Forbidden subgraphs for constant domination number Michitaka Furuya - - PowerPoint PPT Presentation

Michitaka Furuya (Kitasato University)

Forbidden subgraphs for constant domination number

Let 𝐻 be a graph. A set 𝑇 ⊆ 𝑊 𝐻 is a dominating set of 𝐻 if for ∀ 𝑦 ∈ 𝑊 𝐻 − 𝑇, ∃ 𝑧 ∈ 𝑇 s.t. 𝑦𝑧 ∈ 𝐹 𝐻 . The minimum cardinality of a dominating set of 𝐻 is called the domination number of 𝐻, and is denoted by 𝛿 𝐻 .

Domination

Let 𝐻 be a graph. A set 𝑇 ⊆ 𝑊 𝐻 is a dominating set of 𝐻 if for ∀ 𝑦 ∈ 𝑊 𝐻 − 𝑇, ∃ 𝑧 ∈ 𝑇 s.t. 𝑦𝑧 ∈ 𝐹 𝐻 . The minimum cardinality of a dominating set of 𝐻 is called the domination number of 𝐻, and is denoted by 𝛿 𝐻 .

Domination

Domination

Let 𝐻 be a graph. A set 𝑇 ⊆ 𝑊 𝐻 is a dominating set of 𝐻 if for ∀ 𝑦 ∈ 𝑊 𝐻 − 𝑇, ∃ 𝑧 ∈ 𝑇 s.t. 𝑦𝑧 ∈ 𝐹 𝐻 . The minimum cardinality of a dominating set of 𝐻 is called the domination number of 𝐻, and is denoted by 𝛿 𝐻 . 𝛿 𝐻 = 5

Domination

Theorem 1 (Ore, 1962) Let 𝐻 be a conn. graph of order 𝑜 ≥ 2. Then 𝛿 𝐻 ≤ 𝑜/2. Theorem 2 (Fink et al., 1985; Payan and Xuong, 1982) A conn. graph 𝐻 of order 𝑜 satisfies 𝛿 𝐻 = 𝑜/2 if and only if 𝐻 = 𝐷4 or 𝐻 is the corona of a conn. graph.

… …

𝐼 corona of 𝐼

Domination

Theorem 1 (Ore, 1962) Let 𝐻 be a conn. graph of order 𝑜 ≥ 2. Then 𝛿 𝐻 ≤ 𝑜/2. Theorem 2 (Fink et al., 1985; Payan and Xuong, 1982) A conn. graph 𝐻 of order 𝑜 satisfies 𝛿 𝐻 = 𝑜/2 if and only if 𝐻 = 𝐷4 or 𝐻 is the corona of a conn. graph.

… …

𝐼 corona of 𝐼

Forbidden subgraph

Let ℋ be a set of conn. graphs. A graph 𝐻 is ℋ-free if 𝐻 has no graph in ℋ as an induced subgraph. (If 𝐻 is 𝐼 -free, then 𝐻 is simply said to be 𝐼-free.) In this context, graphs in ℋ are called forbidden subgraphs. 𝐿1,3-free graph (𝐿1,3 : )

Forbidden subgraph

Let ℋ be a set of conn. graphs. A graph 𝐻 is ℋ-free if 𝐻 has no graph in ℋ as an induced subgraph. (If 𝐻 is 𝐼 -free, then 𝐻 is simply said to be 𝐼-free.) In this context, graphs in ℋ are called forbidden subgraphs. Let ℋ1 and ℋ2 be sets of conn. graphs. We write ℋ1 ≤ ℋ2 if for ∀𝐼2 ∈ ℋ2, ∃𝐼1 ∈ ℋ1 s.t. 𝐼1 is an induced subgraph of 𝐼2. Remark If ℋ1 ≤ ℋ2, then every ℋ1-free graph is ℋ2-free.

Domination and forbidden subgraph

Theorem 3 (Cockayne et al., 1985) Let 𝐻 be a conn. {𝐿1,3, }-free graph of order 𝑜. Then 𝛿 𝐻 ≤ ⌈𝑜/3⌉. Let 𝛿pr 𝐻 be the minimum cardinality of a dominating set 𝑇 of 𝐻 s.t. 𝐻 𝑇 has a perfect matching. Theorem 4 (Dorbec et al., 2006) Let 𝐻 be a conn. 𝐿1,𝑛-free graph of order 𝑜 ≥ 2. Then 𝛿pr 𝐻 ≤ 2 𝑛𝑜 + 1 / 2𝑛 + 1 . Theorem 5 (Dorbec and Gravier, 2008) Let 𝐻 be a conn. 𝑄5-free graph of order 𝑜 ≥ 2. If 𝐻 ≠ 𝐷5, then 𝛿pr 𝐻 ≤ 𝑜/2 + 1.

Domination and forbidden subgraph

Theorem 3 (Cockayne et al., 1985) Let 𝐻 be a conn. {𝐿1,3, }-free graph of order 𝑜. Then 𝛿 𝐻 ≤ ⌈𝑜/3⌉. Let 𝛿pr 𝐻 be the minimum cardinality of a dominating set 𝑇 of 𝐻 s.t. 𝐻 𝑇 has a perfect matching. Theorem 4 (Dorbec et al., 2006) Let 𝐻 be a conn. 𝐿1,𝑛-free graph of order 𝑜 ≥ 2. Then 𝛿pr 𝐻 ≤ 2 𝑛𝑜 + 1 / 2𝑛 + 1 . Theorem 5 (Dorbec and Gravier, 2008) Let 𝐻 be a conn. 𝑄5-free graph of order 𝑜 ≥ 2. If 𝐻 ≠ 𝐷5, then 𝛿pr 𝐻 ≤ 𝑜/2 + 1.

Domination and forbidden subgraph

We focus on the most effective case, that is, sets ℋ of connected graphs satisfying the following: ∃ const. 𝑑 = 𝑑 ℋ s.t. for ∀ conn. ℋ-free graph 𝐻, 𝛿 𝐻 ≤ 𝑑. What graphs do belong to ℋ?? Let 𝐿𝑑+1

∗

be the corona of 𝐿𝑑+1. Since 𝛿 𝐿𝑑+1

∗

= 𝑑 + 1, 𝐿𝑑+1

∗

is not ℋ-free. ➡ ∃𝐼 ∈ ℋ s.t. 𝐼 is an induced subgraph of 𝐿𝑑+1

∗

. ➡ ℋ ≤ 𝐿𝑑+1

∗

. By similar argument, ℋ ≤ , 𝑄3𝑑+1 .

…

𝐿𝑑+1

…

𝑑 + 1 𝐿𝑑+1

∗

：

Domination and forbidden subgraph

We focus on the most effective case, that is, sets ℋ of connected graphs satisfying the following: ∃ const. 𝑑 = 𝑑 ℋ s.t. for ∀ conn. ℋ-free graph 𝐻, 𝛿 𝐻 ≤ 𝑑. What graphs do belong to ℋ?? Let 𝐿𝑑+1

∗

be the corona of 𝐿𝑑+1. Since 𝛿 𝐿𝑑+1

∗

= 𝑑 + 1, 𝐿𝑑+1

∗

is not ℋ-free. ➡ ∃𝐼 ∈ ℋ s.t. 𝐼 is an induced subgraph of 𝐿𝑑+1

∗

. ➡ ℋ ≤ 𝐿𝑑+1

∗

. By similar argument, ℋ ≤ , 𝑄3𝑑+1 .

…

𝐿𝑑+1

…

𝑑 + 1 𝐿𝑑+1

∗

：

Domination and forbidden subgraph

We focus on the most effective case, that is, sets ℋ of connected graphs satisfying the following: ∃ const. 𝑑 = 𝑑 ℋ s.t. for ∀ conn. ℋ-free graph 𝐻, 𝛿 𝐻 ≤ 𝑑. What graphs do belong to ℋ?? Let 𝐿𝑑+1

∗

be the corona of 𝐿𝑑+1. Since 𝛿 𝐿𝑑+1

∗

= 𝑑 + 1, 𝐿𝑑+1

∗

is not ℋ-free. ➡ ∃𝐼 ∈ ℋ s.t. 𝐼 is an induced subgraph of 𝐿𝑑+1

∗

. ➡ ℋ ≤ 𝐿𝑑+1

∗

. By similar argument, ℋ ≤ , 𝑄3𝑑+1 .

…

𝐿𝑑+1

…

𝑑 + 1 𝐿𝑑+1

∗

：

Main result

Theorem Let ℋ be a set of conn. graphs. Then ∃ const. 𝑑 = 𝑑 ℋ s.t. for ∀ conn. ℋ-free graph 𝐻, 𝛿 𝐻 ≤ 𝑑 if and only if ℋ ≤ 𝐿𝑙

∗, 𝑇𝑚 ∗, 𝑄 𝑛

for some positive integers 𝑙, 𝑚 and 𝑛 where 𝐿𝑙

∗ =

and 𝑇𝑚

∗ =

.

…

𝐿𝑙

…

𝑚

Outline of proof of main result

We show that 𝛿 𝐻 ≤ 1 +

2≤𝑗≤𝑛−2

𝑔

𝑙,𝑚 𝑗 𝑆 𝑙, 𝑚

for ∀conn. 𝐿𝑙

∗, 𝑇𝑚 ∗, 𝑄 𝑛 -free graph 𝐻, where

𝑔

𝑙,𝑚 𝑗 : = 1

𝑗 = 1 𝑆 𝑙, 𝑚 − 1 𝑔

𝑙,𝑚 𝑗 − 1 + 1 − 1

𝑗 ≥ 2 . Let 𝑦 ∈ 𝑊 𝐻 , and let 𝑌𝑗 = 𝑧 ∈ 𝑊 𝐻 ∶ dist 𝑦, 𝑧 = 𝑗 . Then 𝑊 𝐻 = 0≤𝑗≤𝑛−2 𝑌𝑗. Key Lemma For 𝑗 ≥ 2, the set 𝑌𝑗 is dominated by at most 𝑔

𝑙,𝑚 𝑗 𝑆 𝑙, 𝑚 vertices.

Outline of proof of main result

We show that 𝛿 𝐻 ≤ 1 +

2≤𝑗≤𝑛−2

𝑔

𝑙,𝑚 𝑗 𝑆 𝑙, 𝑚

for ∀conn. 𝐿𝑙

∗, 𝑇𝑚 ∗, 𝑄 𝑛 -free graph 𝐻, where

𝑔

𝑙,𝑚 𝑗 : = 1

𝑗 = 1 𝑆 𝑙, 𝑚 − 1 𝑔

𝑙,𝑚 𝑗 − 1 + 1 − 1

𝑗 ≥ 2 . Let 𝑦 ∈ 𝑊 𝐻 , and let 𝑌𝑗 = 𝑧 ∈ 𝑊 𝐻 ∶ dist 𝑦, 𝑧 = 𝑗 . Then 𝑊 𝐻 = 0≤𝑗≤𝑛−2 𝑌𝑗. Key Lemma For 𝑗 ≥ 2, the set 𝑌𝑗 is dominated by at most 𝑔

𝑙,𝑚 𝑗 𝑆 𝑙, 𝑚 vertices.

Outline of proof of main result

We show that 𝛿 𝐻 ≤ 1 +

2≤𝑗≤𝑛−2

𝑔

𝑙,𝑚 𝑗 𝑆 𝑙, 𝑚

for ∀conn. 𝐿𝑙

∗, 𝑇𝑚 ∗, 𝑄 𝑛 -free graph 𝐻, where

𝑔

𝑙,𝑚 𝑗 : = 1

𝑗 = 1 𝑆 𝑙, 𝑚 − 1 𝑔

𝑙,𝑚 𝑗 − 1 + 1 − 1

𝑗 ≥ 2 . Let 𝑦 ∈ 𝑊 𝐻 , and let 𝑌𝑗 = 𝑧 ∈ 𝑊 𝐻 ∶ dist 𝑦, 𝑧 = 𝑗 . Then 𝑊 𝐻 = 0≤𝑗≤𝑛−2 𝑌𝑗. Key Lemma For 𝑗 ≥ 2, the set 𝑌𝑗 is dominated by at most 𝑔

𝑙,𝑚 𝑗 𝑆 𝑙, 𝑚 vertices.

Suppose that 𝑌𝑗 is independent. Let 𝑉 ⊆ 𝑌𝑗−1 be a smallest set dominating 𝑌𝑗. If 𝑉 is “large”…  If ∃large clique 𝑉1 ⊆ 𝑉 …  If ∃large indep. set 𝑉2 ⊆ 𝑉 …

Outline of proof of main result

𝑉 𝑌𝑗 𝑌𝑗−1

…

𝑉1

∃𝐿𝑙

∗

𝑉 𝑌𝑗 𝑌𝑗−1

…

𝑉2

𝑌𝑗−2

∃𝑇𝑙

∗

Suppose that 𝑌𝑗 is independent. Let 𝑉 ⊆ 𝑌𝑗−1 be a smallest set dominating 𝑌𝑗. If 𝑉 is “large”…  If ∃large clique 𝑉1 ⊆ 𝑉 …  If ∃large indep. set 𝑉2 ⊆ 𝑉 …

Outline of proof of main result

𝑉 𝑌𝑗 𝑌𝑗−1

…

𝑉1

∃𝐿𝑙

∗

𝑉 𝑌𝑗 𝑌𝑗−1

…

𝑉2

𝑌𝑗−2

∃𝑇𝑙

∗

Extension of main result

Corollary Let 𝜈 be an invariant of graphs s.t. 𝑑1𝛿 𝐻 ≤ 𝜈 𝐻 ≤ 𝑑2𝛿 𝐻 for ∀ conn. graph 𝐻 of suff. large order.

• ---- (＊)

Let ℋ be a set of conn. graphs. Then ∃ const. 𝑑 = 𝑑 ℋ s.t. for ∀ conn. ℋ-free graph 𝐻, 𝜈 𝐻 ≤ 𝑑 if and only if ℋ ≤ 𝐿𝑙

∗, 𝑇𝑚 ∗, 𝑄 𝑛

for some positive integers 𝑙, 𝑚 and 𝑛. Many domination-like invariants satisfy (＊). (total domination 𝛿𝑢, paired domination 𝛿pr, etc…)

Extension of main result

Corollary Let 𝜈 be an invariant of graphs s.t. 𝑑1𝛿 𝐻 ≤ 𝜈 𝐻 ≤ 𝑑2𝛿 𝐻 for ∀ conn. graph 𝐻 of suff. large order.

• ---- (＊)

Let ℋ be a set of conn. graphs. Then ∃ const. 𝑑 = 𝑑 ℋ s.t. for ∀ conn. ℋ-free graph 𝐻, 𝜈 𝐻 ≤ 𝑑 if and only if ℋ ≤ 𝐿𝑙

∗, 𝑇𝑚 ∗, 𝑄 𝑛

for some positive integers 𝑙, 𝑚 and 𝑛. Many domination-like invariants satisfy (＊). (total domination 𝛿𝑢, paired domination 𝛿pr, etc…)

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