On the zero forcing process Jephian C.-H. Lin Department of Applied - PowerPoint PPT Presentation

On the zero forcing process Jephian C.-H. Lin Department of Applied Mathematics, National Sun Yat-sen University Department of Mathematics and Statistics, University of Victoria May 9, 2018 Taiwan-Vietnam Workshop on Mathematics, Kaohsiung, Taiwan On the zero forcing process 1/17 NSYSU & UVic

Zero forcing Zero forcing process: ◮ Start with a given set of blue vertices. ◮ If for some x , the closed neighbourhood N G [ x ] are all blue except for one vertex y and y � = x , then y turns blue. An initial blue set that can make the whole graph blue is called a zero forcing set. The zero forcing number Z ( G ) of a graph G is the minimum size of a zero forcing set. On the zero forcing process 2/17 NSYSU & UVic

Zero forcing Zero forcing process: ◮ Start with a given set of blue vertices. ◮ If for some x , the closed neighbourhood N G [ x ] are all blue except for one vertex y and y � = x , then y turns blue. An initial blue set that can make the whole graph blue is called a zero forcing set. The zero forcing number Z ( G ) of a graph G is the minimum size of a zero forcing set. On the zero forcing process 2/17 NSYSU & UVic

Zero forcing Zero forcing process: ◮ Start with a given set of blue vertices. ◮ If for some x , the closed neighbourhood N G [ x ] are all blue except for one vertex y and y � = x , then y turns blue. An initial blue set that can make the whole graph blue is called a zero forcing set. The zero forcing number Z ( G ) of a graph G is the minimum size of a zero forcing set. On the zero forcing process 2/17 NSYSU & UVic

Zero forcing Zero forcing process: ◮ Start with a given set of blue vertices. ◮ If for some x , the closed neighbourhood N G [ x ] are all blue except for one vertex y and y � = x , then y turns blue. An initial blue set that can make the whole graph blue is called a zero forcing set. The zero forcing number Z ( G ) of a graph G is the minimum size of a zero forcing set. On the zero forcing process 2/17 NSYSU & UVic

Zero forcing Zero forcing process: ◮ Start with a given set of blue vertices. ◮ If for some x , the closed neighbourhood N G [ x ] are all blue except for one vertex y and y � = x , then y turns blue. An initial blue set that can make the whole graph blue is called a zero forcing set. The zero forcing number Z ( G ) of a graph G is the minimum size of a zero forcing set. On the zero forcing process 2/17 NSYSU & UVic

Zero forcing Zero forcing process: ◮ Start with a given set of blue vertices. ◮ If for some x , the closed neighbourhood N G [ x ] are all blue except for one vertex y and y � = x , then y turns blue. An initial blue set that can make the whole graph blue is called a zero forcing set. The zero forcing number Z ( G ) of a graph G is the minimum size of a zero forcing set. On the zero forcing process 2/17 NSYSU & UVic

Zero forcing Zero forcing process: ◮ Start with a given set of blue vertices. ◮ If for some x , the closed neighbourhood N G [ x ] are all blue except for one vertex y and y � = x , then y turns blue. An initial blue set that can make the whole graph blue is called a zero forcing set. The zero forcing number Z ( G ) of a graph G is the minimum size of a zero forcing set. On the zero forcing process 2/17 NSYSU & UVic

Zero forcing Zero forcing process: ◮ Start with a given set of blue vertices. ◮ If for some x , the closed neighbourhood N G [ x ] are all blue except for one vertex y and y � = x , then y turns blue. An initial blue set that can make the whole graph blue is called a zero forcing set. The zero forcing number Z ( G ) of a graph G is the minimum size of a zero forcing set. On the zero forcing process 2/17 NSYSU & UVic

Z ( G ) = 1 Z ( G ) = 1 if and only if G is a path. On the zero forcing process 3/17 NSYSU & UVic

Z ( G ) = 1 Z ( G ) = 1 if and only if G is a path. On the zero forcing process 3/17 NSYSU & UVic

Z ( G ) = 1 Z ( G ) = 1 if and only if G is a path. On the zero forcing process 3/17 NSYSU & UVic

Z ( G ) = 1 Z ( G ) = 1 if and only if G is a path. On the zero forcing process 3/17 NSYSU & UVic

Z ( G ) = 1 Z ( G ) = 1 if and only if G is a path. On the zero forcing process 3/17 NSYSU & UVic

Z ( G ) = 1 Z ( G ) = 1 if and only if G is a path. On the zero forcing process 3/17 NSYSU & UVic

Z ( G ) = n or n − 1 Z ( G ) = n = ⇒ P 2 -free Z ( G ) = n − 1 = ⇒ P 3 -free Let G be a graph on n vertices. ◮ Then Z ( G ) = n if and only if G is the union of isolated vertices. ◮ And Z ( G ) = n − 1 if and only if G is K r ˙ ∪ K n − r , r � = 1. On the zero forcing process 4/17 NSYSU & UVic

Generalised adjacency matrix Let G be a simple graph on n vertices. The family S ( G ) consists � � of all n × n real symmetric matrix M = M i , j with  M i , j = 0 if i � = j and { i , j } is not an edge ,   M i , j � = 0 if i � = j and { i , j } is an edge ,  M i , j ∈ R if i = j .   0 1 0   1 − 1 0   2 0 . 1 0   ,  ,  , · · · S ( ) ∋ 1 0 1 − 1 2 − 1 0 . 1 1 π    0 1 0 0 − 1 1 0 0 π On the zero forcing process 5/17 NSYSU & UVic

Why zero forcing?  − 2 0 0 7 0   x 1   0  0 1 0 0 − 9 x 2 0 1 4             0 0 0 3 4 = 0 x 3 3             7 0 3 − 4 5 x 4 0 2 5       0 − 9 4 5 0 0 x 5 ◮ Pick a matrix A ∈ S ( G ) and consider A x = 0 . ◮ Each vertex represents a variable. Each vertex also represents an equation where appearing variables are the neighbours and possibly itself. ◮ Blue means zero. White means unknown. On the zero forcing process 6/17 NSYSU & UVic

Hidden triangle in a system 1. − 2 x 1 +7 x 4 = 0 2. 1 x 2 − 9 x 5 = 0 1 4 3. 3 x 4 +4 x 5 = 0 3 4. 7 x 1 +3 x 3 − 4 x 4 +5 x 5 = 0 2 5 5. − 9 x 2 +4 x 3 +5 x 4 = 0 Given x 1 = x 2 = 0, Given 1 and 2 blue, 1. = ⇒ x 4 = 0 , 1 → 4 , 2. = ⇒ x 5 = 0 , 2 → 5 , 4. = ⇒ x 3 = 0 . 4 → 3 . On the zero forcing process 7/17 NSYSU & UVic

Hidden triangle in a system 1. − 2 x 1 +7 x 4 = 0 2. 1 x 2 − 9 x 5 = 0 1 4 3. 3 x 4 +4 x 5 = 0 3 4. 7 x 1 +3 x 3 − 4 x 4 +5 x 5 = 0 2 5 5. − 9 x 2 +4 x 3 +5 x 4 = 0 Given x 1 = x 2 = 0, Given 1 and 2 blue, 1. = ⇒ x 4 = 0 , 1 → 4 , 2. = ⇒ x 5 = 0 , 2 → 5 , 4. = ⇒ x 3 = 0 . 4 → 3 . On the zero forcing process 7/17 NSYSU & UVic

Hidden triangle in a system 1. − 2 x 1 +7 x 4 = 0 2. 1 x 2 − 9 x 5 = 0 1 4 3. 3 x 4 +4 x 5 = 0 3 4. 7 x 1 +3 x 3 − 4 x 4 +5 x 5 = 0 2 5 5. − 9 x 2 +4 x 3 +5 x 4 = 0 Given x 1 = x 2 = 0, Given 1 and 2 blue, 1. = ⇒ x 4 = 0 , 1 → 4 , 2. = ⇒ x 5 = 0 , 2 → 5 , 4. = ⇒ x 3 = 0 . 4 → 3 . On the zero forcing process 7/17 NSYSU & UVic

Hidden triangle in a system 1. 7 x 4 − 2 x 1 = 0 2. − 9 x 5 +1 x 2 = 0 1 4 4. − 4 x 4 +5 x 5 +3 x 3 +7 x 1 = 0 3 3. 3 x 4 +4 x 5 = 0 2 5 5. 5 x 4 +4 x 3 − 9 x 2 = 0 Given x 1 = x 2 = 0, Given 1 and 2 blue, 1. = ⇒ x 4 = 0 , 1 → 4 , 2. = ⇒ x 5 = 0 , 2 → 5 , 4. = ⇒ x 3 = 0 . 4 → 3 . On the zero forcing process 7/17 NSYSU & UVic

Hidden triangle in a system 1. 7 x 4 0 0 − 2 x 1 = 0 2. − 9 x 5 0 +1 x 2 = 0 1 4 4. − 4 x 4 +5 x 5 +3 x 3 +7 x 1 = 0 3 3. 3 x 4 +4 x 5 = 0 2 5 5. 5 x 4 +4 x 3 − 9 x 2 = 0 Given x 1 = x 2 = 0, Given 1 and 2 blue, 1. = ⇒ x 4 = 0 , 1 → 4 , 2. = ⇒ x 5 = 0 , 2 → 5 , 4. = ⇒ x 3 = 0 . 4 → 3 . As long as the red terms has nonzero coefficients and the orange terms are zero, the same argument always works. On the zero forcing process 7/17 NSYSU & UVic

Triangle number ◮ A pattern is a matrix whose entries are in { 0 , ∗ , ? } . ◮ A triangle is a submatrix of a pattern that can be permuted to a lower triangular matrix with ∗ on the diagonal.  ? 0 0 ∗ 0      0 ? 0 0 ∗ 0 ∗ 0 ∗ 0 0    →   0 0 ? ∗ ∗ 0 0 ∗ 0 ∗ 0 triangle        ∗ 0 ∗ ? ∗ ∗ ? ∗ ? ∗ ∗   0 ∗ ∗ ∗ ? On the zero forcing process 8/17 NSYSU & UVic

Triangle number ◮ A pattern is a matrix whose entries are in { 0 , ∗ , ? } . ◮ A triangle is a submatrix of a pattern that can be permuted to a lower triangular matrix with ∗ on the diagonal.  ? 0 0 ∗ 0    0 ? 0 0 ∗ ? 0 0     0 0 ? ∗ ∗ 0 ? 0 not a triangle       ∗ 0 ∗ ? ∗ 0 0 ?   0 ∗ ∗ ∗ ? On the zero forcing process 8/17 NSYSU & UVic

Triangle number ◮ A pattern is a matrix whose entries are in { 0 , ∗ , ? } . ◮ A triangle is a submatrix of a pattern that can be permuted to a lower triangular matrix with ∗ on the diagonal.  ? 0 0 ∗ 0      0 ? 0 0 ∗ 0 0 ∗ ∗ 0 0    →   0 0 ? ∗ ∗ ∗ ? ∗ ? ∗ 0 triangle        ∗ 0 ∗ ? ∗ 0 ∗ ? ∗ ? ∗   0 ∗ ∗ ∗ ? On the zero forcing process 8/17 NSYSU & UVic