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

Forbidden subgraphs for constant domination number Michitaka Furuya (Kitasato University)

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 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 . 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 = 𝐷 4 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 = 𝐷 4 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. 𝐿 1,3 -free graph ( 𝐿 1,3 : )

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 ℋ 1 and ℋ 2 be sets of conn. graphs. We write ℋ 1 ≤ ℋ 2 if for ∀𝐼 2 ∈ ℋ 2 , ∃𝐼 1 ∈ ℋ 1 s.t. 𝐼 1 is an induced subgraph of 𝐼 2 . Remark If ℋ 1 ≤ ℋ 2 , then every ℋ 1 -free graph is ℋ 2 -free.

Domination and forbidden subgraph Theorem 3 (Cockayne et al., 1985) Let 𝐻 be a conn. { 𝐿 1,3 , }-free graph of order 𝑜 . Then 𝛿 𝐻 ≤ ⌈𝑜/3⌉ . Let 𝛿 pr 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. 𝐿 1,𝑛 -free graph of order 𝑜 ≥ 2 . Then 𝛿 pr 𝐻 ≤ 2 𝑛𝑜 + 1 / 2𝑛 + 1 . Theorem 5 (Dorbec and Gravier, 2008) Let 𝐻 be a conn. 𝑄 5 -free graph of order 𝑜 ≥ 2 . If 𝐻 ≠ 𝐷 5 , then 𝛿 pr 𝐻 ≤ 𝑜/2 + 1 .

Domination and forbidden subgraph Theorem 3 (Cockayne et al., 1985) Let 𝐻 be a conn. { 𝐿 1,3 , }-free graph of order 𝑜 . Then 𝛿 𝐻 ≤ ⌈𝑜/3⌉ . Let 𝛿 pr 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. 𝐿 1,𝑛 -free graph of order 𝑜 ≥ 2 . Then 𝛿 pr 𝐻 ≤ 2 𝑛𝑜 + 1 / 2𝑛 + 1 . Theorem 5 (Dorbec and Gravier, 2008) Let 𝐻 be a conn. 𝑄 5 -free graph of order 𝑜 ≥ 2 . If 𝐻 ≠ 𝐷 5 , then 𝛿 pr 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 ?? ∗ ： 𝐿 𝑑+1 ∗ Let 𝐿 𝑑+1 be the corona of 𝐿 𝑑+1 . … ∗ ∗ Since 𝛿 𝐿 𝑑+1 = 𝑑 + 1 , 𝐿 𝑑+1 is not ℋ -free. 𝐿 𝑑+1 ∗ ➡ ∃𝐼 ∈ ℋ s.t. 𝐼 is an induced subgraph of 𝐿 𝑑+1 . ∗ ➡ ℋ ≤ 𝐿 𝑑+1 . 𝑑 + 1 By similar argument, ℋ ≤ , 𝑄 3𝑑+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 ?? ∗ ： 𝐿 𝑑+1 ∗ Let 𝐿 𝑑+1 be the corona of 𝐿 𝑑+1 . … ∗ ∗ Since 𝛿 𝐿 𝑑+1 = 𝑑 + 1 , 𝐿 𝑑+1 is not ℋ -free. 𝐿 𝑑+1 ∗ ➡ ∃𝐼 ∈ ℋ s.t. 𝐼 is an induced subgraph of 𝐿 𝑑+1 . ∗ ➡ ℋ ≤ 𝐿 𝑑+1 . 𝑑 + 1 By similar argument, ℋ ≤ , 𝑄 3𝑑+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 ?? ∗ ： 𝐿 𝑑+1 ∗ Let 𝐿 𝑑+1 be the corona of 𝐿 𝑑+1 . … ∗ ∗ Since 𝛿 𝐿 𝑑+1 = 𝑑 + 1 , 𝐿 𝑑+1 is not ℋ -free. 𝐿 𝑑+1 ∗ ➡ ∃𝐼 ∈ ℋ s.t. 𝐼 is an induced subgraph of 𝐿 𝑑+1 . ∗ ➡ ℋ ≤ 𝐿 𝑑+1 . 𝑑 + 1 By similar argument, ℋ ≤ , 𝑄 3𝑑+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 ∗ , 𝑇 𝑚 ∗ , 𝑄 ℋ ≤ 𝐿 𝑙 for some positive integers 𝑙 , 𝑚 and 𝑛 𝑛 𝑚 ∗ = ∗ = where 𝐿 𝑙 and 𝑇 𝑚 . … … 𝐿 𝑙

Outline of proof of main result We show that 𝛿 𝐻 ≤ 1 + 𝑔 𝑙,𝑚 𝑗 𝑆 𝑙, 𝑚 2≤𝑗≤𝑛−2 ∗ , 𝑇 𝑚 ∗ , 𝑄 for ∀ conn. 𝐿 𝑙 𝑛 -free graph 𝐻 , where 𝑙,𝑚 𝑗 : = 1 𝑗 = 1 𝑔 𝑗 ≥ 2 . 𝑆 𝑙, 𝑚 − 1 𝑔 𝑙,𝑚 𝑗 − 1 + 1 − 1 Let 𝑦 ∈ 𝑊 𝐻 , and let 𝑌 𝑗 = 𝑧 ∈ 𝑊 𝐻 ∶ dist 𝑦, 𝑧 = 𝑗 . Then 𝑊 𝐻 = 0≤𝑗≤𝑛−2 𝑌 𝑗 . Key Lemma For 𝑗 ≥ 2 , the set 𝑌 𝑗 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 + 𝑔 𝑙,𝑚 𝑗 𝑆 𝑙, 𝑚 2≤𝑗≤𝑛−2 ∗ , 𝑇 𝑚 ∗ , 𝑄 for ∀ conn. 𝐿 𝑙 𝑛 -free graph 𝐻 , where 𝑙,𝑚 𝑗 : = 1 𝑗 = 1 𝑔 𝑗 ≥ 2 . 𝑆 𝑙, 𝑚 − 1 𝑔 𝑙,𝑚 𝑗 − 1 + 1 − 1 Let 𝑦 ∈ 𝑊 𝐻 , and let 𝑌 𝑗 = 𝑧 ∈ 𝑊 𝐻 ∶ dist 𝑦, 𝑧 = 𝑗 . Then 𝑊 𝐻 = 0≤𝑗≤𝑛−2 𝑌 𝑗 . Key Lemma For 𝑗 ≥ 2 , the set 𝑌 𝑗 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 + 𝑔 𝑙,𝑚 𝑗 𝑆 𝑙, 𝑚 2≤𝑗≤𝑛−2 ∗ , 𝑇 𝑚 ∗ , 𝑄 for ∀ conn. 𝐿 𝑙 𝑛 -free graph 𝐻 , where 𝑙,𝑚 𝑗 : = 1 𝑗 = 1 𝑔 𝑗 ≥ 2 . 𝑆 𝑙, 𝑚 − 1 𝑔 𝑙,𝑚 𝑗 − 1 + 1 − 1 Let 𝑦 ∈ 𝑊 𝐻 , and let 𝑌 𝑗 = 𝑧 ∈ 𝑊 𝐻 ∶ dist 𝑦, 𝑧 = 𝑗 . Then 𝑊 𝐻 = 0≤𝑗≤𝑛−2 𝑌 𝑗 . Key Lemma For 𝑗 ≥ 2 , the set 𝑌 𝑗 is dominated by at most 𝑔 𝑙,𝑚 𝑗 𝑆 𝑙, 𝑚 vertices.

Outline of proof of main result Suppose that 𝑌 𝑗 is independent. Let 𝑉 ⊆ 𝑌 𝑗−1 be a smallest set dominating 𝑌 𝑗 . If 𝑉 is “large”…  If ∃ large clique 𝑉 1 ⊆ 𝑉 … 𝑌 𝑗 ∗ ∃𝐿 𝑙 … 𝑌 𝑗−1 𝑉 1 𝑉 𝑌 𝑗  If ∃ large indep. set 𝑉 2 ⊆ 𝑉 … … 𝑌 𝑗−1 𝑉 2 𝑉 ∗ ∃𝑇 𝑙 𝑌 𝑗−2

Outline of proof of main result Suppose that 𝑌 𝑗 is independent. Let 𝑉 ⊆ 𝑌 𝑗−1 be a smallest set dominating 𝑌 𝑗 . If 𝑉 is “large”…  If ∃ large clique 𝑉 1 ⊆ 𝑉 … 𝑌 𝑗 ∗ ∃𝐿 𝑙 … 𝑌 𝑗−1 𝑉 1 𝑉 𝑌 𝑗  If ∃ large indep. set 𝑉 2 ⊆ 𝑉 … … 𝑌 𝑗−1 𝑉 2 𝑉 ∗ ∃𝑇 𝑙 𝑌 𝑗−2

Extension of main result Corollary Let 𝜈 be an invariant of graphs s.t. 𝑑 1 𝛿 𝐻 ≤ 𝜈 𝐻 ≤ 𝑑 2 𝛿 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 ∗ , 𝑇 𝑚 ∗ , 𝑄 ℋ ≤ 𝐿 𝑙 for some positive integers 𝑙 , 𝑚 and 𝑛 . 𝑛 Many domination-like invariants satisfy ( ＊ ). (total domination 𝛿 𝑢 , paired domination 𝛿 pr , etc …)

Extension of main result Corollary Let 𝜈 be an invariant of graphs s.t. 𝑑 1 𝛿 𝐻 ≤ 𝜈 𝐻 ≤ 𝑑 2 𝛿 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 ∗ , 𝑇 𝑚 ∗ , 𝑄 ℋ ≤ 𝐿 𝑙 for some positive integers 𝑙 , 𝑚 and 𝑛 . 𝑛 Many domination-like invariants satisfy ( ＊ ). (total domination 𝛿 𝑢 , paired domination 𝛿 pr , etc …) Thank you for your attention!