gj a is a connected layer in G, which is hierarchically ( ) convex in G. Then φ Gj a is hierarchically convex in H. ( ) So φ = F 1 (V 1 ) F m 1 (V m 1 ) F m, where G a j F i H i for 1 i m and ( V) i W i for 1 i m 1. We know G j = G a j = φ Gj a are prime. So( F k ) = {b k } for all but one index, call it π(j). φ Gj a = {b 1 } F π(j) (V π(j) ) {b m } ( ) So φ Gj a H φ(a) π(j).the reverse argument shows ( ) φ Gj a = H φ(a) π(j).

71 Hierarchical Product Additional Questions What information does this tell us about the Cartesian product?

72 Hierarchical Product Additional Questions What information does this tell us about the Cartesian product? What can be said about α(g 1 (U 1 ) G 2 )?

73 Hierarchical Product Additional Questions What information does this tell us about the Cartesian product? What can be said about α(g 1 (U 1 ) G 2 )? i(g 1 (U 1 ) G 2 )?

74 Hierarchical Product Additional Questions What information does this tell us about the Cartesian product? What can be said about α(g 1 (U 1 ) G 2 )? i(g 1 (U 1 ) G 2 )? ρ 2 (G 1 (U 1 ) G 2 )?

75 Hierarchical Product Additional Questions What information does this tell us about the Cartesian product? What can be said about α(g 1 (U 1 ) G 2 )? i(g 1 (U 1 ) G 2 )? ρ 2 (G 1 (U 1 ) G 2 )? Is there a polynomial time algorithm for finding the prime factors of a hierarchical product?

76 Hierarchical Product Additional Questions What information does this tell us about the Cartesian product? What can be said about α(g 1 (U 1 ) G 2 )? i(g 1 (U 1 ) G 2 )? ρ 2 (G 1 (U 1 ) G 2 )? Is there a polynomial time algorithm for finding the prime factors of a hierarchical product? Can you find γ(g 1 (U 1 ) G 2 )?

77 Hierarchical Product References R. Hammack, W. Imrich, and S. Klavžar. Handbook of product graphs. Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, second edition, L. Barrière, C. Dalfó, M.A. Fiol, and M. Mitjana. The generalized hierarchical product of graphs. Discrete Math., 309(12): , V.G. Vizing. The Cartesian product of graphs (Russian). Vyčisl. Sistemy, 9, 30-43, English translation in Comp. El. Syst. 2: , G. Sabidussi. Graph Multiplication. Math. Zeitschr., 72: , 1960.

78 Hierarchical Product Thank you!

Prime Factorization and Domination in the Hierarchical Product of Graphs

Prime Factorization and Domination in the Hierarchical Product of Graphs S. E. Anderson 1, Y. Guo 2, A. Rubin 2 and K. Wash 2 1 Department of Mathematics, University of St. Thomas, St. Paul, MN 55105 2