RICE UNIVERSITY ON CHARACTERIZING GRAPHS WITH BRANCHWIDTH AT MOST FOUR by Kymb erly Dawn Riggins A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree Master of Science Approved Thesis Committee Dr Nathaniel Dean Chairman ciate Professor Asso Computational and Applied Mathematics Dr Richard A Tapia Noah Harding Professor Computational and Applied Mathematics Dr Matthew DeVos Visiting Assistan t Professor Computational and Applied Mathematics Houston Texas April ON CHARACTERIZING GRAPHS WITH BRANCHWIDTH AT MOST FOUR Kymberly Dawn Riggins Abstract There are several ways in which we can characterize classes of graphs One such way of classifying graphs is by their branchwidth In working to characterize the class of graphs with branchwidth at most four we have found a set of reductions that reduces members of to the zero graph We have also computed several pla rs of the obstruction set O for graphs with branchwidth at most four nar membe This thesis will summarize previous results on branchwidth and reveal the previously mentioned new results Acknowledgments I would like to thank my committee members My advisor Dr Nathaniel Dean for his b elief and p ersistence inpushingmetodowhatheknew I could do I appreciate your honesty and candor as well as your exp ectation that I p erform at a level ab ove what I am used to Dr Richard Tapia for his guidance and inspirational and thought provoking comments Words can not express how much of a p ositive impact you have made on my life Dr Matthew Devos for his willingness to be on my committee and enthusiasm while doing so I also want to thank former graduate students now tenure track professors Dr IllyaV Hicks and William C Christian Their constantnudging and fatherly concern has b een noted and appreciated and has had a ma jor role in who Iamtoday I would also like to acknowledge my parents grandparents greatgrandparents and extended family for the sacrices they made so that I can be who I am Who I am and what I will accomplish is due to your love guidance supp ort and investment I am eternally grateful Last but not least I would like to thank my Lord and Savior Jesus Christ for of me so that I can p erform to completion the talents and gifts He has placed inside the tasks He has assigned to me iv This research work was supp orted by the following National Science Foundation Fellowships Diversity Graduate Program for Science and Engineering and the GK Fellows Program Table of Contents Abstract ii Acknowledgments iii List of Figures vii Introduction Preliminaries Motivation Branchwidth Treewidth and Branchwidth Denitions Tangles and Separations Previous Branchwidth Results Obstruction Sets Obstruction Sets and Treewidth Obstruction Sets for Bounded Branchwidth Graphs Obstruction Sets for Other Classes of Graphs Computing Obstruction Sets Reductions Reduction Rules for Graphs with Branchwidth at Most Three Reduction Rules for Graphs with Branchwidth At Most Four Memb ers of O vi Planar Members of O Results Conclusion and Future Work Analysis of the Results Potential NonPlanar Members of O Bibliography List of Figures A graph and a corresp onding tree decomp osition K and a corresp onding branch decomp osition Middle set example A separation of the Petersen graph A Tangle of order ofthe Cub e An example Halin graph All complete graphs are chordal graphs H O Reduction Rules for Graphs with G Reduction Rules r r r Reduction Rules R nd r and r r a Planar Obstruction Planar Obstruction Planar Obstruction Planar Obstruction Planar Obstruction viii Planar Obstruction Planar Obstruction Planar Obstruction Planar Obstruction Planar Obstruction Planar Obstruction Planar Obstruction Planar Obstruction Do decahedron Icosahedron Triplex Tangle of order for b oth Basket and Triplex Triplex and a branch decomp osition of width Basket and a branch decomp osition of width Chapter Intro duction Life is full of discrete optimization problems waiting to b e solved on real life instances that can b e mo deled on graphs There are those problems for which no solution exists but fortunately there are some of lifes problems that do actually have solutions Unfortunately for any given instance of the solvable problems not all of them can b e solved in a reasonable amountof timeon it The question then b ecomes For which instances can lifes really dicult yet solvable problems be solved in a reasonable amountof time A tree is one of those sp ecial types of graph whose structure makes it easier to solve extremely dicult problems on If a graph can be easily decomp osed into a tree like graph then a dicult graph problem can be solved on that graph in a erty that can indicate how reasonable amount of time Branchwidth is a graph prop close a graph is to b eing a tree The lower the branchwidth of a graph the closer the graph is to a tree Thus characterizing which graphs have small branchwidth would be advantageous to those who need to determine how long it would take to solve a NPhard graph problem This pap er will summarize the work done in that area as well as provide additional work that furthers this eld of interest b ehind this research as well as denitions and Chapter includes the motivation imp ortant concepts needed to pro ceed Chapter fo cuses on the graph prop erty branchwidth Chapter covers obstruction sets and the use of obstruction sets to characterize graph classes Chapter introduces reductions that can be used to reduce a graph with branchwidth at most four down to the empty graph Chapter introduces members of the obstruction set of graphs with branchwidth at most four Chapter isthe conclusion and includes future work and plans Preliminaries We consider a graph G V GEG V E to b e a collection of n no des and m edges where n is the order and m is the size of the graph Eachedgee E G can be represented as an unordered pair e u v In this case u and v are considered to be adjacent to each other and e is incident with b oth u and v A graph G is simple if no no de is adjacent to itself and at most one edge has the same two ends A graph is connected if and only if it has a path between every pair of no des If there exists two no des in a graph for which there is no path between them than that graph is disconnected A graph is k connected if at least k no des must be removed to disconnect the graph A graph is k edgeconnected if at least k edges of order n has myst b e removed to disconnect the graph The complete graph K n every pair of distinct no des adjacent to one another A subgraph G V E of G V E is a graph for which V V and E E A clique is a complete graph that is a prop er subgraph of a graph The class of graph problems for which a solution to each problem can be veried eciently ie in p olynomial time is nondeterministic p olynomial time solvable Problems in this class are often referred to as problems in NP If an algorithm is olynomial time than known that actually nds the solution to a NP problem in p that particular problem is considered to be deterministic p olynomial time solvable these problems belong to a subset of NP called P NPhard problems are those problems which are lab eled as inherently dicult If a problem is NPhard but is also in NP than that problem b elongs to the NPcomplete set Motivation Fortunately many graph problems which are NPhard on general graphs can b e solved in p olynomial and even linear time on graphs with b ounded branchwidth Even though determining the branchwidth of a arbitrary graph has b een proven to b e NP hard we do know that for xed k determining whether or not a graph has branchwidth at most k is p olynomial Unfortunately the pro of of that result do es not provide a way to construct such an algorithm but it merely proves existence of one From the work of Bo dlander and Thilikos we know that an eective decidable algorithm to determine membership in the class of graphs with branchwidth at most k can b e constructed and solved in linear time This algorithm is highly exp onential in a p olynomial in k and thus is quite impractical for actual implementation and use Being as such other ways are needed to characterize graphs with branchwidth at most k Although there are several dierent ways to characterize a graph class we have chosen to fo cus our attention on the concept of using obstruction sets to characterize graph classes We have chosen