Balloons, Cut-edges, Matchings, and Total domination in Regular graphs of Odd Degree

Transcription

1 Balloons, Cut-edges, Matchings, and Total domination in Regular graphs of Odd Degree Suil O and Douglas B. West Department of Mathematics University of Illinois at Urbana-Champaign 47th Midwest Graph Theory Conference, 2008

2 Table of Contents Previous Results Previous Results

3 Previous Results - the Questions In 1891, Petersen proved that every cubic graph without cut-edges has a perfect matching.

4 Previous Results - the Questions In 1891, Petersen proved that every cubic graph without cut-edges has a perfect matching. If there are cut-edges in a cubic graph, then what will happen?

5 Previous Results - the Questions In 1891, Petersen proved that every cubic graph without cut-edges has a perfect matching. If there are cut-edges in a cubic graph, then what will happen? More generally, we consider connected (2r + 1)-regular graphs.

6 Previous Results - the Questions In 1891, Petersen proved that every cubic graph without cut-edges has a perfect matching. If there are cut-edges in a cubic graph, then what will happen? More generally, we consider connected (2r + 1)-regular graphs. First, how many cut-edges can a connected (2r + 1)-regular graph with n vertices have?

7 Previous Results - the Questions In 1891, Petersen proved that every cubic graph without cut-edges has a perfect matching. If there are cut-edges in a cubic graph, then what will happen? More generally, we consider connected (2r + 1)-regular graphs. First, how many cut-edges can a connected (2r + 1)-regular graph with n vertices have? Second, how small can the matching number α (G) be in a connected (2r + 1)-regular graph with n vertices?

8 Previous Results - the Questions In 1891, Petersen proved that every cubic graph without cut-edges has a perfect matching. If there are cut-edges in a cubic graph, then what will happen? More generally, we consider connected (2r + 1)-regular graphs. First, how many cut-edges can a connected (2r + 1)-regular graph with n vertices have? Second, how small can the matching number α (G) be in a connected (2r + 1)-regular graph with n vertices? Third, can we characterize when equality holds? (i.e. when do we have the smallest matching number?)

9 The Tools - Balloons and B r B 1, and a cubic graph with two balloons A balloon in a graph G is a maximal 2-edge-connected subgraph of G that is incident to exactly one cut-edge of G.

10 The Tools - Balloons and B r B 1, and a cubic graph with two balloons A balloon in a graph G is a maximal 2-edge-connected subgraph of G that is incident to exactly one cut-edge of G. B r : the unique graph with 2r + 3 vertices having 2r + 2 vertices of degree 2r + 1 and one vertex of degree 2r. (B r = the complement of P 3 + rk 2 )

11 The Tools - Balloons and B r B 1, and a cubic graph with two balloons A balloon in a graph G is a maximal 2-edge-connected subgraph of G that is incident to exactly one cut-edge of G. B r : the unique graph with 2r + 3 vertices having 2r + 2 vertices of degree 2r + 1 and one vertex of degree 2r. (B r = the complement of P 3 + rk 2 ) B r is the smallest possible balloon in (2r + 1)-regular graphs.

12 The construction - T r and H r Graphs in T 1 and H 1, respectively T r : the family of trees such that every non-leaf vertex has degree 2r + 1 and all the leaves have the same color in a proper 2-coloring.

13 The construction - T r and H r Graphs in T 1 and H 1, respectively T r : the family of trees such that every non-leaf vertex has degree 2r + 1 and all the leaves have the same color in a proper 2-coloring. H r : the family of (2r + 1)-regular graphs obtained from trees in T r by identifying each leaf of such a tree with the vertex of degree 2r in a copy of B r.

14 Previous Results - Cut-edges and Matchcings Theorem ( 2008+) Every n-vertex (2r + 1)-regular graph has at most rn (2r2 +4r+1) 2r 2 +2r 1 cut-edges, which reduces to n 7 3 for cubic graphs. Equality holds infinitely often.

15 Previous Results - Cut-edges and Matchcings Theorem ( 2008+) Every n-vertex (2r + 1)-regular graph has at most rn (2r2 +4r+1) 2r 2 +2r 1 cut-edges, which reduces to n 7 3 for cubic graphs. Equality holds infinitely often. Theorem (Henning and Yeo 2007, 2008+) If G F n, then α (G) n 2 r 2 G H r. (2r 1)n+2 (2r+1)(2r 2 +2r 1), with equality for

16 Previous Results - Cut-edges and Matchcings Theorem ( 2008+) Every n-vertex (2r + 1)-regular graph has at most rn (2r2 +4r+1) 2r 2 +2r 1 cut-edges, which reduces to n 7 3 for cubic graphs. Equality holds infinitely often. Theorem (Henning and Yeo 2007, 2008+) If G F n, then α (G) n 2 r 2 G H r. Theorem ( 2008+) (2r 1)n+2 (2r+1)(2r 2 +2r 1), with equality for G has the smallest matching number over all connected (2r + 1)- regular graphs if and only if G is in H r.

17 Simple Approach to Previous Results and New Results By maximizing the number of balloons in a connected (2r + 1)-regular graph with n vertices, the maximum number of cut-edges and minimum size of a maximum matching in such graphs were determined, and the graphs with the smallest maximum matchings were characterized.

18 Simple Approach to Previous Results and New Results By maximizing the number of balloons in a connected (2r + 1)-regular graph with n vertices, the maximum number of cut-edges and minimum size of a maximum matching in such graphs were determined, and the graphs with the smallest maximum matchings were characterized. Balloons also help in the study of the total domination number γ t (G).

19 Main Result Previous Results A vertex subset S is a dominating set in a graph G if every vertex outside S has a neighbor in S.

20 Main Result A vertex subset S is a dominating set in a graph G if every vertex outside S has a neighbor in S. A vertex subset S is a total dominating set in a graph G if every vertex in V (G) has a neighbor in S. The total domination number,denoted γ t (G), is the minimum size of a total dominating set.

21 Main Result A vertex subset S is a dominating set in a graph G if every vertex outside S has a neighbor in S. A vertex subset S is a total dominating set in a graph G if every vertex in V (G) has a neighbor in S. The total domination number,denoted γ t (G), is the minimum size of a total dominating set. Let b(g) denote the number of balloons in a graph G.

22 Main Result A vertex subset S is a dominating set in a graph G if every vertex outside S has a neighbor in S. A vertex subset S is a total dominating set in a graph G if every vertex in V (G) has a neighbor in S. The total domination number,denoted γ t (G), is the minimum size of a total dominating set. Let b(g) denote the number of balloons in a graph G. Main Result : For 3-regular graphs, we improve the known bound γ t (G) α (G) by showing that γ t (G) α (G) b(g)/6 (except possibly when b(g) 3).

23 Tools for the domination result Lemma (Henning 2007) If G is a graph with n vertices and m edges, then γ t (G) n m (G).

24 Tools for the domination result Lemma (Henning 2007) If G is a graph with n vertices and m edges, then γ t (G) n m (G). Lemma ( 2008+) If B is a balloon with p vertices in a cubic graph G, then γ t (B) p 1 2. Also, B has a dominating (p 1)/2-set that contains the neck of B and a neighbor of every vertex other than the neck.

25 Tools for the domination result Lemma (Henning 2007) If G is a graph with n vertices and m edges, then γ t (G) n Lemma ( 2008+) m (G). If B is a balloon with p vertices in a cubic graph G, then γ t (B) p 1 2. Also, B has a dominating (p 1)/2-set that contains the neck of B and a neighbor of every vertex other than the neck. By Henning s Lemma, γ t (B) p 3(p 1)/2 3 = p/2 + 1/6 = (p 1)/2.

26 Tools for the domination result Lemma (Henning 2007) If G is a graph with n vertices and m edges, then γ t (G) n Lemma ( 2008+) m (G). If B is a balloon with p vertices in a cubic graph G, then γ t (B) p 1 2. Also, B has a dominating (p 1)/2-set that contains the neck of B and a neighbor of every vertex other than the neck. By Henning s Lemma, γ t (B) p 3(p 1)/2 3 = p/2 + 1/6 = (p 1)/2. For the set containing the neck v, apply Henning s Lemma to B N B [v] if (B N B [v]) = 3. If (B v B [v]) 2,then there are two special graphs (We skip this case).

27 Examples of dominating sets in balloons

28 Definitions Previous Results A dominating set S in a graph G is a semitotal dominating set (abbreviated SD-set) if every vertex with maximum degree in G has a neighbor in S.

29 Definitions Previous Results A dominating set S in a graph G is a semitotal dominating set (abbreviated SD-set) if every vertex with maximum degree in G has a neighbor in S. A hypergraph H is k-uniform if every edge in H has size k. The transversal number τ(h) of a hypergraph H is the minimum size of a set of vertices that intersects every edge.

30 Construction of semitotal dominating sets Theorem (Chvátal and McDiarmid 1992) If H is a k-uniform hypergraph with n vertices and m edges, then τ(h) k/2 m+n 3k/2.

31 Construction of semitotal dominating sets Theorem (Chvátal and McDiarmid 1992) If H is a k-uniform hypergraph with n vertices and m edges, then τ(h) k/2 m+n 3k/2. Corollary ( 2008+) If G is an n-vertex graph in which every vertex has degree 2r + 1 or 2r, then G has an SD-set of size at most rn+n 3r+1.

32 Construction of semitotal dominating sets Theorem (Chvátal and McDiarmid 1992) If H is a k-uniform hypergraph with n vertices and m edges, then τ(h) k/2 m+n 3k/2. Corollary ( 2008+) If G is an n-vertex graph in which every vertex has degree 2r + 1 or 2r, then G has an SD-set of size at most rn+n 3r+1. (Proof) Form a (2r + 1)-uniform hypergraph H to get V (H) = V (G ) by letting the edges be the open nbhds of vertices with degree 2r + 1 and the closed nbhds of vertices with degree 2r. A transversal T has a neighbor of every vertex with degree 2r + 1 and dominates every vertex with degree 2r.Thus,T is SD-set in G.

33 Construction of semitotal dominating sets Theorem ( 2008+) If G is a connected n-vertex graph with maximum degree at most 3, and n > 1, then G has an SD-set of size at most n/2.

34 Construction of semitotal dominating sets Theorem ( 2008+) If G is a connected n-vertex graph with maximum degree at most 3, and n > 1, then G has an SD-set of size at most n/2. (Idea of Proof) Basis step : When (G ) < 3, an ordinary dominating set suffices.

35 Construction of semitotal dominating sets Theorem ( 2008+) If G is a connected n-vertex graph with maximum degree at most 3, and n > 1, then G has an SD-set of size at most n/2. (Idea of Proof) Basis step : When (G ) < 3, an ordinary dominating set suffices. If (G ) = 3 and δ(g ) 2, then the prevous theorem provides the desired SD-set.

36 Construction of semitotal dominating sets Theorem ( 2008+) If G is a connected n-vertex graph with maximum degree at most 3, and n > 1, then G has an SD-set of size at most n/2. (Idea of Proof) Basis step : When (G ) < 3, an ordinary dominating set suffices. If (G ) = 3 and δ(g ) 2, then the prevous theorem provides the desired SD-set. If (G ) = 3 and d(u) = 1, then delete u and its neighbor v and apply the induction hypothesis. In each of three cases, adding u or v completes the desired SD-set.

37 Example on SD-set in G The red vertices form an SD-set of size (n 1)/2

38 The effectiveness of Semitotal Dominating sets Main Theorem ( 2008+) If G is a connected cubic graph with n vertices, then γ t (G) n 2 b(g) 2 (except that γ t (G) n/2 1 when b(g) = 3 and the three balloons have a common neighbor), and this is sharp for all even values of b(g).

39 The effectiveness of Semitotal Dominating sets Main Theorem ( 2008+) If G is a connected cubic graph with n vertices, then γ t (G) n 2 b(g) 2 (except that γ t (G) n/2 1 when b(g) = 3 and the three balloons have a common neighbor), and this is sharp for all even values of b(g). (Idea of Proof) Form G by deleting all vertices in balloons in G.

40 The effectiveness of Semitotal Dominating sets Main Theorem ( 2008+) If G is a connected cubic graph with n vertices, then γ t (G) n 2 b(g) 2 (except that γ t (G) n/2 1 when b(g) = 3 and the three balloons have a common neighbor), and this is sharp for all even values of b(g). (Idea of Proof) Form G by deleting all vertices in balloons in G. If G has more than one vertex, then we apply the previous theorem to obtain an SD-set S in G of size at most n /2.

41 SD-set in the reduced graph G 9 vertices, SD-set of size 4

42 Add dominating sets in balloons to SD-set in G G Neck of balloon is in S if and only if neighbor is in SD-set in G. γ t (G) n /2 + b(g) i=1 (p i 1)/2

43 Exceptional case and Sharpness example If G = K 1, then G consists of three balloons and their common neighbor. Sharpness : Let G be a cycle plus a pendant edge at each vertex (in G, two balloons are adjacent to each leaf of G ). Then γ t (G) = n/2 b(g)/2.

44 Improvement of Earlier Result Theorem (Henning, Kang, Shan and Yeo 2008) When G is regular with degree at least 3, γ t (G) α (G).

45 Improvement of Earlier Result Theorem (Henning, Kang, Shan and Yeo 2008) When G is regular with degree at least 3, γ t (G) α (G). Corollary ( 2008+) If G is a connected n-vertex cubic graph, then γ t (G) α (G) b(g)/6, except when b(g) = 3 and there is exactly one vertex outside the balloons, in which case still γ t (G) α (G).

46 Improvement of Earlier Result Theorem (Henning, Kang, Shan and Yeo 2008) When G is regular with degree at least 3, γ t (G) α (G). Corollary ( 2008+) If G is a connected n-vertex cubic graph, then γ t (G) α (G) b(g)/6, except when b(g) = 3 and there is exactly one vertex outside the balloons, in which case still γ t (G) α (G). (Idea of Proof) γ t (G) n/2 b/2.

47 Improvement of Earlier Result Theorem (Henning, Kang, Shan and Yeo 2008) When G is regular with degree at least 3, γ t (G) α (G). Corollary ( 2008+) If G is a connected n-vertex cubic graph, then γ t (G) α (G) b(g)/6, except when b(g) = 3 and there is exactly one vertex outside the balloons, in which case still γ t (G) α (G). (Idea of Proof) γ t (G) n/2 b/2. n/2 b/3 α (G) (This is the way we proved the lower bound on α (G).)

48 Improvement of Earlier Result Theorem (Henning, Kang, Shan and Yeo 2008) When G is regular with degree at least 3, γ t (G) α (G). Corollary ( 2008+) If G is a connected n-vertex cubic graph, then γ t (G) α (G) b(g)/6, except when b(g) = 3 and there is exactly one vertex outside the balloons, in which case still γ t (G) α (G). (Idea of Proof) γ t (G) n/2 b/2. n/2 b/3 α (G) (This is the way we proved the lower bound on α (G).) Exceptional case (α (G) n/2 1 γ t (G)).

49 Previous Results What is the smallest matching number for the family of t-edge-connected (2r + 1)-regular graphs with n vertices? (Also for 2r-regular graphs.)

50 Previous Results What is the smallest matching number for the family of t-edge-connected (2r + 1)-regular graphs with n vertices? (Also for 2r-regular graphs.) What is the best upper bound for the length of chinese postman tours in t-edge-conneced (2r + 1)-regular graphs with n vertices?