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Hypercubes, Median Graphs and Products of Graphs: Some Algorithmic and Combinatorial Results Pranava Kumar Jha Iowa State University

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Hypercubes, Median Graphs and Products of Graphs: Some Algorithmic and Combinatorial Results Pranava Kumar Jha Iowa State University Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1990 Hypercubes, median graphs and products of graphs: some algorithmic and combinatorial results Pranava Kumar Jha Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Computer Sciences Commons Recommended Citation Jha, Pranava Kumar, "Hypercubes, median graphs and products of graphs: some algorithmic and combinatorial results " (1990). Retrospective Theses and Dissertations. 9445. https://lib.dr.iastate.edu/rtd/9445 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS The most advanced technology has been used to photograph and reproduce this manuscript from the microfîhn master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. Hie quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. 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Ml 40106-1346 USA 313 761-4700 800 521-0600 Order Number 9101366 Hypercubes, median graphs and products of graphs: Some algorithmic and combinational results Jha, Pranava Kumar, Ph.D. Iowa State University, 1990 UMI 300N.ZeebRd. Ann Aibor, MI 48106 Hypercubes, median graphs and products of graphs: Some algorithmic and combinatorial results by Pranava Kumar Jha A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Major: Computer Science Approved: Signature was redacted for privacy. Signature was redacted for privacy. )r the Major Departmmt Signature was redacted for privacy. For the Graduate College Iowa State University Ames, Iowa 1990 ii TABLE OF CONTENTS 1. INTRODUCTION 1 1.1 Basic Terminology 1 1.2 Hypercubes, Median Graphs and Products of Graphs 6 1.3 What Follows 15 2. MINIMUM CARDINALITY MAXIMAL INDEPENDENT SETS OF A HYPERCUBE 17 2.1 Preliminaries 18 2.2 Main Result 19 2.3 An Upper Bound 24 3. EFFICIENT ALGORITHMS FOR RECOGNIZING MEDIAN GRAPHS 27 3.1 Preliminaries 28 3.2 Recognition Scheme based on Bandelt's Characterization 31 3.2.1 Bandelt's proposition 31 3.2.2 Constructing the sets V, U and Z 33 3.3 Recognition Scheme based on Mulder's Characterization 39 3.3.1 Theoretical background 39 3.3.2 The convex-expansions algorithm 44 3.4 Concluding Remarks 48 Ill 4. ISOMETRIC EMBEDDING OF MEDIAN GRAPHS IN THE HYPERCUBE 52 4.1 Theoretical Background 54 4.2 Embedding Scheme based on Bandelt's Characterization 56 4.3 Embedding Scheme based on Mulder's Characterization 60 4.4 Concluding Remarks 64 5. CHARACTERIZATIONS FOR THE PLANARITY AND OUTER- PLAN ARITY OF PRODUCT GRAPHS 66 5.1 Preliminaries 67 5.2 Planarity 69 5.2.1 Planarity of the Strong product 70 5.3 Outerplanarity 74 5.3.1 Outerplanarity of the Cartesian product 74 5.3.2 Outerplanarity of the Kronecker product 76 5.4 Concluding Remarks 91 6. TOPOLOGICAL INVARIANTS OF PRODUCT GRAPHS . 93 6.1 Chromatic Numbers 94 6.2 Independence Numbers 98 6.2.1 Bounds on a{G\ 0G2) 98 6.2.2 Bounds on oc{Gi x 102 6.2.3 Bounds on «(Crj B Gg) 103 6.3 Domination Numbers 104 6.4 Clique Numbers 106 6.5 Haxniltonian Paths/Cycles 109 6.6 Concluding Remarks 112 iv 7. DISCUSSION AND OPEN PROBLEMS 114 8. BIBLIOGRAPHY 122 9. ACKNOWLEDGMENTS 128 V LIST OF FIGURES Figure 1.1: An example of an elementary contraction 5 Figure 1.2; The hypercube of dimension three 7 Figure 1.3: The graphs Cg and P2 10 Figure 1.4; The graph C5OP2 H Figure 1.5; The graph C5 x P2 11 Figure 1.6: The graph C5 B P2 12 Figure 3.1: The recognition algorithm RECOG-B 34 Figure 3.2: The iterative procedure ITER-B 36 Figure 3.3; The recognition algorithm RECOG-M 46 Figure 3.4: The iterative procedure ITER-M 48 Figure 4.1: Inclusion relationships among certain classes of graphs .... 53 Figure 4.2; The isometric embedding algorithm EMBED-B 58 Figure 4.3: The isometric embedding algorithm EMBED-M 63 Figure 5.1; The graphs and fTg g 67 Figure 5.2; The graphs and j^2,3 68 Figure 5.3; The graph +x 69 Figure 5.4; A planar embedding of the graph B . 72 Figure 5.5: The graph Pg B P2 73 vi Figure 5.6: The graph Ki^'^UK2 76 Figure 5.7: The graph 76 Figure 5.8: An ahnost bipartite graph 78 Figure 5.9: Graphs illustrating the construction in the proof of Lemma 5.3.4 82 Figure 5.10: Graphs H', H" aaà H'" 83 Figure 5.11: The graph X K2 83 Figure 5.12: The graph if'' x K2 84 Figure 5.13: The graph JEf''' x K2 84 Figure 5.14: The graph P4 x P4 87 Figure 5.15: An outerplanar embedding of the graph Pqx 88 Figure 5.16: The graphs C3 x P2 and C5 x P2 89 Figure 5.17: The graphs C4 x P2 and CqX P2 89 Figure 5.18: The graph g x P2 90 Figure 6.1: The recursive procedure R-INDEP 100 Figure 6.2: The iterative procedure I-INDEP 100 Figure 6.3: An illustration of the construction in the proof of Theorem 6.5.3 Ill 1 1. INTRODUCTION In this chapter, we present (i) the basic terminology and essential notations, (ii) the definitions and simple characteristics of hypercubes, median graphs, Cartesian product of graphs, Kronecker product of graphs and Strong product of graphs, and (iii) an outline of the organization of the subsequent chapters of this dissertation. 1.1 Basic Terminology This section introduces the important set- and graph-theoretical notation and terminology. For any undefined terms, we refer to [8]. Let S and T be sets. We write s G 5 to indicate that s is an element of S. The cardinality of S is denoted by |5| and the empty set is written il), S C T indicates that 5 is a subset of T while S C T indicates proper inclusion. The union and intersection of S and T are denoted by S'UT and 5(1^ respectively. S \T consists of those elements of S which are not in T, and is called the set difference. The Cartesian product (or simply product) of S and T is denoted by S" x T and is defined as the set of all ordered pairs (s,<) such that s Ç. S and t E T. We also use the following logical connectives: negation ( ), implication ( ==^ ) and equivalence ( )• A graph G = (V, E) consists of a set V of vertices and a set E of edges. Earh edge consists of a pair of vertices called its end points or end vertices. All graphs 2 we consider are assumed to be finite, undirected and simple, i.e., without loops and multiple edges. For a graph G, we also use V(G) and E(G) to denote its vertex set and edge set respectively. I£ e = {u, v} is an edge of G, then we say that e is incident on u and u, and that u, V are adjacent to each other, or « and v are neighbors. For a vertex v of G, we use degQ(v) to denote the degree of v, i.e., the number of neighbors of « in G. When the identity of the graph G is clear from the context, we will use deg(y) instead of degGiv). By A(G), we will denote the maximum degree of a vertex of G. A vertex V is isolated if deg{v) = 0 and is a terminal vertex if deg(u) = 1. We say that a graph G is regular if each vertex of G is of the same degree. A path is a sequence of vertices P = vi,.,,,vn, n > 1, such that {v^, } 6 E{G), 1 < I < n. The length of P is its number of edges. P is a simple path if it contains no repeated vertices, a closed path if = vn, and a cycle if it is closed and all vertices except and vn are distinct. We say that G is connected if there is a path between any two distinct vertices of G.
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