Characterization by forbidden induced graphs of some subclasses of chordal graphs

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1 Characterization by forbidden induced graphs of some subclasses of chordal graphs Sérgio H. Nogueira 1,2 e Vinicius F. dos Santos 1,3 1 PPGMMC, CEFET-MG 2 Instituto de Ciências Exatas e Tecnológicas, UFV 3 Departamento de Ciência da Computação, UFMG La Plata, Argentina November, 2016 Nogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

2 Definitions Definitions Chordal graph Nogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

3 Definitions Definitions Chordal graph Definition A graph is chordal if every cycle of length greater than three has a chord*. *An edge connecting two nonconsecutive vertices. Nogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

4 Definitions Definitions Chordal graph Definition A graph is chordal if every cycle of length greater than three has a chord*. *An edge connecting two nonconsecutive vertices. Minimal separators Nogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

5 Definitions Definitions Chordal graph Definition A graph is chordal if every cycle of length greater than three has a chord*. *An edge connecting two nonconsecutive vertices. Minimal separators Definition A set S V (G) disconnects a vertex a from b in G if every path of G between a and b contains a vertex from S. A non-empty set S V (G) is a minimal separator of G if there exist a and b such that S disconnects a from b in G and no proper subset of S disconnects a from b in G. Nogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

6 Definitions Clique tree ogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

7 Definitions Clique tree Definition A clique tree of a connected chordal graph is any tree T whose vertices are the maximal cliques of G such that for every two cliques C 1, C 2 each clique on the path from C 1 to C 2 in T contains C 1 C 2. ogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

8 Definitions Clique tree Definition A clique tree of a connected chordal graph is any tree T whose vertices are the maximal cliques of G such that for every two cliques C 1, C 2 each clique on the path from C 1 to C 2 in T contains C 1 C 2. Separating pair ogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

9 Definitions Clique tree Definition A clique tree of a connected chordal graph is any tree T whose vertices are the maximal cliques of G such that for every two cliques C 1, C 2 each clique on the path from C 1 to C 2 in T contains C 1 C 2. Separating pair Definition Two maximal cliques C 1, C 2 of G form a separating pair if C 1 C 2 is non-empty, and every path in G from a vertex of C 1 \C 2 to a vertex of C 2 \C 1 contains a vertex of C 1 C 2. ogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

10 Definitions Clique tree Definition A clique tree of a connected chordal graph is any tree T whose vertices are the maximal cliques of G such that for every two cliques C 1, C 2 each clique on the path from C 1 to C 2 in T contains C 1 C 2. Separating pair Definition Two maximal cliques C 1, C 2 of G form a separating pair if C 1 C 2 is non-empty, and every path in G from a vertex of C 1 \C 2 to a vertex of C 2 \C 1 contains a vertex of C 1 C 2. ogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

11 Previous results Main results Theorem A graph G is chordal if and only if every minimal separator of G is a clique. (Golumbic, 2004) [4] Theorem Let G be a chordal graph. The multiset S of minimal separators of vertices of G is the same for every clique tree T of G.(Blair and Peyton, 1992) [1] Theorem A set S is a minimal separator of a chordal graph G if and only if there exist maximal cliques C 1, C 2 forming a separating pair such that S = C 1 C 2. (Habib and Stacho, 2012) [5] Nogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

12 Goals Goal Our goal is to characterize a subclass of chordal graphs by the intersection of minimal separators and forbidden induced subgraphs. Nogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

13 Results Theorem Theorem Let G be a chordal graph and let S = {S 1, S 2,,..., S n } the multiset of minimal separators of G. Then: (i) For every S i, S j S, i j, S i S j = G is (claw, gem)-free. Nogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

14 Results Theorem Theorem Let G be a chordal graph and let S = {S 1, S 2,,..., S n } the multiset of minimal separators of G. Then: (i) For every S i, S j S, i j, S i S j = G is (claw, gem)-free.. Other results:. (ii) (For every S i, S j S, S i S j S i = S j ) G is (dart, gem)-free. (strictly chordal graphs) Markezon and Waga, 2015) [6], [7]. (iii) (For every S i, S j S, i j, S i S j S i S j or S j S i ) G is gem-free. (iv) Π 4 : For every S i, S j S, i j, S i S j S i S j and S j S i. A chordal graph is Π 4 hereditary G is dart-free. (( ) De Caria and Gutiérrez, 2016) [3] Nogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

15 Results Proof S i S j = G is (claw, gem)-free. Nogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

16 Results Proof S i S j = G is (claw, gem)-free. Nogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

17 Results ( ) Suppose that G has a claw or a gem. If it has a claw, let x, y, z, t be the vertices of the claw and let C 1, C 2, C 3 cliques containing {x, y}, {x, z}, {x, t}, respectively. Nogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

18 Results We can have other situations, as But it is analogous. Nogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

19 Results Without loss of generality, we may consider only the case Then x S 1 = C 1 C 2 e x S 2 = C 2 C 3. Then x S 1 S 2. If G has a gem, analogously let C 1, C 2, C 3 be cliques containing {x, y, z}, {x, z, t}, {x, t, w}, respectively. Then the minimal separators are S 1 = C 1 C 2 {xz}, S 2 = C 2 C 3 {xt}. Therefore x S 1 S 2. Nogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

20 Results ( ) Conversely, suppose S 1 S 2. We may assume that S 1 and S 2 are adjacent (if this is not the case, since the intersection is no empty, there exist adjacent cliques that give us the same result). In this case let C 1, C 2, C 3 be cliques such that S 1 = C 1 C 2, S 2 = C 2 C 3 e x S 1 S 2, i.e., x C 1 C 2 C 3. Since the cliques are distinct and maximal then a C 1 \C 2 b C 3 \C 2 C 2 \C 1 C 2 \C 3 We have two cases: Nogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

21 Results Case 1 : c C 2 \(C 1 C 3 ) In this case we have a claw. Nogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

22 Results Case 2: C 2 \(C 1 C 3 ) =. e C 2 \C 1 C 3 d C 2 \C 3 C 1 In this case we have a gem. Nogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

23 Future researchs Future researchs combination of intersections Nogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

24 Future researchs Future researchs combination of intersections minimal separators and Helly property Nogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15

25 References References Blair, J.R.S., Peyton, B., An Introduction to Chordal Graphs an Clique Trees. Graph Theory and Sparc Matrix Computation, IMA 56, De Caria, P., Gutiérrez, M., Introducing subclasses of basic chordal graphs, Eletronic Notes in Discrete Mathematics, 44, 2014, De Caria, P., Gutiérrez, M., On basic chordal graphs and some of its subclasses, Discrete Applied Mahematics, 210, 2016, Golumbic, M. C. Algorithmic Graph Theory and Perfect Graphs (2nd edition), North Holland, (2004). Habib, M., Stacho, L., Reduced clique graphs of chordal graphs, European Journal of Combinatorics, 33, (2012), Markenzon, L., Waga, C.F.E.M.,Strictly Cordal Graphs:Characterization and Linear Time Recongnition, Eletronic Notes in Discrete Mathematics, 52, (2016), Markenzon, L., Waga, C.F.E.M., New results on ptolemaic graphs, Discrete Applied Mathematics, 195,(2015), Nogueira e dos Santos (CEFET-MG, UFV, DCC-UFMG) Characterization of subclasses of chordal graphs November, / 15