Zero forcing number, Grundy domination number and variants Jephian C.-H. Lin Department of Applied Mathematics, National Sun Yat-sen University July 12, 2019 2019 Meeting of the International Linear Algebra Society, Rio de Janeiro, Brazil Zero forcing vs Grundy domination 1/15 NSYSU
Domination number Let G be a graph. The domination number γ ( G ) is the minimum cardinality of a set X such that � N G [ x ] = V ( G ) . x ∈ X The total domination number γ t ( G ) is the minimum cardinality of a set X such that � N G ( x ) = V ( G ) . x ∈ X γ t ( P 3 ) = 2 γ ( P 3 ) = 1 Zero forcing vs Grundy domination 2/15 NSYSU
Greedy algorithm ◮ Greedy algorithm makes the locally optimal choice at each stage with the hope of finding a global optimum. ◮ Maze: You may keep going straight at fork. But it might lead you to a dead end. ◮ Graph coloring: You may keep using the smallest free number to color the next vertex, showing χ ( G ) ≤ ∆( G ) + 1. ◮ Greedy algorithm for domination number: When X are chosen and not yet dominate the whole graph, pick a vertex v such that � N G [ v ] \ N G [ x ] � = ∅ . x ∈ X Zero forcing vs Grundy domination 3/15 NSYSU
Greedy algorithm ◮ Greedy algorithm makes the locally optimal choice at each stage with the hope of finding a global optimum. ◮ Maze: You may keep going straight at fork. But it might lead you to a dead end. ◮ Graph coloring: You may keep using the smallest free number to color the next vertex, showing χ ( G ) ≤ ∆( G ) + 1. ◮ Greedy algorithm for domination number: When X are chosen and not yet dominate the whole graph, pick a vertex v such that � N G [ v ] \ N G [ x ] � = ∅ . x ∈ X Zero forcing vs Grundy domination 3/15 NSYSU
Greedy algorithm ◮ Greedy algorithm makes the locally optimal choice at each stage with the hope of finding a global optimum. ◮ Maze: You may keep going straight at fork. But it might lead you to a dead end. ◮ Graph coloring: You may keep using the smallest free number to color the next vertex, showing χ ( G ) ≤ ∆( G ) + 1. ◮ Greedy algorithm for domination number: When X are chosen and not yet dominate the whole graph, pick a vertex v such that � N G [ v ] \ N G [ x ] � = ∅ . x ∈ X Zero forcing vs Grundy domination 3/15 NSYSU
Greedy algorithm ◮ Greedy algorithm makes the locally optimal choice at each stage with the hope of finding a global optimum. ◮ Maze: You may keep going straight at fork. But it might lead you to a dead end. ◮ Graph coloring: You may keep using the smallest free number to color the next vertex, showing χ ( G ) ≤ ∆( G ) + 1. ◮ Greedy algorithm for domination number: When X are chosen and not yet dominate the whole graph, pick a vertex v such that � N G [ v ] \ N G [ x ] � = ∅ . x ∈ X Zero forcing vs Grundy domination 3/15 NSYSU
Grundy domination number The Grundy domination number γ gr ( G ) is the length of the longest sequence ( v 1 , v 2 , . . . , v k ) such that i − 1 � N G [ v i ] \ N G [ v j ] � = ∅ . j =1 Zero forcing vs Grundy domination 4/15 NSYSU
Grundy domination number The Grundy domination number γ gr ( G ) is the length of the longest sequence ( v 1 , v 2 , . . . , v k ) such that i − 1 � N G [ v i ] \ N G [ v j ] � = ∅ . j =1 Zero forcing vs Grundy domination 4/15 NSYSU
Grundy domination number The Grundy domination number γ gr ( G ) is the length of the longest sequence ( v 1 , v 2 , . . . , v k ) such that i − 1 � N G [ v i ] \ N G [ v j ] � = ∅ . j =1 Zero forcing vs Grundy domination 4/15 NSYSU
Grundy domination number The Grundy domination number γ gr ( G ) is the length of the longest sequence ( v 1 , v 2 , . . . , v k ) such that i − 1 � N G [ v i ] \ N G [ v j ] � = ∅ . j =1 Zero forcing vs Grundy domination 4/15 NSYSU
Grundy domination number The Grundy domination number γ gr ( G ) is the length of the longest sequence ( v 1 , v 2 , . . . , v k ) such that i − 1 � N G [ v i ] \ N G [ v j ] � = ∅ . j =1 Zero forcing vs Grundy domination 4/15 NSYSU
Grundy domination number The Grundy domination number γ gr ( G ) is the length of the longest sequence ( v 1 , v 2 , . . . , v k ) such that i − 1 � N G [ v i ] \ N G [ v j ] � = ∅ . j =1 Zero forcing vs Grundy domination 4/15 NSYSU
Grundy domination number The Grundy domination number γ gr ( G ) is the length of the longest sequence ( v 1 , v 2 , . . . , v k ) such that i − 1 � N G [ v i ] \ N G [ v j ] � = ∅ . j =1 So γ gr ( G ) = 5. Zero forcing vs Grundy domination 4/15 NSYSU
Grundy total domination number The Grundy total domination number γ t gr ( G ) is the length of the longest sequence ( v 1 , v 2 , . . . , v k ) such that i − 1 � N G ( v i ) \ N G ( v j ) � = ∅ . j =1 Zero forcing vs Grundy domination 5/15 NSYSU
Grundy total domination number The Grundy total domination number γ t gr ( G ) is the length of the longest sequence ( v 1 , v 2 , . . . , v k ) such that i − 1 � N G ( v i ) \ N G ( v j ) � = ∅ . j =1 Zero forcing vs Grundy domination 5/15 NSYSU
Grundy total domination number The Grundy total domination number γ t gr ( G ) is the length of the longest sequence ( v 1 , v 2 , . . . , v k ) such that i − 1 � N G ( v i ) \ N G ( v j ) � = ∅ . j =1 Zero forcing vs Grundy domination 5/15 NSYSU
Grundy total domination number The Grundy total domination number γ t gr ( G ) is the length of the longest sequence ( v 1 , v 2 , . . . , v k ) such that i − 1 � N G ( v i ) \ N G ( v j ) � = ∅ . j =1 Zero forcing vs Grundy domination 5/15 NSYSU
Grundy total domination number The Grundy total domination number γ t gr ( G ) is the length of the longest sequence ( v 1 , v 2 , . . . , v k ) such that i − 1 � N G ( v i ) \ N G ( v j ) � = ∅ . j =1 Zero forcing vs Grundy domination 5/15 NSYSU
Grundy total domination number The Grundy total domination number γ t gr ( G ) is the length of the longest sequence ( v 1 , v 2 , . . . , v k ) such that i − 1 � N G ( v i ) \ N G ( v j ) � = ∅ . j =1 So γ t gr ( G ) = 4. Zero forcing vs Grundy domination 5/15 NSYSU
Other dominating sequences (Breˇ sar et al. 2017) The Z -Grundy domination number γ Z gr ( G ) is the length of the longest sequence ( v 1 , v 2 , . . . , v k ) such that i − 1 � N G ( v i ) \ N G [ v j ] � = ∅ . j =1 The L -Grundy domination number γ L gr ( G ) is the length of the longest sequence ( v 1 , v 2 , . . . , v k ) such that i − 1 � N G [ v i ] \ N G ( v j ) � = ∅ . j =1 [Note: v 1 , . . . , v k have to be distinct vertices.] Zero forcing vs Grundy domination 6/15 NSYSU
Theorem (Breˇ sar et al. 2017) For any graph G, Z ( G ) = n − γ Z gr ( G ) , where Z ( G ) is the zero forcing number. Zero forcing vs Grundy domination 7/15 NSYSU
Main theorem Theorem (L 2019) Let G be a graph and | V ( G ) | = n. Then 1. Z ( G ) = n − γ Z 3. Z − ( G ) = n − γ t gr ( G ) , gr ( G ) , 4. Z L ( G ) = n − γ L 2. Z ˙ ℓ ( G ) = n − γ gr ( G ) , gr ( G ) . Zero forcing vs Grundy domination 8/15 NSYSU
Maximum nullity and minimum rank For a graph G , define S ( G ) as the collection of all real symmetric matrices whose � = 0 if ij ∈ E ( G ) , i � = j ; i , j -entry = = 0 if ij / ∈ E ( G ) , i � = j ; ∈ R if i = j ; ◮ minimum rank mr( G ) = smallest possible rank among matrices in S ( G ) ◮ maximum nullity M ( G ) = largest possible nullity among matrices in S ( G ) ◮ M ( G ) = n − mr( G ) for any graph G on n vertices. Zero forcing vs Grundy domination 9/15 NSYSU
Rank bound Theorem (L 2017) Let G be a graph. Then γ gr ( G ) ≤ rank( A ) for any A ∈ S ( G ) with diagonal entries all nonzero; and γ t gr ( G ) ≤ rank( A ) for any A ∈ S ( G ) with zero diagonal. Zero forcing vs Grundy domination 10/15 NSYSU
Let P be the Petersen graph. Consider � C − I � � C � I 5 I 5 A = and B = , C ′ − I − C ′ I 5 I 5 where C and C ′ are the adjacency matrix of C 5 and C 5 , respectively. Then γ gr ( P ) ≤ rank( A ) = 5 and the sequence (1 , 2 , 3 , 4 , 5) is optimal. 5 1 4 10 9 6 8 7 2 3 Zero forcing vs Grundy domination 11/15 NSYSU
Let P be the Petersen graph. Consider � C − I � � C � I 5 I 5 A = and B = , C ′ − I − C ′ I 5 I 5 where C and C ′ are the adjacency matrix of C 5 and C 5 , Then γ t respectively. gr ( G ) ≤ rank( B ) = 6 and the sequence (9 , 1 , 2 , 3 , 4 , 5) is optimal. 5 1 4 10 9 6 8 7 2 3 Zero forcing vs Grundy domination 11/15 NSYSU
Proof of the rank bound ◮ Goal: Show γ gr ( G ) ≤ rank( A ) for all A ∈ S ( G ) with nonzero diagonal entries. ◮ Key: Permutation does not change the rank, and the dominating sequence gives an echelon form. Pick an optimal sequence ( v 1 , . . . , v k ) and a matrix A . Let N i be the vertices dominated by v i but not any vertex before v i . N 1 N 2 N k ··· ∗ · · · ∗ 0 · · · 0 v 1 . . ? ∗ · · · ∗ 0 . v 2 . ... . . ? ? 0 ? · · · ? ∗ · · · ∗ v k ? ? ? ? other vertices Zero forcing vs Grundy domination 12/15 NSYSU
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