Zero forcing number, Grundy domination number and variants Jephian - - PowerPoint PPT Presentation

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

• x∈X

NG[x] = V (G). The total domination number γt(G) is the minimum cardinality of a set X such that

• x∈X

NG(x) = V (G). γ(P3) = 1 γt(P3) = 2

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 NG[v] \

• x∈X

NG[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 NG[v] \

• x∈X

NG[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 NG[v] \

• x∈X

NG[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 NG[v] \

• x∈X

NG[x] = ∅.

Zero forcing vs Grundy domination 3/15 NSYSU

Grundy domination number

The Grundy domination number γgr(G) is the length of the longest sequence (v1, v2, . . . , vk) such that NG[vi] \

i−1

• j=1

NG[vj] = ∅.

Zero forcing vs Grundy domination 4/15 NSYSU

Grundy domination number

The Grundy domination number γgr(G) is the length of the longest sequence (v1, v2, . . . , vk) such that NG[vi] \

i−1

• j=1

NG[vj] = ∅.

Zero forcing vs Grundy domination 4/15 NSYSU

Grundy domination number

The Grundy domination number γgr(G) is the length of the longest sequence (v1, v2, . . . , vk) such that NG[vi] \

i−1

• j=1

NG[vj] = ∅.

Zero forcing vs Grundy domination 4/15 NSYSU

Grundy domination number

The Grundy domination number γgr(G) is the length of the longest sequence (v1, v2, . . . , vk) such that NG[vi] \

i−1

• j=1

NG[vj] = ∅.

Zero forcing vs Grundy domination 4/15 NSYSU

Grundy domination number

The Grundy domination number γgr(G) is the length of the longest sequence (v1, v2, . . . , vk) such that NG[vi] \

i−1

• j=1

NG[vj] = ∅.

Zero forcing vs Grundy domination 4/15 NSYSU

Grundy domination number

The Grundy domination number γgr(G) is the length of the longest sequence (v1, v2, . . . , vk) such that NG[vi] \

i−1

• j=1

NG[vj] = ∅.

Zero forcing vs Grundy domination 4/15 NSYSU

Grundy domination number

The Grundy domination number γgr(G) is the length of the longest sequence (v1, v2, . . . , vk) such that NG[vi] \

i−1

• j=1

NG[vj] = ∅. So γgr(G) = 5.

Zero forcing vs Grundy domination 4/15 NSYSU

Grundy total domination number

The Grundy total domination number γt

gr(G) is the length of the

longest sequence (v1, v2, . . . , vk) such that NG(vi) \

i−1

• j=1

NG(vj) = ∅.

Zero forcing vs Grundy domination 5/15 NSYSU

Grundy total domination number

The Grundy total domination number γt

gr(G) is the length of the

longest sequence (v1, v2, . . . , vk) such that NG(vi) \

i−1

• j=1

NG(vj) = ∅.

Zero forcing vs Grundy domination 5/15 NSYSU

Grundy total domination number

The Grundy total domination number γt

gr(G) is the length of the

longest sequence (v1, v2, . . . , vk) such that NG(vi) \

i−1

• j=1

NG(vj) = ∅.

Zero forcing vs Grundy domination 5/15 NSYSU

Grundy total domination number

The Grundy total domination number γt

gr(G) is the length of the

longest sequence (v1, v2, . . . , vk) such that NG(vi) \

i−1

• j=1

NG(vj) = ∅.

Zero forcing vs Grundy domination 5/15 NSYSU

Grundy total domination number

The Grundy total domination number γt

gr(G) is the length of the

longest sequence (v1, v2, . . . , vk) such that NG(vi) \

i−1

• j=1

NG(vj) = ∅.

Zero forcing vs Grundy domination 5/15 NSYSU

Grundy total domination number

The Grundy total domination number γt

gr(G) is the length of the

longest sequence (v1, v2, . . . , vk) such that NG(vi) \

i−1

• j=1

NG(vj) = ∅. So γt

gr(G) = 4.

Zero forcing vs Grundy domination 5/15 NSYSU

Other dominating sequences (Breˇ sar et al. 2017)

The Z-Grundy domination number γZ

gr(G) is the length of the

longest sequence (v1, v2, . . . , vk) such that NG(vi) \

i−1

• j=1

NG[vj] = ∅. The L-Grundy domination number γL

gr(G) is the length of the

longest sequence (v1, v2, . . . , vk) such that NG[vi] \

i−1

• j=1

NG(vj) = ∅. [Note: v1, . . . , vk 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 − γZ

gr(G),

where Z(G) is the zero forcing number.

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Main theorem

Theorem (L 2019)

Let G be a graph and |V (G)| = n. Then

• 1. Z(G) = n − γZ

gr(G),

• 2. Z ˙

ℓ(G) = n − γgr(G),

• 3. Z−(G) = n − γt

gr(G),

• 4. ZL(G) = n − γL

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 i, j-entry =      = 0 if ij ∈ E(G), i = j; = 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 γgr(G) ≤ rank(A) for any A ∈ S(G) with diagonal entries all nonzero; and γt

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 A = C − I I5 I5 C ′ − I

• and B =

C I5 I5 −C ′

• ,

where C and C ′ are the adjacency matrix of C5 and C5,

• respectively. Then γgr(P) ≤ rank(A) = 5 and the sequence

(1, 2, 3, 4, 5) is optimal. 1 6 2 7 3 8 4 9 5 10

Zero forcing vs Grundy domination 11/15 NSYSU

Let P be the Petersen graph. Consider A = C − I I5 I5 C ′ − I

• and B =

C I5 I5 −C ′

• ,

where C and C ′ are the adjacency matrix of C5 and C5, respectively. Then γt

gr(G) ≤ rank(B) = 6 and the sequence

(9, 1, 2, 3, 4, 5) is optimal. 1 6 2 7 3 8 4 9 5 10

Zero forcing vs Grundy domination 11/15 NSYSU

Proof of the rank bound

◮ Goal: Show γgr(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 (v1, . . . , vk) and a matrix A. Let Ni be the vertices dominated by vi but not any vertex before vi.        

N1 N2 ··· Nk v1

∗ · · · ∗ · · ·

v2

? ∗ · · · ∗ . . . . . . ? ? ...

vk

? · · · ? ∗ · · · ∗

• ther vertices

? ? ? ?        

Zero forcing vs Grundy domination 12/15 NSYSU

Proof of the rank bound

◮ Goal: Show γgr(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 (v1, . . . , vk) and a matrix A. Let Ni be the vertices dominated by vi but not any vertex before vi.        

N1 N2 ··· Nk v1

∗ · · · ∗ · · ·

v2

? ∗ · · · ∗ . . . . . . ? ? ...

vk

? · · · ? ∗ · · · ∗

• ther vertices

? ? ? ?        

Zero forcing vs Grundy domination 12/15 NSYSU

M < Z and γgr < mr

Z Z− Z ˙

ℓ

ZL M M0 M ˙

ℓ

ML mrL mr0 mr ˙

ℓ

mr γL

gr

γt

gr

γgr γZ

gr

Thanks!

Zero forcing vs Grundy domination 13/15 NSYSU

M < Z and γgr < mr

Z Z− Z ˙

ℓ

ZL M M0 M ˙

ℓ

ML mrL mr0 mr ˙

ℓ

mr γL

gr

γt

gr

γgr γZ

gr

Thanks!

Zero forcing vs Grundy domination 13/15 NSYSU

References I

• B. Breˇ

sar, Cs. Bujt´ as, T. Gologranc, S. Klavˇ zar, G. Koˇ smrlj,

• B. Patk´
• s, Zs. Tuza, and M. Vizer.

Grundy dominating sequences and zero forcing sets. Discrete Optim., 26:66–77, 2017.

• B. Breˇ

sar, T. Gologranc, M. Milaniˇ c, D. F. Rall, and R. Rizzi. Dominating sequences in graphs. Discrete Math., 336:22–36, 2014.

• B. Breˇ

sar, M. A. Henning, and D. F. Rall. Total dominating sequences in graphs. Discrete Math., 339:1665–1676, 2016.

Zero forcing vs Grundy domination 14/15 NSYSU

References II

• J. C.-H. Lin.

Zero forcing number, Grundy domination number, and their variants. Linear Algebra Appl., 563:240–254, 2019.

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