Thinking Outside the Graph

Transcription

1 Thinking Outside the Graph Terry McKee Wright State University Dayton, Ohio 26 th Clemson mini-conference 27 October 2011

2 Overview 1. The family/tree approach to graphs 2. Application: Protein interaction graphs 3. Various family/2-tree approaches 4. Good applications of part 3?

3 Let F = F (G) be any family of induced subgraphs of a graph G. G: F (G): a c a c a c b d e f b d e c d e d e f or F (G): abc acde cde def

4 Let F = F (G) be any family of induced subgraphs of a graph G. G: F (G): a c abc acde b d e f cde def the intersection graph of F (G): abc acde cde def

5 Suppose T is any spanning tree of the intersection graph of F (G) and, for each v V(G), let T v be the subgraph of T that is induced by the nodes of T that contain v. Call such a T an F tree for G if every T v is connected (i.e., if every T v is a subtree of T ). has two F trees: abc acde abc acde abc acde cde def cde def cde def abc bde but acde def has no F tree

6 Suppose T is any spanning tree of the intersection graph of F (G) and, for each v V(G), let T v be the subgraph of T that is induced by the nodes f T that contain v. Call such a T an F tree for G if every T v is connected (i.e., if every T v is a subtree of T ). Example: If F is the family of all maxcliques of G, then G has an F tree IFF G is chordal. [Such F trees are called clique trees.]

7 A graph G is chordal IFF G has a clique tree. Tutorial G is the intersection graph of subtrees of a tree. G contains no induced cycle C k of length k 4. Every cycle of G large enough to have a chord, does have a chord. V(G) can be ordered as v 1,v 2,,v n such that the closed neighborhood N[v i ] of each v i induces a clique of the subgraph of G induced by {v i,,v n }. Every minimal vertex separator is complete.

8 G a clique tree T for G Theorem: The edges of a clique tree for G correspond exactly to the minimal vertex separators of G. 10 = 26 16

9 Theorem: Suppose T is any tree with vertex set F. Then T is an F tree for G IFF V(G) = F F F ʹ. F V(T ) F F ʹ E(T ) (where holds in general ) Proof: v 1 : 1 V(T v1 ) E(T v1 ) (= iff T v1 is connected) v 2 : 1 V(T v2 ) E(T v2 ) (= iff T v2 is connected) v i : 1 V(T vi ) E(T vi ) (= iff T vi is connected) V(G) F F F ʹ (= iff all T vi connected) F V(T ) F F ʹ E(T )

10 Example: a c abc acde b d e f 6 = 13 7 cde def but not: abc acde cde def 6 < 13 5

11 G the clique intersection graph of G Theorem: If G has an F tree, then the F trees for G are precisely the maximum spanning trees of the weighted F intersection graph.

12 Overview 1. The family/tree approach to graphs 2. Application: Protein interaction graphs 3. Various family/2-tree approaches 4. Good applications of part 3?

13 E. Zotenko, K.S. Guimarães, R. Jothi, and T.M. PRZYTYCKA, Decomposition of overlapping protein complexes: A graph theoretical method for analyzing static and dynamic protein associations, Algorithms for Molecular Biology 1 (2006) #7. Given a graph family, it is usually very useful to be able to represent it using some kind of a tree. Such [a] tree representation exposes a hierarchical organization that a graph may have, allowing for a structured analysis. This [tree] representation shows a smooth transition between functional groups and allows for tracking a proteinʼs path through a cascade of functional groups. Therefore, depending on the nature of the network, the representation may be capable of elucidating temporal relations between functional groups [and capturing] the manner in which proteins enter and leave their enclosing functional groups.

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18 Pattern: If F is the family of all of G, then G has an F tree IFF G is. Hypertheorem: G has an F tree if and only if the hypergraph (V(G), F ) is the dual of a tree hypergraph. (Falk Nicolai) Cograph (= complement-reducible graph): a P 4 -free subgraph (i.e., with no induced 4-vertex paths) Example: If F is the family of all maximal connected cographs of G, then G has an F tree IFF G is distance-hereditary. (F. Nicolai, 1994 [1997])

19 Tutorial A graph G is distance-hereditary IFF All induced paths between the same endpoints are of the same length. The distance between vertices in a connected induced subgraph of G always equals their distance in G. G contains no induced C k with k 5 and no induced house, domino, or gem subgraph. G has a maximal-cograph tree. Every cycle of length 5 has two crossing chords.

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21 Elena Zotenko,, Teresa Przytycka, Decomposition of overlapping protein complexes: A graph theoretical method for analyzing static and dynamic protein associations, Algorithms for Molecular Biology 1 (2006) #7. Conjecture: A protein interaction network G has a Tree of Complexes representation IFF G has an F tree where F is the family of maximal cographs in G. The method is not guaranteed to produce the Tree of Complexes representation for every possible protein interacton network. We conjecture that graphs that admit Tree of Complexes representation are exactly those graphs that admit a clique tree representation, with the nodes being maximal cographs rather than maximal cliques.

22 Elena Zotenko,, Teresa Przytycka, Decomposition of overlapping protein complexes: A graph theoretical method for analyzing static and dynamic protein associations, Algorithms for Molecular Biology 1 (2006) #7. Conjecture: A protein interaction network G has a Tree of Complexes representation IFF G has a maximal cograph tree. So: IFF G is distance-hereditary. But: Their real-data protein interaction graphs (for TNFα/NFκB signaling pathways and pheromone signaling pathways) are not distance-hereditary.

23 Overview 1. The family/tree approach to graphs 2. Application: Protein interaction graphs 3. Various family/2-tree approaches 4. Good applications of part 3?

24 Definition: 2-trees are constructed recursively from an edge by repeatedly identifying an edge of a new triangle with a pre-existing edge. a nonchordal graph G a clique 2-tree for G

25 Definition: 2-trees are constructed recursively from an edge by repeatedly identifying an edge of a new triangle with a pre-existing edge. Lemma: A graph has a clique 2-tree IFF it has a clique partial* 2-tree. *a subgraph of a clique 2-tree IFF it has a clique series-parallel* graph *no subgraph is a subdivision of K 4

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27 Recall: A graph has a clique tree IFF it is chordal. no C 4 is a contraction of an induced subgraph.

28 Theorem: A graph G has a clique 2-tree IFF none of the graphs shown below is a contraction of an induced subgraph of G. Note: Each G i has i triangles and 4 i edge-disjoint claws.

29 Abbr: mop graph = maximal outerplanar graph. (equivalently, a 2-tree with each edge in 2 triangles) outerplanar graph partial mop graph mop graph 2-tree outerplanar graph series-parallel graph Lemma: A graph has a clique mop graph IFF it has a clique outerplanar graph.

30 Theorem: A graph G has a clique mop graph IFF none of the graphs shown below is a contraction of an induced subgraph of G.

31 Questions about graphs with clique 2-trees: Given the graph, how to find the maxcliques? not efficiently Given the maxcliques, how to find a clique 2-tree? not greedily Given a clique 2-tree of G, how to find the minimal vertex separators of G? The edge cutsets of a clique 2-tree are uniquely determined and, among these, the minimal vertex separators of G can be identified, but not elegantly

32 Overview 1. The family/tree approach to graphs 2. Application: Protein interaction graphs 3. Various family/2-tree approaches 4. Good applications of part 3?

33 Good applications must "answer (or can be made to seem to answer, or can be twisted and wrenched and piled into odd shapes until they hint at being somehow perhaps on the verge of answering) a question that someone might conceivably want asked." David Quammen, in The Boilerplate Rhino (2000)

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35 E. Zotenko, K.S. Guimarães, R. Jothi, and T.M. PRZYTYCKA, Decomposition of overlapping protein complexes: A graph theoretical method for analyzing static and dynamic protein associations, Algorithms for Molecular Biology 1 (2006) #7. This [tree] representation shows a smooth transition between functional groups and allows for tracking a protein s path through a cascade of functional groups. Therefore, depending on the nature of the network, the representation may be capable of elucidating temporal relations between functional groups [and capturing] the manner in which proteins enter and leave their enclosing functional groups.

36 Questions about this clique tree application: Justification for using maxcliques? want to use cliquish subgraphs Justification for using trees? not strong; indeed, want to avoid imposing artificial order! So maybe instead use clique (partial) 2-trees? (in other words, clique series-parallel graphs) [or maybe clique outerplanar graphs?]

37 Theorem: [D Antona & Kung, DISCRETE MATH. (1980)] A 2-connected graph is series-parallel IFF the arbitrary orientation of any one edge determines a unique orientation of all the edges such that the set of directed cycles is exactly the set of cycles that contain that one edge. Applying clique 2-trees: Knowing or positing the temporal relation* between any two arbitrary adjacent functional groups will imply the temporal relations between all the functional groups. *the direction in which proteins move Does this answer "a question that someone might conceivably want asked"?