Chordal Graphs

1. Chordal Graphs Bayesian Networks, Winter 2009-2010 YoavHaimovitch & Ariel Raviv

2. Contents • Chordal Graph • Warm up • Theorem 7 • Perfect Vertex Elimination Scheme • Maximal cliques Tree • Bibliography • M.C.Golumbic Algorithmic Graph Theory and Perfect Graphs. Second edition • Pearl, J. Probabilistic Reasoning in Intelligent Systems.

3. Chordal Graph • Definition: Undirected graph G is chordal if every cycle of length greater than 3 has a chord. • Remark: Chordality is a hereditary property, All induced subgraphs of a chordal graph are chordal! • Chordal graphs are also called triangulated and perfecteliminationgraphs.

4. Examples of graphs

5. Lemma 1 • A vertex x of G is called simplicial if its adjacency set Adj(x) is a clique (not necessarily maximal) • Lemma 1[Dirac 1961]: Each chordal graph has a simplicial vertex and if G is not a clique it has two non adjacent simplicial vertices.

6. Proof of Lemma 1 • If G is a clique, done. • If not, assume that be has two non-adjacent nodes a and b and that the lemma is true to all graphs with fewer vertices than G. Let S be a minimalvertex separator for a and b. Let Gaand Gbbe the connected components of a and b respectively.

7. Proof of Lemma 1 (Cont.) • S is a clique – if not, then there exists x and y such that there is no edge between them, and because S is minimal vertex separator there is a cycle <a,…,x,…,b,…,y,…,a>. It is easy to show that there exist a minimal cycle with length no less then 4 with no chords in it, contradiction.

8. Proof of Lemma 1 (Cont.) • S is a clique. GA+S is smaller than G therefore by induction the lemma holds, i.e. GA+S is a clique or has two non adjacent simplicial vertices, one of each must be in GA. Any simplicial vertex in GA is a simplicial vertex in G because all elements of Adj(A) are inside GA+S. Thus from GA and GB we get two simplicial vertices in G.

9. Theorem 7 • Definition: The following are equivalent: • G is chordal • The edges of G can be directed acyclically so that every pair of converging arrows comes from two adjacent vertices. • G has a perfect vertex elimination scheme. • There is a tree T with maximal cliques of G as vertices. Any two cliques containing v are either adjacent in T or connected by a path of cliques that contain v.

10. Proof of Theorem 7 (1→2) • By lemma 1 every chordal graph has a simplicial vertex • Direct all of its edges to it • Repeat the process with the rest of the graph (can be done because being chordal is hereditary) • From the definition of simplicial vertex corollary that each two converging edges coming from adjacent vertices.

11. Proof of Theorem 7 (2→1) • Let there be a cycle of size larger than 3. • There exist a valid direction which is acyclic • There are two converging edges, which come from two adjacent vertices • The cycle has a chord.

12. Vertex Elimination Order • Given a Markov network, we would like to calculate a probability function over it’s variables • What if we want to calculate the probability of one variable?

13. Example 1 P(X1,X2,X3,X4,X5)=K∙f1(X1,X2)f2(X2,X3)f3(X3,X4)f4(X4,X5)f5(X5,X1) P(X1,X2,X3,X4)=ΣX5K∙f1(X1,X2)f2(X2,X3)f3(X3,X4)f4(X4,X5)f5(X5,X1)= =K∙f1(X1,X2)f2(X2,X3)f3(X3,X4) ΣX5f4(X4,X5)f5(X5,X1) = K`∙f1(X1,X2)f2(X2,X3)f3(X3,X4) F(X4, X1) • Explanation: Each time we sum over a variable we automatically add edges between all it’s adjacent vertices (creating a clique)

14. Example 2 • Here is an example of how bad (i.e. big cliques) things can happen with incorrect order of summing P(X1,X2,X3,X4,X5)=ΣX0K∙f1(X0,X1)f2(X0,X2)f3(X0,X3)f4(X0,X4)f5(X0,X5) = K`∙F(X1,X2,X3,X4,X5) • Explanation: For wrong order of summing we get big cliques and thus huge tables of probabilities to store, which are exponential in number of variables.

15. Perfect Vertex Elimination • Let G=(V,E) be an undirected graph and let σ=[v1,v2,…,vn] be an ordering of the vertices. We say that σ is a perfect vertex elimination scheme if for each i: Xi = { vj in Adj(vi) | j>I } is complete.

16. Theorem 7 (Cont.) • Definition: The following are equivalent: • G is chordal • The edges of G can be directed acyclically so that every pair of converging arrows comes from two adjacent vertices. • G has a perfect vertex elimination scheme. • There is a tree T with maximal cliques of G as vertices. Any two cliques containing v are either adjacent in T or connected by a path of cliques that contain v.

17. Proof of Theorem 7 (1→3) • By lemma 1, G has a simplicial vertex. • The subgraph induced after removing this vertex is also chordal. • By induction, there is an elimination order.

18. Proof of Theorem 7 (3→1) • Let us assume a cycle of length 4 or greater. • Let x be the first vertex on the elimination scheme of this cycle. • x is simplicial at this point. • Two vertices adjacent to x from the cycle are also adjacent between themselves.

19. Elimination Algorithm • All vertices of G can be deleted by arranging them in separate piles, one for each clique, and then repeatedly applying the following two operations: • Delete a vertex that occurs in only one pile. • Delete a pile if all its vertices appear in another pile.

20. Elimination Algorithm Example A B C D A B B C C D D E E

21. Elimination Algorithm Example A A B A C B C D B D C D D E E

22. Theorem 7 (Cont.) • Definition: The following are equivalent: • G is chordal • The edges of G can be directed acyclically so that every pair of converging arrows comes from two adjacent vertices. • G has a perfect vertex elimination scheme. • There is a tree T with maximal cliques of G as vertices. Any two cliques containing v are either adjacent in T or connected by a path of cliques that contain v.

23. Clique Tree Example C1 a a b c C1 b c C2 C2 b c C3 e d d C3 b e d

24. Clique Tree Example C1 a b C1 a b C2 C4 C2 C4 a b c d c d C3 C3 c d

25. Proof of Theorem 7 (1→4) • Lemma 2:A vertex is simplicial iff it belongs to exactly one clique. • Let G =(V,E) be a chordal graph and |V|=k • Proof by induction: Assume the claim holds for all graphs of size < k. • By Lemma 1 G has a simplicial vertex v. • By Lemma 2 v belongs to exactly one clique C.

26. Proof of Theorem 7 (1→4) • For G’ =(V\{v},E’) there exists a tree T’ which satisfies the claim. Split into two cases: • C’ = C – {v} is maximal in G’. Add v to C’ to build T from T’. T is a Clique-Tree as needed. a C1 C2 C1 b c a b b c e d C2 e d

27. Proof of Theorem 7 (1→4) • C’ = C – {v} is not maximal in G’. There is a maximal clique P in G’ such that C’⊂P. Add C and (C, P) to T’ to build T. T is a Clique-Tree as needed. C1 C2 a b C2 a a b C1 c d d c d

28. Proof of Theorem 7 (4→1) • Let T be a Clique-Tree of graph G=(V,E) and |V|=k. • Proof by induction: Assume the claim holds for all graphs of size < k. • Let L be a leaf in T, and P its parent in T. • Let v∈L\P (exists from maximality). • v cannot be in any other clique. • by Lemma 2, v is simplicial.

29. Proof of Theorem 7 (4→1) • Let T’ be T with v removed. T’ is a Clique- Tree. • By assumption G’ =(V\{v},E’) is chordal. • By part 3 of Theorem 7, there exists σ’, a Perfect Elimination Order for G’. • Let σ = [v, σ’] be a perfect elimination order for G. • By the converse of part 3, we can conclude that G is chordal