Influence of the edge subdivision on the convex domination number

Transcription

1 Influence of the edge subdivision on the convex domination number Magda Dettlaff, Magdalena Lemańska Faculty of Applied Physics and Mathematics, Gdansk University of Technology, Poland September 18-23, 2011, Szklarska Poreba

2 Basic definitions Domination number

3 Basic definitions Domination number A set D V is a dominating set of G if N G [D] = V.

4 Basic definitions Domination number A set D V is a dominating set of G if N G [D] = V. The domination number of G, denoted γ(g), is the minimum cardinality of a dominating set in G.

5 Basic definitions Domination number A set D V is a dominating set of G if N G [D] = V. The domination number of G, denoted γ(g), is the minimum cardinality of a dominating set in G. Geodesic path

6 Basic definitions Domination number A set D V is a dominating set of G if N G [D] = V. The domination number of G, denoted γ(g), is the minimum cardinality of a dominating set in G. Geodesic path The distance d G (u, v) between two vertices u and v in a connected graph G is the length of the shortest (u v) path in G.

7 Basic definitions Domination number A set D V is a dominating set of G if N G [D] = V. The domination number of G, denoted γ(g), is the minimum cardinality of a dominating set in G. Geodesic path The distance d G (u, v) between two vertices u and v in a connected graph G is the length of the shortest (u v) path in G. A (u v) path of length d G (u, v) is called (u v)-geodesic.

8 Convex domination number Definition

9 Convex domination number Definition A set X is convex in G if vertices from all (a b)-geodesics belong to X for every two vertices a, b X.

10 Convex domination number Definition A set X is convex in G if vertices from all (a b)-geodesics belong to X for every two vertices a, b X. A set X is a convex dominating set if it is convex and dominating.

11 Convex domination number Definition A set X is convex in G if vertices from all (a b)-geodesics belong to X for every two vertices a, b X. A set X is a convex dominating set if it is convex and dominating. The convex domination number γ con(g) of a graph G is the minimum cardinality of a convex dominating set.

12 Convex domination number Definition A set X is convex in G if vertices from all (a b)-geodesics belong to X for every two vertices a, b X. A set X is a convex dominating set if it is convex and dominating. The convex domination number γ con(g) of a graph G is the minimum cardinality of a convex dominating set. The convex domination number was first introduced by Jerzy Topp, (2002) and studied in [1], [3], [4].

13 Example Example Figure: Graph G.

14 Edge subdivision Edge subdivision

15 Edge subdivision Edge subdivision The subdivision of some edge e = uv in a graph G yields to a graph containing one new vertex w, and with an edge set replacing e by two new edges with endpoints uw and wv.

16 Edge subdivision Edge subdivision The subdivision of some edge e = uv in a graph G yields to a graph containing one new vertex w, and with an edge set replacing e by two new edges with endpoints uw and wv. We denote by G uv or G e graph obtained from a graph G by subdivision of an edge e = uv in G.

17 Theorem 1

18 Theorem 1 The difference between γ con(g) and γ con(g uv ) and between γ con(g uv ) and γ con(g) can be arbitrarily large.

19 Theorem 1 The difference between γ con(g) and γ con(g uv ) and between γ con(g uv ) and γ con(g) can be arbitrarily large. Firstly we show that γ con(g uv ) γ con(g) can be arbitrarily large. We show that for any positive integer k there exists a graph G such that γ con(g uv ) γ con(g) = k.

20 Theorem 1 The difference between γ con(g) and γ con(g uv ) and between γ con(g uv ) and γ con(g) can be arbitrarily large. Firstly we show that γ con(g uv ) γ con(g) can be arbitrarily large. We show that for any positive integer k there exists a graph G such that γ con(g uv ) γ con(g) = k. Secondly we show that the difference γ con(g) γ con(g uv ) can be arbitrarily large. We show that for any positive integer k there exists a graph G such that γ con(g) γ con(g uv ) = k.

21 First case Exemple u v u v Figure: Graphs G and G uv for k = 4.

22 Second case Example v 2 u v 1 w 1 w 2 v 3 w 3 z v 2 u v 1 w w 1 w 2 v 3 w 3 z Figure: Graphs G and G uw1 for k = 3.

23 Unicyclic graphs and trees

24 Unicyclic graphs and trees Observation 1 If G is an unicyclic graph with the unique cycle C and if D is a minimum convex dominating set of G, then at most two vertices of C does not belong to D.

25 Unicyclic graphs and trees Observation 1 If G is an unicyclic graph with the unique cycle C and if D is a minimum convex dominating set of G, then at most two vertices of C does not belong to D. Theorem 2 If G is an unicyclic graph with the unique cycle C, then γ con(g) γ con(g uv ) γ con(g) + 3.

26 Unicyclic graphs and trees Corollary 3 Let G be a unicyclic graph with the only cycle C and let D be a minimum convex dominating set of G. If e is not a cyclic edge, then γ con(g e) = γ con(g) + 1; If V (C) D, then for every edge e E(G) we obtain γ con(g e) = γ con(g) + 1; If there is v V (C) such that v D, then C = C 3 or C = C 4 or C = C 5 : If C = C 3 and V (C) D = 1, then for every edge e E(G) we obtain γ con(g e) = γ con(g) + 1. If V (C) D = 2, let us say x, y, z V (C) and x, y D, then γ con(g e) = γ con(g) + 1 for e = xy and γ con(g e) = γ con(g) for e = xz and e = yz; If C = C 4, then V (C) D = 2 and for every edge e E(G) we obtain γ con(g e) = γ con(g) + 1; If C = C 5, then V (C) D = 3 and for every edge e V (C) we obtain γ con(g e) = γ con(g) + 3.

27 The convex domination number increases after subdividing an edge by exactly one.

28 The convex domination number increases after subdividing an edge by exactly one. Proposition 4 If T is a tree of order at least three, then for every edge of T is γ con(t uv ) = γ con(t ) + 1.

29 The convex domination number increases after subdividing an edge by exactly one. Proposition 4 If T is a tree of order at least three, then for every edge of T is γ con(t uv ) = γ con(t ) + 1. Observation 2 Let G be a connected graph with δ(g) = 1 and let u be an end-vertex of G which is adjacent to v. Then γ con(g uv ) = γ con(g) + 1.

30 The convex domination number increases after subdividing an edge by exactly one. Proposition 4 If T is a tree of order at least three, then for every edge of T is γ con(t uv ) = γ con(t ) + 1. Observation 2 Let G be a connected graph with δ(g) = 1 and let u be an end-vertex of G which is adjacent to v. Then γ con(g uv ) = γ con(g) + 1. Observation 3 Let G be a connected graph with (G) = n(g) 1. Then γ con(g uv ) = γ con(g) + 1 for any edge uv E(G).

31 The convex domination number increases after subdividing an edge by exactly one. Proposition 4 If T is a tree of order at least three, then for every edge of T is γ con(t uv ) = γ con(t ) + 1. Observation 2 Let G be a connected graph with δ(g) = 1 and let u be an end-vertex of G which is adjacent to v. Then γ con(g uv ) = γ con(g) + 1. Observation 3 Let G be a connected graph with (G) = n(g) 1. Then γ con(g uv ) = γ con(g) + 1 for any edge uv E(G). Theorem 5 [1] If G is a connected graph with δ(g) 2 and g(g) 6, then γ con(g) = n(g).

32 The convex domination number increases after subdividing an edge by exactly one. Proposition 4 If T is a tree of order at least three, then for every edge of T is γ con(t uv ) = γ con(t ) + 1. Observation 2 Let G be a connected graph with δ(g) = 1 and let u be an end-vertex of G which is adjacent to v. Then γ con(g uv ) = γ con(g) + 1. Observation 3 Let G be a connected graph with (G) = n(g) 1. Then γ con(g uv ) = γ con(g) + 1 for any edge uv E(G). Theorem 5 [1] If G is a connected graph with δ(g) 2 and g(g) 6, then γ con(g) = n(g). Corollary 6 If G is a connected graph with δ(g) 2 and g(g) 6, then γ con(g uv ) = γ con(g) + 1 for any edge uv E(G).

33 The convex domination number increases after subdividing an edge by exactly one.

34 The convex domination number increases after subdividing an edge by exactly one. Theorem 7 If G is a connected graph with g(g) 6, then γ con(g) = n(g) n 1 (G).

35 The convex domination number increases after subdividing an edge by exactly one. Theorem 7 If G is a connected graph with g(g) 6, then γ con(g) = n(g) n 1 (G). Corollary 8 If g(g) 6, then γ con(g uv ) = γ con(g) + 1 for any edge uv E(G).

36 Lemma

37 Lemma Lemma 9 For any integer number k 3 there exists a graph G such that for any edge e E(G) we have γ con(g e) = γ con(g) + k.

38 Lemma Lemma 9 For any integer number k 3 there exists a graph G such that for any edge e E(G) we have γ con(g e) = γ con(g) + k. Construction u 1 u 2 u 3 u 4 u k 3 u k 2 u k 1 v 1 v 2 v 3 v 4 v k 3 v k 2 v k 1 v k Figure: Graph H k

39 Construction

40 Construction Graph G k w 1 w 2 w 3

41 Summary

42 Summary We give the examples of graphs for which for any edge e is γ con(g e) = γ con(g) + k for k = 1 or k 3. The existence of graphs G for which for any edge e is γ con(g e) = γ con(g) + k where k < 1 or k = 0 or k = 2 remains as an open problem.

43 Summary We give the examples of graphs for which for any edge e is γ con(g e) = γ con(g) + k for k = 1 or k 3. The existence of graphs G for which for any edge e is γ con(g e) = γ con(g) + k where k < 1 or k = 0 or k = 2 remains as an open problem. γ con(g e) = γ con(g) 1 for any edge e G Figure: Graph G

44 References 1. J. Cyman, M. Lemanska, J. Raczek, Graphs with convex domination number close to their order, Disscusioness Mathematicae Graph Theory 26 (2006). 2. T.Haynes, S.Hedetniemi, P.Slater, Fundamentals of domination in graphs, Marcel Dekker, Inc. (1998). 3. M. Lemańska, Weakly convex and convex domination numbers, Opuscula Mathematica 24/2 (2004), J. Raczek, NP-completeness of weakly convex and convex dominating set decision problems, Opuscula Mathematica 24 (2004),