A Simple Linear Time Algorithm for the Domatic Partition Problem on Strongly Chordal Graphs

1. A Simple Linear Time Algorithm for the Domatic Partition Problem on Strongly Chordal Graphs Shen-Lung Peng and Maw-Shang Chang Speaker: Chuang-Chieh Lin (林莊傑) Computation Theory Laboratory Advisor: Professor Maw-Shang Chang Information Processing Letters, 43 (1992), pp. 297-300.

2. Outline • Introduction • Strongly Chordal Graphs • Algorithm • Correctness • Complexity • Conclusions • References

3. Outline • Introduction • Strongly Chordal Graphs • Algorithm • Correctness • Complexity • Conclusions • References

4. v7 v7 v1 v1 v5 v5 v4 v4 v8 v8 Dominating set • A graph of vertices D is a dominating set of a graph G = (V, E) if every vertex in V \ D is adjacent to a vertex in D. D = {v1, v4, v5, v7, v8} v2 v6 v3

5. v7 v7 v2 v6 v1 v1 v5 v5 v3 v4 v4 v8 v8 Dominating set • Another dominating set D = {v2, v3, v6} D = {v3, v6} v2 v2 v6 v3 Q: Is there another smaller dominating set?

6. v7 v7 v1 v1 v2 v2 v6 v6 v5 v5 v3 v3 v4 v4 v8 v8 Dominating set • What is the dominating set of the following graph?

7. v7 v1 v2 v6 v3 v4 v5 v8 Domatic number • The domatic number of G, denoted by d(G), is the maximum number of pairwise disjoint dominating sets in G. v7 v1 v2 v6 v3 v4 v5 v8 {v1, v4, v5, v7, v8},{v2, v3, v6}

8. v7 v1 v2 v6 v7 v1 v2 v6 v3 v4 v5 v8 v3 v4 v5 v8 Domatic number • Another family of pairwise disjoint dominating sets in G: {v1, v4, v5, v7, v8},{v3, v6}

9. v7 v1 v2 v6 v3 v4 v5 v8 Domatic number • Another family of pairwise disjoint dominating sets in G: v7 v1 v2 v6 v3 v4 v5 v8 {v3, v5, v7, v8},{v1, v4 , v6} The domatic number of the graph is “2”. (Why?)

10. v7 v1 v2 v6 v7 v1 v2 v6 v3 v4 v5 v8 v3 v4 v5 v8 The domatic partition problem • The domatic partition problem is to partition V into d(G) disjoint dominating sets. {{v1, v4, v5, v7, v8}, {v2, v3, v6}}is called a domatic partition.

11. 5 7 2 1 8 3 6 4 • Another example: { {1, 5}, {2, 6}, {3, 8}, {4, 7}} is a domatic partition.

12. Question: • Does any graph has a domatic partition?

13. The domatic number was defined and studied by E. J. Cockayne and S. T. Hedetniemi [CH77]. • They showed that d(G) ≤  + 1, where  is the minimum degree of vertices in G. • G is domatically full if d(G) =  + 1.

14. Some special graph classes are domatically full, such as : • Trees, [CH77] • Cliques, [CH77] • Maximum outerplanar graphs, [CH77] • Strongly chordal graphs, [F84] • ……

15. 1 3 2 4 5 Chord • An edge is called a chord of a cycle if it connects two vertices of the cycle but is not itself an edge within the cycle. G

16. 1 3 4 5 Chord • The subgraph of G induced by {1,2,3} has a cycle. 2 Any chord exists?

17. 1 3 4 5 Chord • The subgraph of G induced by {1,2,3,4} has cycles. 2 Any chord exists? [2, 4], [1, 3]

18. 1 3 2 4 5 Chord • The subgraph of G induced by {1,2,4,5} has cycles. Any chord exists? [1, 4]

19. 1 3 2 4 5 Hole • A chordless cycle is sometimes also called a graph hole.

20. Chordal graphs • A chordal graph is a simple graph possessing no holes (chordless cycles). • That is, A graph is chordal if and only if every cycle of length greater than three has a chord.

21. 1 1 2 2 3 6 7 7 6 3 1 4 4 2 5 5 4 5 A chordal graph: A non-chordal graph: A hole !

22. Outline • Introduction • Strongly Chordal Graphs • Algorithm • Correctness • Complexity • Conclusions • References

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24. 2 3 4 5 6 1 Odd chords • Let (v1, v2, …, vk) be an even length cycle. Then an edge e = [vi, vj] is an odd chord of the even length cycle if e is a chord and i – j is odd. Any chord else? (1, 2, 3, 4, 5, 6) is an even length cycle.

25. 2 3 4 5 6 1 Can you find an odd chord in the following graph? • Is it a chordal graph?

26. Strongly chordal graphs • A graph is strongly chordal if it is chordal and every even length cycle of length six or more has an odd chord. • Other useful properties that characterize chordal and strongly chordal graphs are introduced in the following. Why six?

27. v w x y v w x y Simplicial vertex • N[v] = Adj(v){v} is called the closed neighborhood of v. • A vertex v is called simplicial if the subgraph induced by N[v] is a clique. N[v] = {v, w, x, y} v is simplicial

28. 1 2 3 6 7 4 5 Where are the simplicial vertices in the graph? {7, 6} Is vertex “1” a simplicial vertex?

29. Perfect elimination ordering • Rose [R70] showed that a graph G is chordal if and only if it is possible to order the vertices (v1, v2,…, vn) in such a way that, for each i{1, 2,…, n}, vi is a simplicial vertex of Gi = G({vi, vi+1,…, vn}), where Gi is a subgraph induced by the vertex set {vi, vi+1,…, vn}. • Such an ordering is called a perfect elimination ordering.

30. 1 2 6 7 3 4 5 4 → 5 3 → 2 → 1 → 6 → 7 → (6, 7, 2, 1, 3, 4, 5) is called a perfect elimination ordering of the given graph.

31. Strong elimination ordering • Let Ni[v] denote the closed neighborhood of v in Gi. • The ordering of vertices (v1, v2,…, vn) is called a strong elimination ordering if it is a perfect elimination ordering and, for each i < j < k, if vj and vk belong to Ni[vi] in Gi then Ni[vj]  Ni[vk]. • For example,

32. 7 3 7 5 5 1 3 2 8 8 6 2 6 9 9 4 4 4 7 3 3 2 6 5 6 5

33. Strong elimination ordering (contd.) • In other words, for each vertex vi we not only require that vi be simplicial in Gi but in addition insist that the ordering of the vertices in Ni[vi] be consistent with the ordering of this nested family (smallest to largest).

34. Why are we concerned for the strong elimination ordering? • Farber [F83] showed that a graph is strongly chordal if and only if it admits a strong elimination ordering. • To date (1992), the best-known time to recognize strongly chordal graphs is O(mlogn). This algorithm additionally determines a strong elimination ordering. [PT87]

35. Outline • Introduction • Strongly Chordal Graphs • Algorithm • Correctness • Complexity • Conclusions • References

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37. Preliminary of the algorithm • A greedy approach is used to design this algorithm. • First, we assume that a strong elimination ordering has been constructed for the strongly chordal graph G = (V, E). • It has the property that the closed neighborhoods of the members of Ni[vi] form a nested family of sets.

38. Preliminary of the algorithm (contd.) • The algorithm maintains +1 disjoint sets. • Initially, these sets are empty. • The algorithm visits vertices in the reverse ordering of the strong elimination ordering. A vertex is included in a set when it is visited.

39. Preliminary of the algorithm (contd.) • We say that a vertex v is dominated by set S if there exists a vertex uS such that uN[v]. • A vertex v is completely dominated if v is dominated by +1 dominating sets. • Now let us see the algorithm.

40. Algorithm DP • 1. Initially, Si =  for 1≤ i ≤ +1. • 2. FORi = n down to 1 DO • 3. Find the largest k such that vk N[vi] and vk is not completely dominated; • 4. Let Sj be a set that does not dominate vk; • 5. Include vi in Sj, i.e., Sj = {vi} Sj; • 6. If no such set exists, then include vi to an arbitrary set. • 7. ENDFOR

41. 1 1 7 3 5 5 3 7 9 6 2 8 8 9 6 2 4 4 N1[2] = {2, 3, 4, 5, 6, 7} N1[5] = {2, 3, 5, 6} N1[6] = {2, 3, 5, 6, 7} A strong elimination ordering: (preprocessing) 1 → 5 → 6 → 2 → 7 → 4 → 3 → 9 → 8

42. 1 5 3 7 2 9 6 4 8 Algorithm DP:  = 2 1 ← 5 ← 6 ← 2 ← 7 ← 4 ← 3 ← 9 ← 8 S1 =  S2 =  S3 = 

43. 1 5 3 7 2 9 6 4 8 Algorithm DP: 1 ← 5 ← 6 ← 2 ← 7 ← 4 ← 3 ← 9 ← 8 S1 =  {8} S2 =  S3 = 

44. 1 5 3 7 2 9 6 4 8 Algorithm DP: 1 ← 5 ← 6 ← 2 ← 7 ← 4 ← 3 ← 9 ← 8 S1 = {8} S2 =  {9} S3 = 

45. 1 5 3 7 2 9 6 4 8 Algorithm DP: 1 ← 5 ← 6 ← 2 ← 7 ← 4 ← 3 ← 9 ← 8 S1 = {8} S2 = {9} S3 =  {3}

46. 1 5 3 7 2 9 6 4 8 Algorithm DP: 1 ← 5 ← 6 ← 2 ← 7 ← 4 ← 3 ← 9 ← 8 S1 = {8} {4, 8} S2 = {9} S3 = {3}

47. 1 5 3 7 2 9 6 4 8 Algorithm DP: 1 ← 5 ← 6 ← 2 ← 7 ← 4 ← 3 ← 9 ← 8 S1 = {4, 8} S2 = {9} {7, 9} S3 = {3}

48. 1 5 3 7 2 9 6 4 8 Algorithm DP: 1 ← 5 ← 6 ← 2 ← 7 ← 4 ← 3 ← 9 ← 8 {2, 4, 8} S1 = {4, 8} S2 = {7, 9} S3 = {3}

49. 1 5 3 7 2 9 6 4 8 Algorithm DP: 1 ← 5 ← 6 ← 2 ← 7 ← 4 ← 3 ← 9 ← 8 S1 = {2, 4, 8} S2 = {7, 9} {6, 7, 9} S3 = {3}

50. 1 5 3 7 2 9 6 4 8 Algorithm DP: 1 ←5← 6 ← 2 ← 7 ← 4 ← 3 ← 9 ← 8 S1 = {2, 4, 8} S2 = {6, 7, 9} S3 = {3} {3, 5}