View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by espace@Curtin School of Science Department of Mathematics and Statistics Connected Domination Critical Graphs Pawaton Kaemawichanurat This thesis is presented for the Degree of Doctor of Philosophy of Curtin University October 2015 Declaration To the best of my knowledge and belief, this thesis contains no material previously published by any other person except where due acknowledgment has been made. This thesis contains no material which has been accepted for the award of any other degree or diploma in any university. .......................................................... Pawaton Kaemawichanurat October 2015 i Dedicated to my father, Weerapol Kaemawichanurat ii Abstract A dominating set D of a graph G is a vertex subset of V(G) which every vertex of G is either in D or adjacent to a vertex in D. The minimum cardinality of a dominating set is called the domination number of G and is denoted by g(G). A dominating set D of G is called a connected dominating set if the subgraph of G induced by D is connected. The minimum cardinality of a connected dominating set is called the connected dom- ination number of G and is denoted by gc(G). A dominating set D of G is called an independent dominating set if D is also an independent set. The minimum cardinality of an independent dominating set is called the independent domination number of G and is denoted by i(G). A vertex subset D is called a total dominating set if every ver- tex of G is adjacent to a vertex in D. The minimum cardinality of a total dominating set is called the total domination number of G and is denoted by gt(G). A graph G is said to be k − g−edge critical if the domination number g(G) = k and g(G + uv) < k for any pair of non-adjacent vertices u and v of G. For the con- nected domination number gc(G) = k, the total domination number gt(G) = k and the independent domination number i(G) = k, a k − gc−edge critical graph, a k − gt−edge critical graph and a k − i−edge critical graph are similarly defined. In the context of vertex removal, a graph G is said to be k − g−vertex critical if the domination number g(G) = k and g(G − v) < k for any vertex v of G.A k − gc−vertex critical graph, a k − i−vertex critical graph, are similarly defined. Moreover, a graph G is said to be k − gt−vertex critical if gt(G) = k and gt(G − v) < k for any vertex v of G which is not adjacent to a vertex of degree one. In this thesis, we investigate the intersection between the classes of critical graphs with respect to different domination numbers. We show that the class of connected k − gt−edge critical graphs is identical to the class of connected k − gc−edge critical graphs if and only if k = 3 or 4. In addition, for the vertex critical case, we prove that the class of 2−connected k − gt−vertex critical graphs is identical to the class of 2−connected k − gc−vertex critical graphs if and only if k = 3 or 4. Moreover, in the class of claw-free graphs, we show that every k − g−edge critical graph is also a k − i−edge critical graph and vice versa. We also have an analogous result for k − iii g−vertex critical graphs and k − i−vertex critical graphs. For k − gc−vertex critical graphs, we establish the order of k − gc−vertex critical graphs in terms of maximum degree D and k. We prove that D + k ≤ n ≤ (D − 1)(k − 1) + 3 and the upper bound is sharp for all integer k ≥ 3 when D is even. It has been proved that every k −gc−vertex critical graph achieving the upper bound is D−regular for k = 2 or 3. For k = 4, we prove that every 4 − gc−vertex critical graph achieving the upper bound is D−regular. We further show that, for k = 2;3 or 4, there exists a k − gc−vertex critical graph of order (D − 1)(k − 1) + 3 if and only if D is even. For k ≥ 5, we show that if G is a k − gc−vertex critical graph of smallest possible order, namely D + k, then G is isomorphic to a cycle of length k + 2. We also establish the realizability of k − gc−vertex critical graphs of maximum degree D whose order is between the bounds when D and k are small. For maximal k −gc−vertex critical graphs (the k −gc−vertex critical graphs whose connected domination number is decreased after any single edge is added), we char- acterize some classes for k = 3. More specifically, we prove that every even order maximal 3 − gc−vertex critical graph is bi-critical. If the order is odd, then every maximal 3 − gc−vertex critical graph is 3−factor critical with exactly one exception. We investigate the hamiltonian properties of k − D−edge critical graphs where D 2 fgc;gt;ig. We prove that if k = 1;2 or 3, then every 2−connected k − gc−edge critical graph is hamiltonian. We provide a class of l−connected k − gc−edge critical n−3 non-hamiltonian graphs for k ≥ 4 and 2 ≤ l ≤ k−1 . Thus, for n ≥ l(k − 1) + 3, the class of l−connected k − gc−edge critical non-hamiltonian graphs of order n is empty if and only if k = 1;2 or 3. In addition, for k − gt−edge critical graphs, we show that these graphs are hamiltonian when k = 2 or 3 and we provide classes of 2−connected k − gt−edge critical non-hamiltonian graphs for k = 4 or 5. For k − i−edge critical graphs, we give a construction for a class of 2−connected non-hamiltonian graphs for k ≥ 3. Further, we investigate on the hamiltonian properties of k−D−edge critical graphs where D 2 fgc;gt;g;ig when the graphs are claw-free. We prove that every 2−connected 4 − gc−edge critical claw-free graph is hamiltonian and show that the claw-free con- dition cannot be relaxed. We further prove that the class of k − gc−edge critical claw- free non-hamiltonian graphs of connectivity two is empty if and only if k = 1;2;3 or 4. We show that every 3−connected k − gc−edge critical claw-free graph is hamilto- nian for 1 ≤ k ≤ 6. For k − gt−edge critical graphs, we show that every 3−connected k − gt−edge critical claw-free graph is hamiltonian for 2 ≤ k ≤ 5. We also show that iv every 3−connected 4 − D−edge critical claw-free graph where D 2 fg;ig is hamilto- nian. v Acknowledgements The research reported in this thesis has been carried out from July 2011 to October 2015. During this period, I was a PhD student in the Department of Mathematics and Statistics, Curtin University. I wish to acknowledge the financial support of Development and Promotion of Science and Technology Talents Project Scholarship for my PhD study. I would like to express my thanks to my supervisor, Professor Louis Caccetta, for his valuable suggestions, illumination on doing research, encouragement and supervi- sion throughout the past four years with remarkable patience and enthusiasm. I would like to thank Associate Professor Nawarat Ananchuen of Silpakorn Uni- versity as well as Associate Professor Watcharaphong Ananchuen of Sukhothai Tham- mathirat Open University for their encouragement and valuable advice during my PhD study. I would like to give thanks to all my colleages and Thai friends for their support and friendship particularly, Dr. Wilaiporn Paisan, Dr. Rinrada Thamchai, Dr. Nattakorn Phewchean, Yaowanuch Raksong, Dr. Nathnarong Khajohnsaksumeth and Elayaraja Aruchunan as well as Lana Dewar and friends from Guild House who make me feel living in the second home. I thank all of the staff in the Department of Mathematics and Statistics for contribut- ing to a friendly working environment. The academic staffs Dr. Nihal Yatawara, Dr. Samy El-Batanouny and Peter Mackinnon and the administrative staffs Joyce Yang, Shuie Liu, Jeannie Darmago and Cheryl Cheng, deserve special thanks for providing kind and professional help on numerous occasions. Finally, on a more personal note, I sincerely thank my dad, my first mathematics teacher, for everything that he has given over 32 years. I am proud to be your son. My mom and Thitima Jiravidhyakul for their understanding and support during my entire period of my PhD candidature in Australia. vi Contents 1 Introduction 1 1.1 Background . 1 1.2 Terminology . 4 1.3 Summary . 8 2 Literature Review 10 2.1 Domination Numbers . 10 2.2 Hamiltonian Graphs . 15 3 Critical Graphs with respect to Domination Numbers 21 3.1 gc−Critical Graphs and gt−Critical Graphs . 22 3.2 g−Critical Graphs and i−Critical Graphs . 32 4 Bounds on the Order of Connected Domination Vertex Critical Graphs 34 4.1 Introduction . 34 4.2 The Upper and Lower Bounds of k − gc−Vertex Critical Graphs . 36 4.3 The k − gc−Vertex Critical Graphs achieving the Upper Bound . 42 4.4 The k − gc−Vertex Critical Graphs achieving the Lower Bound . 51 4.4.1 The First Proof of Lemma 4.4.1 . 53 4.4.2 The Second Proof of Lemma 4.4.1 . 54 4.5 Vertex Critical Graphs of Prescribed Order .