The D-Precoloring Problem for K-Degenerate Graphs∗

Total Page:16

File Type:pdf, Size:1020Kb

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. ∗The work was supported by EU-Project ARACNE, HPRN-CT-1999-00112 and VEGA grant no. 1/3106/06. †Part of this work was done while the author was working at the CAU University of Kiel. 1 1 Introduction Let G = (V,E) be a finite simple graph. For a set S of colors, a proper coloring (or S-coloring) of G is a mapping f : V → S such that f(u) 6= f(v) whenever {u,v} ∈ E. An S-precoloring f ′ of G is any proper coloring f ′ : C → S of the subgraph of G induced by C ⊆ V . An S-precoloring of G is called a (d, S)-precoloring, if each color of S is used for at most d vertices. We say that an S-coloring f of G is an extension of an S-precoloring f ′ : C → S, if f(v)= f ′(v) for each vertex v ∈ C. The subset C ⊆ V is the set of precolored vertices and V \ C is the set of precolorless vertices. The admissible set of colors for a precolorless vertex v is the set of all colors of S which do not occur on precolored neighbors of v. A graph G is k-degenerate if every subgraph of G has a vertex of degree at most k. We can define the class of k-degenerate graphs also inductively as follows: a graph with at most (k +1) vertices is a k-degenerate graph and recursively a k-degenerate graph with (n + 1) vertices can be obtained from a k-degenerate graph with n vertices by adding a new vertex adjacent to at most k vertices. If we start with a (k + 1)-clique and every new vertex has to be adjacent to all vertices of a k-clique, we obtain a k-tree introduced in [1]. A partial k-tree is a subgraph of a k-tree. In this article we investigate the following problem: d-Precoloring Extension Problem (d-PrExt), d is a fixed nonnegative integer Instance: A set S of colors, a graph G, and a (d, S)-precoloring of G. Question (decision): Is there an extension of the (d, S)-precoloring to the entire graph G (using only colors of S)? Task (search): Find an extension to the entire graph G, if there is any. The d-PrExt problem is a subproblem of the Precoloring Extension problem (PrExt). In the PrExt problem there is no restriction on the number of precolored vertices for any color of S. PrExt and d-PrExt arise in practical applications such as scheduling or VLSI-layout and were initiated in [2]. Obviously, 0-PrExt is the problem to decide whether the chromatic number χ(G) is at most |S|. Hence the Graph Coloring problem, which is NP-complete for general graphs ([8]), is a subproblem of the d-PrExt problem. However d-PrExt is NP-complete even for very restricted graph classes: e.g., 1-PrExt problem for bipartite graphs (and |S| = 3) [3], 1-PrExt problem for permutation graphs [12], and 2-PrExt problem for interval graphs [2]. On the other hand, PrExt can be solved in polynomial time for (a) split graphs, (b) complements of bipartite graphs [9], (c) for partial k-trees [11], [13] (if |S| is a constant, in linear time [11]), and in linear time also for trees. Further research was continued in a series of papers, see [10], [14], and also [16] for a survey and references therein. 2 Summary of Results In this paper we consider some combinatorial and algorithmic aspects of the d-PrExt problem restricted to the subclasses of k-degenerate graphs. They include some well studied classes of graphs such as partial k-trees, planar graphs (for k = 5), and outerplanar graphs (for k = 2). In Section 2 we prove that if the answer to the d-PrExt problem for a k-degenerate graph is affirmative then there exists also an extension that uses at most k(d + 2) colors for all precolorless vertices independently of the size of S. On the other hand, we present (d, S)-precolored k-degenerate graphs that require to use at least k(d + 1) + 1 colors of S to extend the precoloring. The similar bounds are presented also for various subclasses of k-degenerate graphs (e.g., planar graphs, outerplanar graphs), with tight bounds for the class of trees. In Section 3 we present a linear time algorithm for both, the decision and the search version of the d-PrExt problem in the following cases: (i) for k-degenerate graphs, if |S| ≥ 2k + 1, (ii) for planar graphs, if |S| ≥ 7, (iii) for partial k-trees without any size restriction on S. The last result improves on the best known result for the d-PrExt problem restricted to the class of partial k-trees, with running time O(|V ||S|k+1) ([11]). 2 Structural Results An interesting combinatorial question relevant to d-PrExt is the following: if there is an extension for every extendable (d, S)-precoloring to the entire graph G (using only colors of S), how many colors of S are necessary to the precolorless vertices of G? Definition. Define the d-chromatic number of a graph G, χd(G), as the minimum number m such that for every set of colors S and every extendable (d, S)-precoloring of G, there is an extension to the entire graph G that uses at most m colors of S for the precolorless vertices. For a class of graphs G, let χd = supG∈G χd(G) (+∞, if χd(G) is unbounded). Firstly, we present the following lemma that expresses a property of k-degenerate graphs. Lemma 1 Let k ≥ 1, M > k be fixed integers. Then every nonempty k-degenerate graph nk with n vertices contains less than M−k vertices with degree at least M. Proof. Due to the fact that any k-degenerate graph is a subgraph of a k-degenerate graph on the same vertex set with all vertices of degree at least k, it is sufficient to prove lemma for k-degenerate graphs with minimum degree k. 3 Let G = (V,E) be a k-degenerate graph with n vertices and minimum degree k. If at nk least M−k vertices have degree at least M in G, then nk nk (1) dG(v) ≥ · M + n − k = 2kn. M − k µ M − k ¶ vX∈V But the sum of vertex degrees in a k-degenerate graph G is bounded by (2) dG(v)= k(k + 1) + 2k(n − k − 1) < 2kn, vX∈V using the inductive construction of k-degenerate graphs. Combining (1) and (2) we obtain a contradiction. ¤ Remark 1 For a planar graph G = (V,E) the inequality |E| ≤ 3|V | − 6 is well known. From this one can prove better upper bound for the number of large degree vertices in planar graphs as follows from Lemma 1 for 5-degenerate graphs (hence also planar graphs). More 3(|V |−4) precisely, a planar graph G contains at most M−3 vertices of degree at least M, M ≥ 4. In the following theorem we prove that for the class of k-degenerate graphs, the value χd can be bounded by a function of d and k. Note that, e.g., for the class of bipartite graphs, χd is unbounded. Theorem 1 Let k ≥ 1, d > 0 be fixed integers. For any (d, S)-precoloring of a k- degenerate graph if there is an extension to the entire graph, then there is also an extension which uses at most k(d + 2) colors of S, i.e., χd ≤ k(d + 2) for the class of k-degenerate graphs. Proof. Suppose to the contrary that the theorem does not hold. The size of a (d, S)-precolored k-degenerate graph G = (V,E) is defined as |V | + |S|. Let G with a (d, S)-precoloring g be a fixed counterexample of smallest size. Then using Lemma 1 it follows that |V | (3) less than d+1 vertices have degree ≥ k(d + 2).
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]
  • 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]
  • On the Colorings of Outerplanar Graphs
    DISCRETE MATHEMATICS ELSEVIER Discrete Mathematics 147 (1995) 257-269 On the colorings of outerplanar graphs Weifan Wang Department of Mathematics, Liaoning University, Shenyang 110036, China Received 1 July 1993; revised 6 April 1994 Abstract In this paper, we have studied seven colorings of outerplanar graphs. Two main conclusions have been proved: if G is an outerplanar graph without cut vertex and A(G)>/6, then (i) Xef(G)= A(G), and (ii) Zvef(G)= A(G) + 1, where gef and Xva are the edge-face chromatic number and the entire chromatic number of G, respectively, and A(G) is the maximum degree of vertices of G. 1. Introduction Let G(V, E, F) be a planar graph, where V(G), E(G) and F(G) are the set of vertices, the set of edges and the set of faces of G, respectively. Two vertices in V(G) are adjacent if they are joined by an edge, two edges in E(G) are adjacent if they have a common vertex and two faces in F(G) are adjacent if their boundaries have at least one common edge. We say that a vertex (an edge) is incident to a face if it forms part of the boundary of the face. Also the vertices u and v are each incident to the edge uv. Definition 1.1. A planar graph G(V,E,F) is k-vertex (edge, face) colorable, if the elements of V(G) (E(G), F(G)) can be colored with k colors such that any two distinct adjacent elements receive different colors.
    [Show full text]