Relation between Graphs

Transcription

1 Max Planck Intitute for Math. in the Sciences, Leipzig, Germany Joint work with Jan Hubička, Jürgen Jost, Peter F. Stadler and Ling Yang SCAC2012, SJTU, Shanghai

2 Outline Motivation and Background 1 Motivation and Background 2 3 R-core R-reduced graph Cocore

3 Motivation Motivation and Background Figure: Domain interaction graph Relation: containing relation {d 2, d 3 p 1 ; d 4 p 2 ; d 1, d 3 p 3 ; d 1 p 4 }

4 Motivation Motivation and Background Figure: Domain interaction graph Relation: containing relation {d 2, d 3 p 1 ; d 4 p 2 ; d 1, d 3 p 3 ; d 1 p 4 } d 1 d 2 d 3 d 4 p 1 p 2 p 3 p 4 Figure: Relation

5 We can define a protein interaction graph by any two proteins have interaction iff there are two domains which belong to them respectively. d 1 d 2 d 3 d 4 p 1 p 2 p 3 p 4

6 We can define a protein interaction graph by any two proteins have interaction iff there are two domains which belong to them respectively. d 1 d 2 p 1 p 2 d 2 d 3 p 1 p2 d 4 d 3 d 4 p 3 p 4 d 1 d 1 d 3 p 4 p 3 Figure: From Domain Interaction Network to Protein Interaction Network

7 Find relation between graphs p 1 p2 p 4 p 3 Figure: Domain interaction network Protein Interaction Network relation R = {(d 2, p 1 ),(d 3, p 1 ),(d 4, p 2 ),(d 1, p 3 ),(d 3, p 3 ),(d 1, p 4 )}

8 Basic Definitions Definition The (binary) relation R between two sets A and B is a subset of A B.

9 Basic Definitions Definition The (binary) relation R between two sets A and B is a subset of A B. Definition Given a graph G = (V G, E G ), a finite set B and a binary relation R V G B, we define a new graph H = G R whose vertex set is B and for arbitrary u, v B, (u, v) E H iff one can find x, y V G, such that (x, u),(y, v) R and (x, y) E G.

10 Basic Definitions Definition The (binary) relation R between two sets A and B is a subset of A B. Definition Given a graph G = (V G, E G ), a finite set B and a binary relation R V G B, we define a new graph H = G R whose vertex set is B and for arbitrary u, v B, (u, v) E H iff one can find x, y V G, such that (x, u),(y, v) R and (x, y) E G. Definition Given two graphs G and H, if there exists a R V G V H, such that G R = H, we say R is a relation from G to H.

11 Definition Graph homomorphism G H is a mapping f : V G V H such that (x, y) E G implies (f(x), f(y)) E H.

12 Definition Graph homomorphism G H is a mapping f : V G V H such that (x, y) E G implies (f(x), f(y)) E H. Definition Graph multihomomorphism is a mapping f : V G 2 V H \ (non-empty subsets of V H ) such that (x, y) E G implies (u, v) E H for every u f(x) and v f(y).

13 Definition Graph homomorphism G H is a mapping f : V G V H such that (x, y) E G implies (f(x), f(y)) E H. Definition Graph multihomomorphism is a mapping f : V G 2 V H \ (non-empty subsets of V H ) such that (x, y) E G implies (u, v) E H for every u f(x) and v f(y). Definition Surjective multihomomorphism is a multihomomorphism such that pre-image of every vertex in H is non-empty and for every edge (u, v) in H we can find an edge (x, y) in G satisfying u f(x), v f(y).

14 Relation VS Surjective multihomomorphism R V G V H f : V G 2 V H \

15 Definition Relation VS Surjective multihomomorphism R V G V H f : V G 2 V H \ R is full if for every x V G, there is (x, u) R. d 1 d 2 d 3 d 4 p 1 p 2 p 3 p 4 Figure: Non-full relation

16 Definition Relation VS Surjective multihomomorphism R V G V H f : V G 2 V H \ R is full if for every x V G, there is (x, u) R. d 1 d 2 d 3 d 4 p 1 p 2 p 3 p 4 Figure: Non-full relation If R is full and G R = H, then R is a surjective multihomomorphism.

17 Relation VS Homomorphism If there is a surjective homomorphism f : G H, then G R = H for R corresponding to f. If G R = H and R is full, then there is a homomorphism f : G H. d 1 d 2 d 3 d 4 p 1 p 2 p 3 p 4 Figure: Homomorphism

18 Relation VS Homomorphism If there is a surjective homomorphism f : G H, then G R = H for R corresponding to f. If G R = H and R is full, then there is a homomorphism f : G H. d 1 d 2 d 3 d 4 p 1 p 2 p 3 p 4 Figure: Homomorphism

19 Compose and Decompose Graph relations compose, i.e., (G R) S = G (R S).

20 Compose and Decompose Graph relations compose, i.e., (G R) S = G (R S). Graph relation R can be decomposed to a relation R D duplicating vertices and a relation R C contracting vertices. d 1 p 1 d 2 p 2 d 3 p 3 d 4 p 4

21 Compose and Decompose Graph relations compose, i.e., (G R) S = G (R S). Graph relation R can be decomposed to a relation R D duplicating vertices and a relation R C contracting vertices. d 1 p 1 d 1 1 d1 p1 d 2 1 d 2 p 2 d2 d 1 2 p2 d 1 3 d 3 p 3 d3 p3 d 2 3 d 4 p 4 d4 d 1 4 p4 Figure: Decomposition

22 d 1 p 1 d 1 1 d1 p1 d 2 1 d 2 p 2 d2 d 1 2 p2 d 1 3 d 3 p 3 d3 p3 d 2 3 d 4 p 4 d4 d 1 4 p4 Figure: R = R D R C

23 d 1 p 1 d 1 1 d1 p1 d 2 1 d 2 p 2 d2 d 1 2 p2 d 1 3 d 3 p 3 d3 p3 d 2 3 d 4 p 4 d4 d 1 4 p4 Figure: R = R D R C d 1 1 d 1 3 d 1 2 d 1 4 d 2 1 d 2 3

24 d 1 p 1 d 1 1 d1 p1 d 2 1 d 2 p 2 d2 d 1 2 p2 d 1 3 d 3 p 3 d3 p3 d 2 3 d 4 p 4 d4 d 1 4 p4 Figure: R = R D R C d 1 1 d 1 3 d 1 2 d 1 4 d 2 d 3 d 4 d 2 1 d 2 3 d 1 d 1 d 3 Figure: G G R D G R D R C

25 relation R G H full homomorphism R + surjective homomorphism RC D G Full homomorphism: preserve both edge and non-edge Surjective homomorphism: contraction

26 Core Motivation and Background R-core R-reduced graph Cocore Definition In graph homomorphism, core of graph G is the smallest subgraph of G which G has homomorphism to. Example Figure: G and its core

27 Core Motivation and Background R-core R-reduced graph Cocore Definition In graph homomorphism, core of graph G is the smallest subgraph of G which G has homomorphism to. Example Figure: G and its core Core of graph G is unique up to isomorphism, so we can it the core of G, denote it by G core. G core is an induced subgraph of G.

28 Why core? Motivation and Background R-core R-reduced graph Cocore G and G core are homomorphically equivalent, i.e.,g G core and G core G. Test whether G has homomorphism to H is equivalent to testing G core H core. Cores are the smallest graphs (in the number of vertices) in the class of homomorphism equivalence.

29 Outline Motivation and Background R-core R-reduced graph Cocore 1 Motivation and Background 2 3 R-core R-reduced graph Cocore

30 R-core R-reduced graph Cocore Definition Two graphs G and H are relationally equivalent, G H, if there are relations R and S such that G R = H and H S = G. If R and S are full, we write G f H. An R-core of a graph G, G R-core, is the smallest graph in the same equivalence class of f as G. Example Figure: G and its R-core

31 R-core R-reduced graph Cocore R-core is unique up to isomorphism. Proposition G R-core is isomorphic to an induced subgraph of G. If G is an R-core, then every relation R such that G R = G satisfies the Hall condition and thus contains a monomorphism.

32 R-core R-reduced graph Cocore Proposition G R-core can be characterized by an algorithm, which removes all vertices v V G such that (1) the neighborhood of v is union of neighborhood of some other vertices v 1, v 2,..., v n and (2) there is vertex u such that N G (v) N G (u). Example Figure: G and its R-core

33 Outline Motivation and Background R-core R-reduced graph Cocore 1 Motivation and Background 2 3 R-core R-reduced graph Cocore

34 R-core R-reduced graph Cocore Definition A graph G R-reduced is R-reduced graph of G if it is the smallest subgraph of G which has relation to G. Proposition G R-reduced coincides with G core. Example Figure: G and its R-reduced graph

35 Outline Motivation and Background R-core R-reduced graph Cocore 1 Motivation and Background 2 3 R-core R-reduced graph Cocore

36 R-core R-reduced graph Cocore There is always homomorphism from subgraph to graph.

37 R-core R-reduced graph Cocore There is always homomorphism from subgraph to graph. This is not always true for relations. We can reverse the notion of core to get...

38 R-core R-reduced graph Cocore There is always homomorphism from subgraph to graph. This is not always true for relations. We can reverse the notion of core to get... Figure: From football to mathematics

39 R-core R-reduced graph Cocore Definition A cocore of graph G is the smallest induced subgraph of G which has relation to G. Cocore of G is unique up to isomorphism, denoted by G cocore Example Figure: G and its cocore

40 R-core R-reduced graph Cocore Proposition G is a cocore if and only if any vertex neighborhood is not the union of other vertex neighborhoods. Proposition Cocore of G is the smallest member of the equivalence class of relational equivalence. Example Figure: G and its cocore

41 R-core R-reduced graph Cocore Figure: G Figure: core cocore R-core

42 Thank you for your attention! R-core R-reduced graph Cocore Figure: R-core = Mountain = Winning Cacau