DOCSLIB.ORG

On K-Trees and Special Classes of K-Trees

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On K-Trees and Special Classes of K-Trees University of Mississippi eGrove Electronic Theses and Dissertations Graduate School 2012 On K-trees and Special Classes of K-trees John Wheless Estes Follow this and additional works at: https://egrove.olemiss.edu/etd Part of the Mathematics Commons Recommended Citation Estes, John Wheless, "On K-trees and Special Classes of K-trees" (2012). Electronic Theses and Dissertations. 101. https://egrove.olemiss.edu/etd/101 This Dissertation is brought to you for free and open access by the Graduate School at eGrove. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of eGrove. For more information, please contact [email protected]. ON k-TREES AND SPECIAL CLASSES OF k-TREES A Thesis presented for the Doctorate of Philosophy Department of Mathematics University of Mississippi JOHN WHELESS ESTES May 2012 Copyright John W. Estes 2011 ALL RIGHTS RESERVED ABSTRACT A tree is a connected graph with no cycles. In 1968 Beineke and Pippet introduced the class of generalized trees known as k-trees [3]. In this dissertation, we classify a subclass of k-trees known as tree-like k-trees and show that tree-like k-trees are a common generalization of paths, maximal outerplanar graphs, and chordal planar graphs with toughness exceeding one. A set I of vertices in a graph G is said to be independent if no pair of vertices of I are incident in G. Let fs = fs(G) be the number of independent sets of cardinality s of G. Pα(G) s Then the polynomial I(G; x) = s≥0 fs(G)x is called the independence polynomial of the graph G. [21]. In this dissertation, all rational roots of the independence polynomials of paths are found, and the exact paths whose independence polynomials have these roots are characterized. Additionally, trees are characterized that have −1=q as a root of their inde- pendence polynomials for 1 ≤ q ≤ 4. The well known vertex and edge reduction identities for independence polynomials are generalized, and the independence polynomials of k-trees are investigated. Additionally, sharp upper and lower bounds for fs of maximal outerplanar graphs, i.e. tree-like 2-trees, are shown along with characterizations of the unique maximal outerplanar graphs that obtain these bounds respectively. These results are extensions of the works of Wingard, Song et al., and Alameddine [1]. The first Zagreb index M1(G) and the second Zagreb index M2(G) of the graph G are P 2 P given by: M1(G) = u2V (G) d(u) , and M2(G) = uv2E(G) d(u)d(v). The study of the ii Zagreb indices M1 and M2 have been an active area of research since the report of Gutman and Trinajsti´c in computational chemistry [23] in 1972. The minimum and maximum M1 and M2 values for k-trees are determined, and the unique k-trees that obtain these minimum and maximum values respectively are characterized. In 2011, Hou, Li, Song, and Wei characterized the Zagreb indices for maximal outerplanar graphs and determined the unique maximal outerplanar graph that obtains minimum M1 and M2 values, respectively, as well as maximum M1 and M2 values respectively [29]. Select works of Hou et al. are extended to all tree-like k-trees. That is, the maximum M1 value for tree-like k-trees is determined, and the unique tree-like k-tree that obtains this maximum values respectively is characterized. Additionally, a partial result for the maximum M2 value for tree-like k-trees is determined, and a conjecture for a full result is presented. iii DEDICATION To Ethan, my son, who is my motivation to become a better person. iv LIST OF SYMBOLS G a graph V the vertex set of a graph G E the edge set of a graph G G[S] the subgraph induced by S G − v G[V − v] G − SG[V − S] G − e G remove an edge e G − FG remove a set of edges F G [ fuvg G add the edge uv G [ H the union of G and H jGj the cardinality of the vertex set of G jjGjj the cardinatlity of the edge set of G N(v) the neighborhood of vertex v N[v] the closed neighborhood of vertex v N(e) the neighborhood of edge e d(v) the degree of v δ(G) the minimum degree of G ∆(G) the maximum degree of G T a tree v d(v; u) the minimum distance from v to u in a graph Kn the complete graph on n vertices Pn the path on n vertices Sn the star on n vertices Kn1;n2 the complete bipartite graph on n1 + n2 vertices k Tn a k-tree on n vertices S1(G) the set of simplicial vertices in G k Pn the k-path on n vertices Sk;n−k the k-star on n vertices k Sn the k-spiral on n vertices k Dn the k-diamond on n vertices k Sh(Tn ) the shell of a k-tree τ(G) the toughness of G α(G) the independence number of G f(G) the fibonacci number of G fs(G) the number of indepedent sets of cardinality s of G I(G; x) the independence polynomial of G M1(G) the first Zagreb index M2(G) the second Zagreb index T (r) a p-descendant or p-ancestor of T vi β fTigi=0 a genealogy of a tree k k β(Tn ) the branching number of Tn vii ACKNOWLEDGEMENTS I thank my wife, Stephany, my source of constant support, encouragement, and inspiration. She is my partner who shares all of my stress, anxieties, laughter, and successes. I also thank my son, Ethan, for revealing to me a depth of love I had not previously known. Honor is due to my parents, Mickey and Suellen Estes, for all of the sacrifices they have made to invest into my future. I would not made it very far if it were not for their guidance, faith, and wisdom. I thank my three lovely sisters, Adrienne, Leslie, and Libby, and their families for all of the support and joyful memories. In addition, I would like to thank my wife's family, the Rangels, for their support as well. I thank Dr. Andrew Lang of Oral Roberts University for striving to mold a young and immature student into a mathematician. I also thank Dr. Bing Wei and Dr. William Staton of the University of Mississippi for the supervision of this dissertation. Their advice and counsel have been integral in developing the material presented here as well as my personal mathematical development. I thank Dr. James Reid, Dr. Haidong Wu, and Dr. Dawn Wilkins for taking time out of their busy schedules to serve on my dissertation committee. Their suggestions were well received. Additionally, I thank the faculty of the Department of Mathematics at the University of Mississippi for introducing many wonderful topics of mathematics to me and providing an environment of thought and growth. viii TABLE OF CONTENTS 1 INTRODUCTION . 1 2 TREE-LIKE k-TREES ............................................................ 9 3 INDEPENDENT SETS OF TREES . 19 4 INDEPENDENT SETS OF k-TREES . 31 5 THE ZAGREB INDICES OF k-TREES . 49 6 THE ZAGREB INDICES OF TREE-LIKE k-TREES . 74 7 TREE GENEALOGIES . 82 8 QUESTIONS FOR FUTURE RESEARCH . 96 LIST OF REFERENCES. 102 LIST OF APENDICES . 106 A TREES ON 1 ≤ n ≤ 10 VERTICES . 107 B INDEPENDENCE POLYNOMIALS OF TREES ON 1 ≤ n ≤ 10 VERTICES . 111 C THE ZAGREB INDICES OF TREE ON 1 ≤ n ≤ 10 VERTICES . 121 VITA................................................................................. 126 ix LIST OF FIGURES 1 A 4-tree on 10 vertices 6 2 The 3-path, 3-star, 3-spiral, and 3-diamond on 7 vertices 8 3 A 2-tree and its shell 9 4 T7;7 30 5 Maximal outerplanar graphs on n = 7 vertices 39 6 A tree-like 3-tree on 11 vertices G 41 7 Tree-like 3-trees on 8 vertices with τ > 1 45 8 A tree-like 4-tree on 10 vertices with toughness exceeding 1 46 9 The path and star on 7 vertices 50 10 The 2-path, 2-diamond, and 2-star on 6 vertices 75 11 The path and star on 7 vertices 82 12 A tree T and T (r) 83 13 A tree with no 1-ancestor 84 14 T5 89 15 T , T (r1), T (r2) 91 0 0 0 16 T , T (r1), T (r2) 92 x 17 A graph G and its shell 96 xi 1. Introduction A graph is chordal if it does not have an induced cycle of length greater than three. A graph is said to be k-degenerate if all of its subgraphs have minimum degree at most k, a concept introduced by Lick and White in 1970 [32], and a graph is maximally k-degenerate if it is k-degenerate and not a spanning subgraph of any other k-degenerate graph. In 1968 Beineke and Pippet introduced the class of generalized trees known as k-trees [3], and these graphs have attracted considerable research as well as many applications [4, 16, 33, 37, 38]. A purpose of this dissertation is to investigate the class of graphs that are both chordal and maximally k-degenerate, and it is shown that a graph is chordal and maximally k-degenerate if and only if it is a k-tree. A major emphasis of this dissertation is to classify a subclass of k-trees based on a \new" parameter known as the shell of a k-tree, which is a reformation of the (k + 1)-line graph first introduced in 2006 by Markenzon et al.
Load more

Recommended publications
  • K-Outerplanar Graphs, Planar Duality, and Low Stretch Spanning Trees
    k-Outerplanar Graphs, Planar Duality, and Low Stretch Spanning Trees Yuval Emek∗ Abstract Low distortion probabilistic embedding of graphs into approximating trees is an extensively studied topic. Of particular interest is the case where the approximating trees are required to be (subgraph) spanning trees of the given graph (or multigraph), in which case, the focus is usually on the equivalent problem of finding a (single) tree with low average stretch. Among the classes of graphs that received special attention in this context are k-outerplanar graphs (for a fixed k): Chekuri, Gupta, Newman, Rabinovich, and Sinclair show that every k-outerplanar graph can be probabilistically embedded into approximating trees with constant distortion regardless of the size of the graph. The approximating trees in the technique of Chekuri et al. are not necessarily spanning trees, though. In this paper it is shown that every k-outerplanar multigraph admits a spanning tree with constant average stretch. This immediately translates to a constant bound on the distortion of probabilistically embedding k-outerplanar graphs into their spanning trees. Moreover, an efficient randomized algorithm is presented for constructing such a low average stretch spanning tree. This algorithm relies on some new insights regarding the connection between low average stretch spanning trees and planar duality. Keywords: planar graphs, outerplanarity, average stretch, planar dual. ∗Microsoft Israel R&D Center, Herzelia, Israel and School of Electrical Engineering, Tel Aviv University, Tel Aviv, Israel. E-mail: [email protected]. Supported in part by the Israel Science Foundation, grants 221/07 and 664/05. 1 Introduction The problem.
    [Show full text]
  • Characterizations of Restricted Pairs of Planar Graphs Allowing Simultaneous Embedding with Fixed Edges
    Characterizations of Restricted Pairs of Planar Graphs Allowing Simultaneous Embedding with Fixed Edges J. Joseph Fowler1, Michael J¨unger2, Stephen Kobourov1, and Michael Schulz2 1 University of Arizona, USA {jfowler,kobourov}@cs.arizona.edu ⋆ 2 University of Cologne, Germany {mjuenger,schulz}@informatik.uni-koeln.de ⋆⋆ Abstract. A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V in the Euclidean plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same simple curve in the simultaneous drawing. Determining in polynomial time which pairs of graphs share a simultaneous embedding with fixed edges (SEFE) has been open. We give a necessary and sufficient condition for when a pair of graphs whose union is homeomorphic to K5 or K3,3 can have an SEFE. This allows us to determine which (outer)planar graphs always an SEFE with any other (outer)planar graphs. In both cases, we provide efficient al- gorithms to compute the simultaneous drawings. Finally, we provide an linear-time decision algorithm for deciding whether a pair of biconnected outerplanar graphs has an SEFE. 1 Introduction In many practical applications including the visualization of large graphs and very-large-scale integration (VLSI) of circuits on the same chip, edge crossings are undesirable. A single vertex set can be used with multiple edge sets that each correspond to different edge colors or circuit layers. While the pairwise union of all edge sets may be nonplanar, a planar drawing of each layer may be possible, as crossings between edges of distinct edge sets are permitted.
    [Show full text]
  • Minor-Closed Graph Classes with Bounded Layered Pathwidth
    Minor-Closed Graph Classes with Bounded Layered Pathwidth Vida Dujmovi´c z David Eppstein y Gwena¨elJoret x Pat Morin ∗ David R. Wood { 19th October 2018; revised 4th June 2020 Abstract We prove that a minor-closed class of graphs has bounded layered pathwidth if and only if some apex-forest is not in the class. This generalises a theorem of Robertson and Seymour, which says that a minor-closed class of graphs has bounded pathwidth if and only if some forest is not in the class. 1 Introduction Pathwidth and treewidth are graph parameters that respectively measure how similar a given graph is to a path or a tree. These parameters are of fundamental importance in structural graph theory, especially in Roberston and Seymour's graph minors series. They also have numerous applications in algorithmic graph theory. Indeed, many NP-complete problems are solvable in polynomial time on graphs of bounded treewidth [23]. Recently, Dujmovi´c,Morin, and Wood [19] introduced the notion of layered treewidth. Loosely speaking, a graph has bounded layered treewidth if it has a tree decomposition and a layering such that each bag of the tree decomposition contains a bounded number of vertices in each layer (defined formally below). This definition is interesting since several natural graph classes, such as planar graphs, that have unbounded treewidth have bounded layered treewidth. Bannister, Devanny, Dujmovi´c,Eppstein, and Wood [1] introduced layered pathwidth, which is analogous to layered treewidth where the tree decomposition is arXiv:1810.08314v2 [math.CO] 4 Jun 2020 required to be a path decomposition.
    [Show full text]
  • NP-Completeness of List Coloring and Precoloring Extension on the Edges of Planar Graphs
    NP-completeness of list coloring and precoloring extension on the edges of planar graphs D´aniel Marx∗ 17th October 2004 Abstract In the edge precoloring extension problem we are given a graph with some of the edges having a preassigned color and it has to be decided whether this coloring can be extended to a proper k-edge-coloring of the graph. In list edge coloring every edge has a list of admissible colors, and the question is whether there is a proper edge coloring where every edge receives a color from its list. We show that both problems are NP-complete on (a) planar 3-regular bipartite graphs, (b) bipartite outerplanar graphs, and (c) bipartite series-parallel graphs. This improves previous results of Easton and Parker [6], and Fiala [8]. 1 Introduction In graph vertex coloring we have to assign colors to the vertices such that neighboring vertices receive different colors. Starting with [7] and [28], a gener- alization of coloring was investigated: in the list coloring problem each vertex can receive a color only from its prescribed list of admissible colors. In the pre- coloring extension problem a subset W of the vertices have preassigned colors and we have to extend this precoloring to a proper coloring of the whole graph, using only colors from a given color set C. It can be viewed as a special case of list coloring: the list of a precolored vertex consists of a single color, while the list of every other vertex is C. A thorough survey on list coloring, precoloring extension, and list chromatic number can be found in [26, 1, 12, 13].
    [Show full text]
  • Star Coloring Outerplanar Bipartite Graphs
    Discussiones Mathematicae Graph Theory 39 (2019) 899–908 doi:10.7151/dmgt.2109 STAR COLORING OUTERPLANAR BIPARTITE GRAPHS Radhika Ramamurthi and Gina Sanders Department of Mathematics California State University San Marcos San Marcos, CA 92096-0001 USA e-mail: [email protected] Abstract A proper coloring of the vertices of a graph is called a star coloring if at least three colors are used on every 4-vertex path. We show that all outerplanar bipartite graphs can be star colored using only five colors and construct the smallest known example that requires five colors. Keywords: chromatic number, star coloring, outerplanar bipartite graph. 2010 Mathematics Subject Classification: 05C15. 1. Introduction A proper r-coloring of a graph G is an assignment of labels from {1, 2,...,r} to the vertices of G so that adjacent vertices receive distinct colors. The minimum r so that G has a proper r-coloring is called the chromatic number of G, denoted by χ(G). The chromatic number is one of the most studied parameters in graph theory, and by convention, the term coloring of a graph is usually used instead of proper coloring. In 1973, Gr¨unbaum [5] considered proper colorings with the additional constraint that the subgraph induced by every pair of color classes is acyclic, i.e., contains no cycles. He called such colorings acyclic colorings, and the minimum r such that G has an acyclic r-coloring is called the acyclic chromatic number of G, denoted by a(G). In introducing the notion of an acyclic coloring, Gr¨unbaum noted that the condition that the union of any two color classes induces a forest can be generalized to other bipartite graphs.
    [Show full text]
  • Surviving Rates of Trees and Outerplanar Graphs for the Firefighter Problem
    Surviving rates of trees and outerplanar graphs for the ¯re¯ghter problem Leizhen Cai¤ Yongxi Chengy Elad Verbinz Yuan Zhouz Abstract The ¯re¯ghter problem is a discrete-time game on graphs introduced by Hartnell in an attempt to model the spread of ¯re, diseases, computer viruses and suchlike in a macro-control level. To measure the defence ability of a graph as a whole, Cai and Wang de¯ned the surviving rate of a graph G for the ¯re¯ghter problem to be the average percentage of vertices that can be saved when a ¯re starts randomly at one vertex of G. In this paper, we prove that the surviving rate of every n-vertex outerplanar graph log n is at least 1 ¡ £( n ), which is asymptotically tight. We also show that the greedy log n strategy of Hartnell and Li for trees saves at least 1 ¡ £( n ) percentage of vertices on average for an n-vertex tree. Key words: ¯re¯ghter problem, surviving rate, tree, outerplanar graph 1 Introduction The study of a discrete-time game, the ¯re¯ghter problem, was initiated by Hartnell [7] in 1995 in an attempt to model the spread of ¯re, diseases, computer viruses and suchlike in a macro-control level. At time 0, a ¯re breaks out at a vertex v of a graph G = (V; E). At each subsequent time, a ¯re¯ghter protects one vertex not yet on ¯re, and the ¯re then spreads from all burning vertices to all their unprotected neighbors. The process ends when the ¯re can no longer spread.
    [Show full text]
  • Fractional and Circular Separation Dimension of Graphs
    Fractional and Circular Separation Dimension of Graphs Sarah J. Loeb∗and Douglas B. West† Revised June 2017 Abstract The separation dimension of a graph G, written π(G), is the minimum number of linear orderings of V (G) such that every two nonincident edges are “separated” in some ordering, meaning that both endpoints of one edge appear before both endpoints of the other. We introduce the fractional separation dimension πf (G), which is the minimum of a/b such that some a linear orderings (repetition allowed) separate every two nonincident edges at least b times. In contrast to separation dimension, fractional separation dimension is bounded: always πf (G) 3, with equality if and only if G contains K4. There is no stronger ≤ 3m bound even for bipartite graphs, since πf (Km,m) = πf (Km+1,m) = m+1 . We also compute πf (G) for cycles and some complete tripartite graphs. We show that πf (G) < √2 when G is a tree and present a sequence of trees on which the value tends to 4/3. Finally, we consider analogous problems for circular orderings, where pairs of non- incident edges are separated unless their endpoints alternate. Let π◦(G) be the number of circular orderings needed to separate all pairs and πf◦(G) be the fractional version. Among our results: (1) π◦(G) = 1 if and only G is outerplanar. (2) π◦(G) 2 when 3 ≤ G is bipartite. (3) π◦(Kn) log2 log3(n 1). (4) πf◦(G) 2 for every graph G, with ≥ − 3m 3 ≤ equality if and only if K4 G.
    [Show full text]
  • Exploring the Earth-Moon Problem for Doubly Outerplanar Graphs
    Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs Maia Wichman Grand Valley State University Partner: Hannah Critchfield, James Madison University Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs Overview 1 Background 2 Earth-Moon Problem 3 Specific Cases of Earth-Moon Graphs Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs A graph is planar if it can be drawn on the plane without edges crossing. Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs A graph G is a set of vertices V together with a set of edges E which are pairs of vertices. Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs A graph is planar if it can be drawn on the plane without edges crossing. Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs A graph G is a set of vertices V together with a set of edges E which are pairs of vertices. Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs A graph G is a set of vertices V together with a set of edges E which are pairs of vertices. A graph is planar if it can be drawn on the plane without edges crossing. Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs A graph G is a set of vertices V together with a set of edges E which are pairs of vertices.
    [Show full text]
  • DISSERTATION GENERALIZED BOOK EMBEDDINGS Submitted By
    DISSERTATION GENERALIZED BOOK EMBEDDINGS Submitted by Shannon Brod Overbay Department of Mathematics In partial fulfillment of the requirements for the degree of Doctor of Philosophy Colorado State University Fort Collins, Colorado Summer 1998 COLORADO STATE UNIVERSITY May 13, 1998 WE HEREBY RECOMMEND THAT THE DISSERTATION PREPARED UNDER OUR SUPERVISION BY SHANNON BROD OVERBAY ENTITLED GENERALIZED BOOK EMBEDDINGS BE ACCEPTED AS FULFILLING IN PART REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSO- PHY. Committee on Graduate Work Adviser Department Head ii ABSTRACT OF DISSERTATION GENERALIZED BOOK EMBEDDINGS An n-book is formed by joining n distinct half-planes, called pages, together at a line in 3-space, called the spine. The book thickness bt(G) of a graph G is the smallest number of pages needed to embed G in a book so that the vertices lie on the spine and each edge lies on a single page in such a way that no two edges cross each other or the spine. In the first chapter, we provide background material on book embeddings of graphs and preview our results on several related problems. In the second chapter, we use a theorem of Bernhart and Kainen and a result of Whitney to present a large class of two-page embeddable planar graphs. In particular, we prove that a graph G that can be drawn in the plane so that G has no triangles other than faces can be embedded in a two-page book. The discussion of planar graphs continues in the third chapter where we define a book with a tree-spine.
    [Show full text]
  • The D-Precoloring Problem for K-Degenerate Graphs∗
    The d-precoloring problem for k-degenerate graphs∗ Janka Chleb´ıkov´a† Faculty of Mathematics, Physics, and Informatics Comenius University, 842 48 Bratislava, Slovakia [email protected] and Klaus Jansen Institut f¨urInformatik und Praktische Mathematik Christian-Albrechts-Universit¨at zu Kiel Olshausenstr. 40, D-24098 Kiel, Germany [email protected] Abstract In this paper we deal with the d-Precoloring Extension problem, (d-PrExt), in various classes of graphs. The d-PrExt problem is the special case of precoloring extension problem where, for a fixed constant d, input instances are restricted to contain at most d precolored vertices for every available color. The goal is to decide if there exists an extension of given precoloring using only available colors or to find it. We present a linear time algorithm for both, the decision and the search version of d-PrExt, in the following cases: (i) restricted to the class of k-degenerate graphs (hence also planar graphs) and with sufficiently large set S of available colors, and (ii) restricted to the class of partial k-trees (without any size restriction on S). We also study the following problem related to d-PrExt: given an instance of the d-PrExt problem which is extendable by colors of S, what is the minimum number of colors of S sufficient to use for precolorless vertices over all such extensions? We establish lower and upper bounds on this value for k-degenerate graphs and its various subclasses (e.g., planar graphs, outerplanar graphs) and prove tight results for the class of trees.
    [Show full text]
  • Large Induced Acyclic and Outerplanar Subgraphs of 2-Outerplanar Graph ∗
    Large induced acyclic and outerplanar subgraphs of 2-outerplanar graph ∗ Glencora Borradaile Hung Le Oregon State University Oregon State University [email protected] [email protected] Melissa Sherman-Bennett University of California, Berkeley m [email protected] Abstract Albertson and Berman conjectured that every planar graph has an induced forest on half of its vertices. The best known lower bound, due to Borodin, is that every planar graph has an induced forest on two fifths of its vertices. In a related result, Chartran and Kronk, proved that the vertices of every planar graph can be partitioned into three sets, each of which induce a forest. We show tighter results for 2-outerplanar graphs. We show that every 2-outerplanar graph has an induced forest on at least half the vertices by showing that its vertices can be partitioned into two sets, each of which induces a forest. We also show that every 2-outerplanar graph has an induced outerplanar graph on at least two-thirds of its vertices. 1 Introduction For many optimization problems, finding subgraphs with certain properties is a key to developing algorithms with efficient running times or bounded approximation ratios. For example, balanced separator subgraphs support the design of divide-and-conquer algorithms for minor-closed graph families [18, 17, 12, 15] and large subgraphs of low-treewidth1 support the design of approximation schemes, also for minor-closed graph families [4, 8, 11]. In the area of graph drawing, one often starts by drawing a subgraph that is somehow easier to draw than the entire graph (such as a planar graph or a tree) and then adds in remaining graph features [5]; the larger the subgraph, the arXiv:1711.00212v1 [math.CO] 1 Nov 2017 bigger the head-start for drawing and the more structure the subgraph has, the easier the subgraph will be to draw.
    [Show full text]
  • Canonical Decomposition of Outerplanar Maps and Application to Enumeration, Coding and Generation
    Journal of Graph Algorithms and Applications http://jgaa.info/ vol. 9, no. 2, pp. 185–204 (2005) Canonical Decomposition of Outerplanar Maps and Application to Enumeration, Coding and Generation Nicolas Bonichon Cyril Gavoille Nicolas Hanusse CNRS-LaBRI Universit´eBordeaux I bonichon,gavoille,hanusse @labri.fr { } Abstract In this article we define a canonical decomposition of rooted outerpla- nar maps into a spanning tree and a list of edges. This decomposition, constructible in linear time in the Word-RAM model, implies the existence of bijection between rooted outerplanar maps with n nodes and bicolored rooted ordered trees with n nodes where all the nodes of the last branch are colored white. As a consequence, for rooted outerplanar maps of n nodes, we derive: n− n • an enumeration formula, and an asymptotic of 23 Θ(log ); • an optimal data structure of asymptotically 3n bits, built in O(n) time, supporting adjacency and degree queries in worst-case constant time and neighbors query of a degree-d node in worst-case O(d) time. • an O(n) expected time uniform random generating algorithm. Article Type Communicated by Submitted Revised Regular paper X. He July 2003 August 2005 An extended abstract of this article appeared in Proceeding of WG’2003, Lecture Notes in Computer Science 2880, pp. 81-92. This work was supported by the ACI Masse de donn´ees grant of French Ministery of Research under Navgraphe project. Bonichon et al., Decomp. of Outerpl. Maps, JGAA, 9(2) 185–204 (2005) 186 1 Introduction A graph is outerplanar if it can be drawn on the plane with non-intersecting edges such that all the nodes lie on the boundary of the infinite face, also called outerface.
    [Show full text]