Universality of Intervals of Line Graph Partial Order cka 2 and Yangjing Long 3 ı Fiala 1 , Jan Hubiˇ Jiˇ r´ 1 Charles University, Czech Republic 2 University of Calgary, Canada 3 Shanghai Jiao Tong University
A new Universality argument of Homomorphism Orders Outline A new Universality argument of Homomorphism Orders What is universality and what are homomorphism orders? Universality of homomorphism order Our new agrument Application: Line Graph Intervals Why and what are line graph intervals? Are line graph intervals universal? What we do not know?
A new Universality argument of Homomorphism Orders Universality Definition A partial order ( P , ≤ P ) can be embedded into a partial order ( Q , ≤ Q ) if there exists a mapping f from ( P , ≤ P ) to ( Q , ≤ Q ) such that a ≤ P b if and only if f ( a ) ≤ Q f ( b ). Figure : No embeddings Definition A partial order P is universal, if any countable order can be embedded into P .
A new Universality argument of Homomorphism Orders Why study universality? If a partial order is universal, we can answer many questions like: ◮ whether there is an infinite chain. ◮ whether there is an infinite antichain (independent elements). ◮ given a particular partial order, whether there is an embedding into it.
A new Universality argument of Homomorphism Orders Homomorphism Orders (1980s): ◮ DiGraphs is the class of all finite directed graphs. ◮ Define (DiGraphs, ≤ H ): for G , H ∈ DiGraphs, G ≤ H H iff G → H . ◮ (DiGraphs , ≤ H ) is a quasi-order: ◮ The relation ≤ H is reflexive (identity is a homomorphism). ◮ The relation ≤ H transitive (composition of two homomorphisms is still a homomorphism).
A new Universality argument of Homomorphism Orders Homomorphism Orders (1980s): ◮ DiGraphs is the class of all finite directed graphs. ◮ Define (DiGraphs, ≤ H ): for G , H ∈ DiGraphs, G ≤ H H iff G → H . ◮ (DiGraphs , ≤ H ) is a quasi-order: ◮ The relation ≤ H is reflexive (identity is a homomorphism). ◮ The relation ≤ H transitive (composition of two homomorphisms is still a homomorphism). ◮ Turn the quasi-order to a partial order: choose a particular representative for each equivalence class, the cores fits perfectly the purpose. ◮ We denote the partial order the same way as the quasi-order, by (DiGraphs , ≤ H ), where DiGraphs is restricted to cores.
A new Universality argument of Homomorphism Orders Is the homomorphism order universal? Theorem (Hedrl´ ın et al., 1980s) The partial order (DiGraphs , ≤ H ) is universal. The proof is complicated. ◮ We consider universality of homomorphism order on graph classes, like on simple graphs, cycles, paths, perfect graphs, planar graphs, etc. ◮ Constrained homomorphism Monomorphisms, surjective homomorphisms, full homomorphisms, locally injective homomorphisms.
A new Universality argument of Homomorphism Orders DiPath: The class of oriented paths. Theorem (Hubiˇ cka, Neˇ setˇ ril, 2003) The homomorphism order on the class of oriented paths (DiPath , ≤ H ) is universal. Figure : Zig Zag
A new Universality argument of Homomorphism Orders DiPath: The class of oriented paths. Theorem (Hubiˇ cka, Neˇ setˇ ril, 2003) The homomorphism order on the class of oriented paths (DiPath , ≤ H ) is universal. Figure : Zig Zag
A new Universality argument of Homomorphism Orders Why Zig Zag is useful? Theorem: Directed paths ordered by homomorphisms are universal. Advantage: Corrollary: Homomorphism order is universal on graphs that are ◮ maximum degree 3, ◮ planar, ◮ have treewidth at most 4, etc. replace all by in each � P k
A new Universality argument of Homomorphism Orders Why Zig Zag is useful? Theorem: Directed paths ordered by homomorphisms are universal. Advantage: Corrollary: Homomorphism order is universal on graphs that are ◮ maximum degree 3, ◮ planar, ◮ have treewidth at most 4, etc. replace all by in each � P k Disadvantage: However, there are some problem which cannot build from zig zag easily, for example, locally injective homomorphism, if you because paths fliping is not locally injective. So we need new techniques.
A new Universality argument of Homomorphism Orders We provide a new and significantly easier method to prove the universality. DiCycles: The class of graphs formed by finite disjoint union of clockwise oriented cycles. · · · − → − → − → → − C 3 C 4 C 5 C 6 Theorem (Fiala, Hubiˇ cka, L., 2012) The partial order (DiCycles , ≤ H ) is universal. ◮ This proof is significantly easier than other proofs. ◮ It gives a new and simple universal class, this can be easily applied to prove the universality of other orders. ◮ More applications: recently Neˇ setˇ ril and Hubiˇ cka applied it to prove the fractal property .
A new Universality argument of Homomorphism Orders Theorem (Fiala, Hubiˇ cka, L., 2012) The partial order (DiCycles , ≤ H ) is universal. ◮ It is a well-known result that any finite order can be represented by finite sets ordered by the containedness. ◮ We generalize it on up-finite order (infinite): Any up-finite order can be represented by sets ordered by the containedness, represent every element by its up-set. 5 3 11 7 ◮ Sets containedness contains no infinite increasing chain. Idea: to use the sets of sets to represent a partial order.
A new Universality argument of Homomorphism Orders On-line Embedding Game ◮ Bob has a secret partial order, each round he gives Alice one element and the orders to the previous elements. ◮ If Alice recover Bob’s partial order on sets of sets, she wins. Alice’s wining strategy: ◮ Each round gives a set and a set of sets, when given order is in forwarding, add it to the set of sets, when given order is in backwarding, add it to set. ◮ Order the sets of sets by set dominate: A ≤ B iff for any a ∈ A there is a set b ∈ B such that a ⊇ b . 5 3 11 7 3 { 3 } and {{ 3 }}
A new Universality argument of Homomorphism Orders On-line Embedding Game ◮ Bob has a secret partial order, each round he gives Alice one element and the orders to the previous elements. ◮ If Alice recover Bob’s partial order on sets of sets, she wins. Alice’s wining strategy: ◮ Each round gives a set and a set of sets, when given order is in forwarding, add it to the set of sets, when given order is in backwarding, add it to set. ◮ Order the sets of sets by set dominate: A ≤ B iff for any a ∈ A there is a set b ∈ B such that a ⊇ b . 5 3 11 7 3 { 3 } and {{ 3 }} 5 and 3 → 5 { 5 } and {{ 3 } , { 5 }}
A new Universality argument of Homomorphism Orders On-line Embedding Game ◮ Bob has a secret partial order, each round he gives Alice one element and the orders to the previous elements. ◮ If Alice recover Bob’s partial order on sets of sets, she wins. Alice’s wining strategy: ◮ Each round gives a set and a set of sets, when given order is in forwarding, add it to the set of sets, when given order is in backwarding, add it to set. ◮ Order the sets of sets by set dominate: A ≤ B iff for any a ∈ A there is a set b ∈ B such that a ⊇ b . 5 3 11 7 3 { 3 } and {{ 3 }} 5 and 3 → 5 { 5 } and {{ 3 } , { 5 }} 7 and 7 → 3, 7 → 5 { 3 , 5 , 7 } and {{ 3 , 5 , 7 }}
A new Universality argument of Homomorphism Orders On-line Embedding Game ◮ Bob has a secret partial order, each round he gives Alice one element and the orders to the previous elements. ◮ If Alice recover Bob’s partial order on sets of sets, she wins. Alice’s wining strategy: ◮ Each round gives a set and a set of sets, when given order is in forwarding, add it to the set of sets, when given order is in backwarding, add it to set. ◮ Order the sets of sets by set dominate: A ≤ B iff for any a ∈ A there is a set b ∈ B such that a ⊇ b . 5 {{ 3 } , { 5 }} 3 11 {{ 3 }} {{ 3 , 5 , 7 } , { 5 , 11 }} WIN! � {{ 3 , 5 , 7 }} 7 3 { 3 } and {{ 3 }} 5 and 3 → 5 { 5 } and {{ 3 } , { 5 }} 7 and 7 → 3, 7 → 5 { 3 , 5 , 7 } and {{ 3 , 5 , 7 }} 11 and 11 → 5, 7 → 11 { 11 , 5 } and {{ 3 , 5 , 7 } , { 5 , 11 }}
A new Universality argument of Homomorphism Orders We show the proof in two layers, first represent any partial order by another order on sets , then transfer it on DiCycles graphs . embedding in ( P fin ( N ), ← − dom ( P , ≤ P ) | ) embedding in ( DiCycles , ≤ Hom ) N { 3, 5 } � C 105 ∪ � { 3 } { 105, 55 } C 55 � { 105 } C 105
Application: Line Graph Intervals Outline A new Universality argument of Homomorphism Orders What is universality and what are homomorphism orders? Universality of homomorphism order Our new agrument Application: Line Graph Intervals Why and what are line graph intervals? Are line graph intervals universal? What we do not know?
Application: Line Graph Intervals Why and what are line graph intervals? ◮ Figure : A graph and its line graph ◮ If the maximal degree of G is n , then 1. K n → L ( G ). 2. (Vising Theorem) The chromatic numer of L ( G ) is bounded by n + 1, i.e., L ( G ) → K n +1 . ◮ From Vising theorem, if a graph G has maximal degree n , then K n → L ( G ) → K n +1 . A line graph L ( G ) is in the line graph interval [ K n , K n +1 ] L if K n → L ( G ) → K n +1 . Question [Roberson’s thesis 2013 Waterloo] Whether ([ K n , K n +1 ] L , ≤ H ) are universal?
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