Applications of zero forcing number to the minimum rank problem - PowerPoint PPT Presentation

Reduction Formula for P ( G ) A vertex v is doubly terminal if v is a one-vertex path in some optimal path cover. A vertex v is simply terminal if v is an endpoint of a path in some optimal path cover and v is not doubly terminal. The path spread of G on v is p v ( G ) = P ( G ) − P ( G − v ) . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Reduction Formula for P ( G ) A vertex v is doubly terminal if v is a one-vertex path in some optimal path cover. A vertex v is simply terminal if v is an endpoint of a path in some optimal path cover and v is not doubly terminal. The path spread of G on v is p v ( G ) = P ( G ) − P ( G − v ) . If G = G 1 ⊕ v G 2 , then ⎧ − 1 , ⎪ ⎪ ⎪ if v is simply terminal p v ( G ) = ⎨ ⎪ of G 1 and G 2 ; ⎪ ⎪ min { p v ( G 1 ) , p v ( G 2 )} , otherwise . [ 5 ] ⎩ Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Reduction Formula for Z ( G ) A vertex v is doubly terminal if v is a one-vertex maximal chain in some optimal chronological list. Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Reduction Formula for Z ( G ) A vertex v is doubly terminal if v is a one-vertex maximal chain in some optimal chronological list. A vertex v is simply terminal if v is an endpoint of a maximal chain in some optimal chronological list and v is not doubly terminal. Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Reduction Formula for Z ( G ) A vertex v is doubly terminal if v is a one-vertex maximal chain in some optimal chronological list. A vertex v is simply terminal if v is an endpoint of a maximal chain in some optimal chronological list and v is not doubly terminal. The zero spread of G on v is z v ( G ) = Z ( G ) − Z ( G − v ) . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Reduction Formula for Z ( G ) A vertex v is doubly terminal if v is a one-vertex maximal chain in some optimal chronological list. A vertex v is simply terminal if v is an endpoint of a maximal chain in some optimal chronological list and v is not doubly terminal. The zero spread of G on v is z v ( G ) = Z ( G ) − Z ( G − v ) . If G = G 1 ⊕ v G 2 , then ⎧ − 1 ⎪ ⎪ ⎪ if v is simply terminal z v ( G ) = ⎨ ⎪ of G 1 and G 2 ; ⎪ ⎪ min { z v ( G 1 ) , z v ( G 2 )} ⎩ otherwise . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Sketch of Proof ⎧ − 1 ≤ z v ( G ) ≤ 1 . ⎪ ⎪ ⎪ ⎨ v is doubly terminal ⇔ z v = 0 . ⎪ ⎪ ⎪ v is simply terminal ⇒ z v = 0 . ⎩ Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Sketch of Proof ⎧ − 1 ≤ z v ( G ) ≤ 1 . ⎪ ⎪ ⎪ v is doubly terminal ⇔ z v = 0 . ⎨ ⎪ ⎪ ⎪ v is simply terminal ⇒ z v = 0 . ⎩ If v is simply terminal for G 1 and G 2 , then z v ( G ) = − 1, z v ( G 1 ) = z v ( G 2 ) = 0. Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Sketch of Proof ⎧ − 1 ≤ z v ( G ) ≤ 1 . ⎪ ⎪ ⎪ v is doubly terminal ⇔ z v = 0 . ⎨ ⎪ ⎪ ⎪ v is simply terminal ⇒ z v = 0 . ⎩ If v is simply terminal for G 1 and G 2 , then z v ( G ) = − 1, z v ( G 1 ) = z v ( G 2 ) = 0. Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Sketch of Proof If G = G 1 ⊕ v G 2 , then Z ( G ) ≤ Z ( G 1 ) + Z ( G 2 − v ) , Z ( G ) ≤ Z ( G 1 − v ) + Z ( G 2 ) , Z ( G ) ≥ Z ( G 1 ) + Z ( G 2 ) − 1 . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Sketch of Proof If G = G 1 ⊕ v G 2 , then Z ( G ) ≤ Z ( G 1 ) + Z ( G 2 − v ) , Z ( G ) ≤ Z ( G 1 − v ) + Z ( G 2 ) , Z ( G ) ≥ Z ( G 1 ) + Z ( G 2 ) − 1 . If G = G 1 ⊕ v G 2 , then z v ( G ) ≤ min { z v ( G 1 ) , z v ( G 2 )} , z v ( G ) ≥ z v ( G 1 ) + z v ( G 2 ) − 1 . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Sketch of Proof If G = G 1 ⊕ v G 2 , then Z ( G ) ≤ Z ( G 1 ) + Z ( G 2 − v ) , Z ( G ) ≤ Z ( G 1 − v ) + Z ( G 2 ) , Z ( G ) ≥ Z ( G 1 ) + Z ( G 2 ) − 1 . If G = G 1 ⊕ v G 2 , then z v ( G ) ≤ min { z v ( G 1 ) , z v ( G 2 )} , z v ( G ) ≥ z v ( G 1 ) + z v ( G 2 ) − 1 . z v ( G ) = − 1, z v ( G 1 ) = z v ( G 2 ) = 0 is the only possibility. This implies v is simply terminal for G 1 and G 2 . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Comparison of Reduction Formulae Denote m v ( G ) = M ( G ) − M ( G − v ) , p v ( G ) = P ( G ) − P ( G − v ) , and z v ( G ) = Z ( G ) − Z ( G − v ) . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Comparison of Reduction Formulae Denote m v ( G ) = M ( G ) − M ( G − v ) , p v ( G ) = P ( G ) − P ( G − v ) , and z v ( G ) = Z ( G ) − Z ( G − v ) . − 1 ≤ m v , p v , r v ≤ 1. Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Comparison of Reduction Formulae Denote m v ( G ) = M ( G ) − M ( G − v ) , p v ( G ) = P ( G ) − P ( G − v ) , and z v ( G ) = Z ( G ) − Z ( G − v ) . − 1 ≤ m v , p v , r v ≤ 1. If G = G 1 ⊕ v G 2 , they have similar behavior. m v ( G 1 / G 2 ) − 1 0 1 − 1 − 1 − 1 − 1 − 1 − 1 0 0 − 1 1 0 1 , p v , z v ( G 1 / G 2 ) − 1 0 1 − 1 − 1 − 1 − 1 − 1 − 1 / 0 0 0 − 1 1 0 1 . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Comparison of Reduction Formulae Denote m v ( G ) = M ( G ) − M ( G − v ) , p v ( G ) = P ( G ) − P ( G − v ) , and z v ( G ) = Z ( G ) − Z ( G − v ) . − 1 ≤ m v , p v , r v ≤ 1. If G = G 1 ⊕ v G 2 , they have similar behavior. m v ( G 1 / G 2 ) − 1 0 1 − 1 − 1 − 1 − 1 − 1 − 1 0 0 − 1 1 0 1 , p v , z v ( G 1 / G 2 ) − 1 0 1 − 1 − 1 − 1 − 1 − 1 − 1 / 0 0 0 − 1 1 0 1 . Hard to apply on induction. Chin-Hung Lin Applications of z. f. number to the minimum rank problem

The PZ condition Recall that P ( G ) ≤ Z ( G ) . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

The PZ condition Recall that P ( G ) ≤ Z ( G ) . A graph G satisfies the PZ condition iff P ( G ) = Z ( G ) . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

The PZ condition Recall that P ( G ) ≤ Z ( G ) . A graph G satisfies the PZ condition iff P ( G ) = Z ( G ) . PZ condition is not hereditary. 1 2 3 4 5 6 Chin-Hung Lin Applications of z. f. number to the minimum rank problem

The PZ condition Recall that P ( G ) ≤ Z ( G ) . A graph G satisfies the PZ condition iff P ( G ) = Z ( G ) . PZ condition is not hereditary. PZ condition does not preserve under vertex-sum operation. v v v G 1 G 2 G 1 ⊕ v G 2 Chin-Hung Lin Applications of z. f. number to the minimum rank problem

The Strong PZ condition A graph G satisfies the strong PZ condition iff each path cover is the set of maximal chain for some zero forcing process. Chin-Hung Lin Applications of z. f. number to the minimum rank problem

The Strong PZ condition A graph G satisfies the strong PZ condition iff each path cover is the set of maximal chain for some zero forcing process. Stong PZ condition ⇒ PZ condition. 1 2 3 4 5 6 Chin-Hung Lin Applications of z. f. number to the minimum rank problem

The Strong PZ condition A graph G satisfies the strong PZ condition iff each path cover is the set of maximal chain for some zero forcing process. Stong PZ condition ⇒ PZ condition. Strong PZ condition is hereditary. Chin-Hung Lin Applications of z. f. number to the minimum rank problem

The Strong PZ condition A graph G satisfies the strong PZ condition iff each path cover is the set of maximal chain for some zero forcing process. Stong PZ condition ⇒ PZ condition. Strong PZ condition is hereditary. Chin-Hung Lin Applications of z. f. number to the minimum rank problem

The Strong PZ condition A graph G satisfies the strong PZ condition iff each path cover is the set of maximal chain for some zero forcing process. Stong PZ condition ⇒ PZ condition. Strong PZ condition is hereditary. Strong PZ condition preserves under vertex-sum operation. Chin-Hung Lin Applications of z. f. number to the minimum rank problem

The Strong PZ condition A graph G satisfies the strong PZ condition iff each path cover is the set of maximal chain for some zero forcing process. Stong PZ condition ⇒ PZ condition. Strong PZ condition is hereditary. Strong PZ condition preserves under vertex-sum operation. Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Cactus graphs A cactus is a graph whose blocks are all K 2 or C n . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Cactus graphs A cactus is a graph whose blocks are all K 2 or C n . A cactus G satisfies the strong PZ condition. Hence we have P ( G ) = Z ( G ) . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Large Z ( G ) − M ( G ) Let G k be the k 5-sun sequence. Then P ( G k ) = Z ( G k ) = 2 k + 1 and M ( G k ) = k + 1. 1(5) 2(5) k (5) k (6) 2(6) 1(6) k (1) 2(1) 1(1) k (10) 2(3) k (3 1(3) k (9) 1(9) 2(9) 1(4) 2(4) k (4) 2(7) k (7) 1(7) 1(2) k (8) 1(8) 2(2) 2(8) k (2) Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Large Z ( G ) − M ( G ) Let G k be the k 5-sun sequence. Then P ( G k ) = Z ( G k ) = 2 k + 1 and M ( G k ) = k + 1. Actually, for all 1 ≤ p ≤ q ≤ 2 p − 1, there is a graph G such that M ( G ) = p and Z ( G ) = q . 1(5) 2(5) k (5) k (6) 2(6) 1(6) k (1) 2(1) 1(1) k (10) 2(3) k (3 1(3) k (9) 1(9) 2(9) 1(4) 2(4) k (4) 2(7) k (7) 1(7) 1(2) k (8) 1(8) 2(2) 2(8) k (2) Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Large Z ( G ) − M ( G ) Let G k be the k 5-sun sequence. Then P ( G k ) = Z ( G k ) = 2 k + 1 and M ( G k ) = k + 1. Actually, for all 1 ≤ p ≤ q ≤ 2 p − 1, there is a graph G such that M ( G ) = p and Z ( G ) = q . Q: Will the inequality Z ( G ) ≤ 2 M ( G ) − 1 holds for all G ? 1(5) 2(5) k (5) k (6) 2(6) 1(6) k (1) 2(1) 1(1) k (10) 2(3) k (3 1(3) k (9) 1(9) 2(9) 1(4) 2(4) k (4) 2(7) k (7) 1(7) 1(2) k (8) 1(8) 2(2) 2(8) k (2) Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Minimum Rank of A Pattern A sign set is { 0 , ∗ , u } . A real number r matchs 0 if r = 0, ∗ if r ≠ 0, while u if r matchs 0 or ∗ . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Minimum Rank of A Pattern A sign set is { 0 , ∗ , u } . A real number r matchs 0 if r = 0, ∗ if r ≠ 0, while u if r matchs 0 or ∗ . A pattern matrix Q is a matrix over S . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Minimum Rank of A Pattern A sign set is { 0 , ∗ , u } . A real number r matchs 0 if r = 0, ∗ if r ≠ 0, while u if r matchs 0 or ∗ . A pattern matrix Q is a matrix over S . The minimum rank of a pattern Q is mr ( Q ) = min { rank A ∶ A ≅ Q } . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Example for Minmum Rank of A Pattern The pattern Q = (∗ u ) 0 0 ∗ u must have rank at least 2. Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Example for Minmum Rank of A Pattern The pattern Q = (∗ u ) 0 0 ∗ u must have rank at least 2. The rank 2 is achievable. Hence mr ( Q ) = 2. Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Operation on S Define addition “ + ” and scalar multiplication “ × ” on S . +∶ S × S → S + ∗ 0 u ∗ 0 0 u ∗ ∗ u u u u u u ×∶ { 0 , ∗ } × S → S × 0 ∗ u 0 0 0 0 ∗ ∗ 0 u Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Independence A sign vector is a tuple with entris on S . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Independence A sign vector is a tuple with entris on S . We say a sign vector v ∼ 0 iff v contains no ∗ . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Independence A sign vector is a tuple with entris on S . We say a sign vector v ∼ 0 iff v contains no ∗ . A set of sign vectors { v 1 , v 2 ,..., v n } is independent iff c 1 v 1 + c 2 v 2 + ⋯ c n v n ∼ 0 implies c 1 = c 2 = ⋯ = c n = 0. Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Independence A sign vector is a tuple with entris on S . We say a sign vector v ∼ 0 iff v contains no ∗ . A set of sign vectors { v 1 , v 2 ,..., v n } is independent iff c 1 v 1 + c 2 v 2 + ⋯ c n v n ∼ 0 implies c 1 = c 2 = ⋯ = c n = 0. The rank of a pattern is the maximum number of independent row sign vectors. Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Independence in different senses Lemma Suppose V = { v 1 , v 2 ,..., v n } is a set of sign vectors, and W = { w 1 , w 2 ,..., w n } is a set of sign vectors such that w i is obtained from v i by replacing entries u by 0 or ∗ . If V is linearly independent, then so is W . Suppose R = { r 1 , r 2 ,..., r n } is a set of real vectors such that each entry in each vector matches the corresponding entry in elements of W . If W is linearly independent, then R is linearly independent as real vectors. Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Independence in different senses Lemma Suppose V = { v 1 , v 2 ,..., v n } is a set of sign vectors, and W = { w 1 , w 2 ,..., w n } is a set of sign vectors such that w i is obtained from v i by replacing entries u by 0 or ∗ . If V is linearly independent, then so is W . Suppose R = { r 1 , r 2 ,..., r n } is a set of real vectors such that each entry in each vector matches the corresponding entry in elements of W . If W is linearly independent, then R is linearly independent as real vectors. Theorem If Q is a pattern matrix and U is the set of all pattern matrices obtained from Q by replacing u by 0 or ∗ , then rank ( Q ) ≤ min Q ′ ∈ U { rank ( Q ′ )} ≤ mr ( Q ) . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Zero Forcing Number with Banned Edges And Given Support Let G be a graph and B is a subset of E ( G ) called the set of banned edge or banned set. Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Zero Forcing Number with Banned Edges And Given Support Let G be a graph and B is a subset of E ( G ) called the set of banned edge or banned set. The zero forcing process on G banned by B is the coloring process by following rules. Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Zero Forcing Number with Banned Edges And Given Support Let G be a graph and B is a subset of E ( G ) called the set of banned edge or banned set. The zero forcing process on G banned by B is the coloring process by following rules. Each vertex of G is either black or white initially. Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Zero Forcing Number with Banned Edges And Given Support Let G be a graph and B is a subset of E ( G ) called the set of banned edge or banned set. The zero forcing process on G banned by B is the coloring process by following rules. Each vertex of G is either black or white initially. If x is a black vertex and y is the only white neighbor of x and xy ∉ B , then change the color of y to black. Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Zero Forcing Number with Banned Edges And Given Support Let G be a graph and B is a subset of E ( G ) called the set of banned edge or banned set. The zero forcing process on G banned by B is the coloring process by following rules. Each vertex of G is either black or white initially. If x is a black vertex and y is the only white neighbor of x and xy ∉ B , then change the color of y to black. Zero forcing set banned by B F : F can force V ( G ) banned by B . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Zero Forcing Number with Banned Edges And Given Support Let G be a graph and B is a subset of E ( G ) called the set of banned edge or banned set. The zero forcing process on G banned by B is the coloring process by following rules. Each vertex of G is either black or white initially. If x is a black vertex and y is the only white neighbor of x and xy ∉ B , then change the color of y to black. Zero forcing set banned by B F : F can force V ( G ) banned by B . Zero forcing number banned by B Z ( G , B ) : minimum size of F . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Zero Forcing Number with Banned Edges And Given Support Let G be a graph and B is a subset of E ( G ) called the set of banned edge or banned set. The zero forcing process on G banned by B is the coloring process by following rules. Each vertex of G is either black or white initially. If x is a black vertex and y is the only white neighbor of x and xy ∉ B , then change the color of y to black. Zero forcing set banned by B F : F can force V ( G ) banned by B . Zero forcing number banned by B Z ( G , B ) : minimum size of F . Zero forcing number banned by B with support W Z W ( G , B ) : minimum size of F ⊇ W . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Zero Forcing Number with Banned Edges And Given Support Let G be a graph and B is a subset of E ( G ) called the set of banned edge or banned set. The zero forcing process on G banned by B is the coloring process by following rules. Each vertex of G is either black or white initially. If x is a black vertex and y is the only white neighbor of x and xy ∉ B , then change the color of y to black. Zero forcing set banned by B F : F can force V ( G ) banned by B . Zero forcing number banned by B Z ( G , B ) : minimum size of F . Zero forcing number banned by B with support W Z W ( G , B ) : minimum size of F ⊇ W . When W and B is empty, Z W ( G , B ) = Z ( G ) . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Natural Relation between Patterns and Bipartites Q is a given m × n pattern. G = ( X ∪ Y , E ) is the related bipartite defined by X = { a 1 , a 2 ,..., a m } , Y = { b 1 , b 2 ,..., b n } , E = { a i b j ∶ Q ij ≠ 0 } . X Y b 1 a 1 � � 0 0 ∗ b 2 u ∗ u a 2 b 3 Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Natural Relation between Patterns and Bipartites Q is a given m × n pattern. G = ( X ∪ Y , E ) is the related bipartite defined by X = { a 1 , a 2 ,..., a m } , Y = { b 1 , b 2 ,..., b n } , E = { a i b j ∶ Q ij ≠ 0 } . B = { a i b j ∶ Q ij = u } . X Y b 1 a 1 � � 0 0 ∗ b 2 u ∗ u a 2 b 3 Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Main Theorem Theorem For a given m × n pattern matrix Q, If G = ( X ∪ Y , E ) is the graph and B is the set of banned edges defined above, then rank ( Q ) + Z Y ( G , B ) = m + n . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Main Theorem Theorem For a given m × n pattern matrix Q, If G = ( X ∪ Y , E ) is the graph and B is the set of banned edges defined above, then rank ( Q ) + Z Y ( G , B ) = m + n . Each initial white vertex represent a sign vector. X Y  0  u ∗ ∗ u ∗ a 1 b 1     0 ∗ u     a 2 b 2 u ∗ ∗  + c 2 u  ∼ 0 c 1       0 ∗ a 3 b 3 Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Main Theorem Theorem For a given m × n pattern matrix Q, If G = ( X ∪ Y , E ) is the graph and B is the set of banned edges defined above, then rank ( Q ) + Z Y ( G , B ) = m + n . Each initial white vertex represent a sign vector. The set of initial white vertices is independent iff it will be forced. X Y  0  u ∗ ∗ u ∗ a 1 b 1     0 ∗ u     a 2 b 2 u ∗ ∗  + c 2 u  ∼ 0 c 1       0 ∗ a 3 b 3 Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Main Theorem Theorem For a given m × n pattern matrix Q, If G = ( X ∪ Y , E ) is the graph and B is the set of banned edges defined above, then rank ( Q ) + Z Y ( G , B ) = m + n . Each initial white vertex represent a sign vector. The set of initial white vertices is independent iff it will be forced. X Y  0  u ∗ ∗ u ∗ a 1 b 1     0 ∗ u     a 2 b 2 u ∗ ∗  + 0 u  ∼ 0 c 1       0 ∗ a 3 b 3 Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Main Theorem Theorem For a given m × n pattern matrix Q, If G = ( X ∪ Y , E ) is the graph and B is the set of banned edges defined above, then rank ( Q ) + Z Y ( G , B ) = m + n . Each initial white vertex represent a sign vector. The set of initial white vertices is independent iff it will be forced. X Y  0  u ∗ ∗ u ∗ a 1 b 1     0 ∗ u         a 2 b 2 u ∗ 0 u ∗  + ∗ u  + ∗ ∗  = u  ∼ 0 ∗             0 ∗ u u a 3 b 3 Chin-Hung Lin Applications of z. f. number to the minimum rank problem

The Exhaustive Zero Forcing Number Recall that rank ( Q ) ≤ min Q ′ ∈ U { rank ( Q ′ )} ≤ mr ( Q ) . The middle term is called the exhaustive rank of Q . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

The Exhaustive Zero Forcing Number Recall that rank ( Q ) ≤ min Q ′ ∈ U { rank ( Q ′ )} ≤ mr ( Q ) . The middle term is called the exhaustive rank of Q . For a given graph G , there is a corresponding pattern Q whose diagonal entries are all u . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

The Exhaustive Zero Forcing Number Recall that rank ( Q ) ≤ min Q ′ ∈ U { rank ( Q ′ )} ≤ mr ( Q ) . The middle term is called the exhaustive rank of Q . For a given graph G , there is a corresponding pattern Q whose diagonal entries are all u . Let I ⊆ [ n ] and Q I be the pattern replace those u in ii -entry by ∗ if i ∈ I and 0 if i ∉ I . Then U = { Q I ∶ I ⊆ [ n ]} . Define ̃ G I to be the bipartite given by Q I . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

The Exhaustive Zero Forcing Number Recall that rank ( Q ) ≤ min Q ′ ∈ U { rank ( Q ′ )} ≤ mr ( Q ) . The middle term is called the exhaustive rank of Q . For a given graph G , there is a corresponding pattern Q whose diagonal entries are all u . Let I ⊆ [ n ] and Q I be the pattern replace those u in ii -entry by ∗ if i ∈ I and 0 if i ∉ I . Then U = { Q I ∶ I ⊆ [ n ]} . Define ̃ G I to be the bipartite given by Q I . The inequality become I ⊆[ n ] Z Y (̃ G I ) − n ≤ Z Y (̃ M ( G ) ≤ max G [ n ] , B ) − n . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

The Exhaustive Zero Forcing Number Recall that rank ( Q ) ≤ min Q ′ ∈ U { rank ( Q ′ )} ≤ mr ( Q ) . The middle term is called the exhaustive rank of Q . For a given graph G , there is a corresponding pattern Q whose diagonal entries are all u . Let I ⊆ [ n ] and Q I be the pattern replace those u in ii -entry by ∗ if i ∈ I and 0 if i ∉ I . Then U = { Q I ∶ I ⊆ [ n ]} . Define ̃ G I to be the bipartite given by Q I . The inequality become I ⊆[ n ] Z Y (̃ G I ) − n ≤ Z Y (̃ M ( G ) ≤ max G [ n ] , B ) − n . The second term is called the exhaustive zero forcing number of G . Denote it by ̃ Z ( G ) . The third term could be proven to equal Z ( G ) . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

The Exhaustive Zero Forcing Number Recall that rank ( Q ) ≤ min Q ′ ∈ U { rank ( Q ′ )} ≤ mr ( Q ) . The middle term is called the exhaustive rank of Q . For a given graph G , there is a corresponding pattern Q whose diagonal entries are all u . Let I ⊆ [ n ] and Q I be the pattern replace those u in ii -entry by ∗ if i ∈ I and 0 if i ∉ I . Then U = { Q I ∶ I ⊆ [ n ]} . Define ̃ G I to be the bipartite given by Q I . The inequality become I ⊆[ n ] Z Y (̃ G I ) − n ≤ Z Y (̃ M ( G ) ≤ max G [ n ] , B ) − n . The second term is called the exhaustive zero forcing number of G . Denote it by ̃ Z ( G ) . The third term could be proven to equal Z ( G ) . Hence M ( G ) ≤ ̃ Z ( G ) ≤ Z ( G ) . Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Example of Exhaustive Zero Forcing Number For G = P 3 , the pattern is ∗ ⎛ ⎞ u 0 ⎜ ∗ ∗ ⎟ u Q = ⎝ ⎠ . ∗ 0 u Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Example of Exhaustive Zero Forcing Number For G = P 3 , the pattern is ∗ ⎛ ⎞ u 0 ⎜ ∗ ∗ ⎟ u Q = ⎝ ⎠ . ∗ 0 u For I = { 1 , 3 } ⊆ [ 3 ] , the pattern is ∗ ∗ ⎛ ⎞ 0 ⎜ ∗ ∗ ⎟ Q = 0 ⎝ ⎠ . ∗ ∗ 0 Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Example of Exhaustive Zero Forcing Number For G = P 3 , the pattern is ∗ ⎛ ⎞ u 0 ⎜ ∗ ∗ ⎟ u Q = ⎝ ⎠ . ∗ 0 u For I = { 1 , 3 } ⊆ [ 3 ] , the pattern is ∗ ∗ ⎛ ⎞ 0 ⎜ ∗ ∗ ⎟ Q = 0 ⎝ ⎠ . ∗ ∗ 0 1 = M ( P 3 ) ≤ ̃ Z ( P 3 ) ≤ Z ( P 3 ) = 1. Hence ̃ Z ( G ) = 1. Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Bipartites related to P 3 4 3 4 3 3 4 3 4 Chin-Hung Lin Applications of z. f. number to the minimum rank problem

Row Rank and Column Rank Theorem If G is the bipartite given by a pattern Q, then Z Y ( G , B ) = Z X ( G , B ) . Row rank: maximum number of rows; Column rank: maximum number of columns. X Y X Y b 1 b 1 a 1 a 1 b 2 b 2 a 2 a 2 b 3 b 3 Chin-Hung Lin Applications of z. f. number to the minimum rank problem