<Emphasis Type="Italic">L</Emphasis>(<Emphasis Type="Italic">h</Emphasis>, 1, 1)-labeling of outerplanar graphs

An L(h, 1, 1)-labeling of a graph is an assignment of labels from the set of integers {0, . . . , λ} to the nodes of the graph such that adjacent nodes are assigned integers of at least distance h ≥ 1 apart and all nodes of distance three or less must be assigned different labels. The aim of the L(h, 1, 1)-labeling problem is to minimize λ, denoted by λ _{h, 1, 1} and called span of the L(h, 1, 1)-labeling. As outerplanar graphs have bounded treewidth, the L(1, 1, 1)-labeling problem on outerplanar graphs can be exactly solved in O(n ³), but the multiplicative factor depends on the maximum degree Δ and is too big to be of practical use. In this paper we give a linear time approximation algorithm for computing the more general L(h, 1, 1)-labeling for outerplanar graphs that is within additive constants of the optimum values. This research is partially supported by the European Research Project Algorithmic Principles for Building Efficient Overlay Computers (AEOLUS) and was done during the visit of Richard B. Tan at the Department of Computer Science, University of Rome “Sapienza”, supported by a visiting fellowship from the University of Rome “Sapienza”.     

Some properties of graph centroids

The concept of a branch weight centroid has been extended in [12] so that it can be defined for an arbitrary finite setX with a distinguished familyC of "convex" subsets ofX. In particular, the centroid of a graphG was defined forX to be the vertex setV(G) ofG andU V(G) is convex if it is the vertex set of a chordless path inG. In this paper, which is an extended version of [13], we give necessary and sufficient conditions for a graph to be a centroid of another graph as well as of itself. Then, we apply these results to some particular classes of graphs: chordal, Halin, series-parallel and outerplanar.This research has been partly supported by Grant RP.I.09 from the Institute of Computer Science, University of Warsaw. This paper was completed when the second author was at Fachbereich 3 -Mathematik, Technische Universität Berlin, supported also by the Alexander von Humboldt-Stiftung (Bonn).     

Approximation of pathwidth of outerplanar graphs

There exists a polynomial time algorithm to compute the pathwidth of outerplanar graphs, but the large exponent makes this algorithm impractical. In this paper, we give an algorithm that, given a biconnected outerplanar graph G, finds a path decomposition of G of pathwidth at most twice the pathwidth of G plus one. To obtain the result, several relations between the pathwidth of a biconnected outerplanar graph and its dual are established.     

On Group Choosability of Total Graphs

In this paper, we study the group and list group colorings of total graphs and present group coloring versions of the total and list total colorings conjectures. We establish the group coloring version of the total coloring conjecture for the following classes of graphs: graphs with small maximum degree, two-degenerate graphs, planner graphs with maximum degree at least 11, planner graphs without certain small cycles, outerplanar graphs and near outerplanar graphs with maximum degree at least 4. In addition, the group version of the list total coloring conjecture is established for forests, outerplanar graphs and graphs with maximum degree at most two.     

A Polynomial-Time Algorithm for Finding Regular Simple Paths in Outerplanar Graphs

Let G be a labeled directed graph with arc labels drawn from alphabet Σ, R be a regular expression over Σ, and x and y be a pair of nodes from G. The regular simple path (RSP) problem is to determine whether there is a simple path p in G from x to y, such that the concatenation of arc labels along p satisfies R. Although RSP is known to be NP-hard in general, we show that it is solvable in polynomial time when G is outerplanar. The proof proceeds by presenting an algorithm which gives a polynomial-time reduction of RSP for outerplanar graphs to RSP for directed acyclic graphs, a problem which has been shown to be solvable in polynomial time.     

Enumeration of labeled outerplanar bicyclic and tricyclic graphs

The class of outerplanar graphs is used for testing the average complexity of algorithms on graphs. A random labeled outerplanar graph can be generated by a polynomial algorithm based on the results of an enumeration of such graphs. By a bicyclic (tricyclic) graph we mean a connected graph with cyclomatic number 2 (respectively, 3). We find explicit formulas for the number of labeled connected outerplanar bicyclic and tricyclic graphs with n vertices and also obtain asymptotics for the number of these graphs for large n. Moreover, we obtain explicit formulas for the number of labeled outerplanar bicyclic and tricyclic n-vertex blocks and deduce the corresponding asymptotics for large n.     

Independent packings in structured graphs

Packing a maximum number of disjoint triangles into a given graph G is NP-hard, even for most classes of structured graphs. In contrast, we show that packing a maximum number of independent (that is, disjoint and nonadjacent) triangles is polynomial-time solvable for many classes of structured graphs, including weakly chordal graphs, asteroidal triple-free graphs, polygon-circle graphs, and interval-filament graphs. These classes contain other well-known classes such as chordal graphs, cocomparability graphs, circle graphs, circular-arc graphs, and outerplanar graphs. Our results apply more generally to independent packings by members of any family of connected graphs. Research of both authors is supported by the Natural Sciences and Engineering Research Council of Canada.     

Chromatic polynomials,polygon trees,and outerplanar graphs

It is proved that all classes of polygon trees are characterized by their chromatic polynomials, and a characterization is given of those polynominals that are chromatic polynomials of outerplanar graphs. The first result yields an alternative proof that outerplanar graphs are recognizable from their vertex-deleted subgraphs. © 1929 John Wiley & Sons, Inc.     

Computation of the center and diameter of outerplanar graphs

The center of a graph is the set of vertices with minimum eccentricity. An outerplanar graph is a planar segmentation of a polygon. We define a notion of edge eccentricities for the edges of an outerplanar graph. We present an algorithm which efficiently computes these edge eccentricities. Knowledge of the edge eccentricities allows subsequent linear time computation of the center and diameter of outerplanar graphs. The computation of edge eccentricities is shown to require linear time for certain subclasses of outerplanar graphs.     

A characterization of α-outerplanar graphs

Chartrand and Harary have shown that if G is a non-outerplanar graph such that, for every edge e, both the deletion G\e and the contraction G/e of e from G are outerplanar, then G is isomorphic to K₄ or K_2,3. An α-outerplanar graph is a graph which is not outerplanar such that, for some edge α, both G\α and G/α are outerplanar. We describe various results for the class of α-outerplanar graphs and obtain a characterization of the class. © 1996 John Wiley & Sons, Inc.     

Star Edge Coloring of Some Classes of Graphs

A star edge coloring of a graph is a proper edge coloring without bichromatic paths and cycles of length four. In this article, we establish tight upper bounds for trees and subcubic outerplanar graphs, and derive an upper bound for outerplanar graphs.     

Invalid proofs on incidence coloring

Incidence coloring of a graph G is a mapping from the set of incidences to a color-set C such that adjacent incidences of G are assigned distinct colors. Since 1993, numerous fruitful results as regards incidence coloring have been proved. However, some of them are incorrect. We remedy the error of the proof in [R.A. Brualdi, J.J.Q. Massey, Incidence and strong edge colorings of graphs, Discrete Math. 122 (1993) 51-58] concerning complete bipartite graphs. Also, we give an example to show that an outerplanar graph with Δ=4 is not 5-incidence colorable, which contradicts [S.D. Wang, D.L. Chen, S.C. Pang, The incidence coloring number of Halin graphs and outerplanar graphs, Discrete Math. 256 (2002) 397-405], and prove that the incidence chromatic number of the outerplanar graph with Δ≥7 is Δ+1. Moreover, we prove that the incidence chromatic number of the cubic Halin graph is 5. Finally, to improve the lower bound of the incidence chromatic number, we give some sufficient conditions for graphs that cannot be (Δ+1)-incidence colorable.     

Excluded-Minor Characterization of Apex-Outerplanar Graphs

The class of outerplanar graphs is minor-closed and can be characterized by two excluded minors: \(K_4\) and \(K_{2,3}\). The class of graphs that contain a vertex whose removal leaves an outerplanar graph is also minor-closed. We provide the complete list of 57 excluded minors for this class.     

Game chromatic number of outerplanar graphs

This note proves that the game chromatic number of an outerplanar graph is at most 7. This improves the previous known upper bound of the game chromatic number of outerplanar graphs. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 67–70, 1999     

Small Drawings of Outerplanar Graphs,Series-Parallel Graphs,and Other Planar Graphs

In this paper, we study small planar drawings of planar graphs. For arbitrary planar graphs, Θ(n ²) is the established upper and lower bound on the worst-case area. A long-standing open problem is to determine for what graphs a smaller area can be achieved. We show here that series-parallel graphs can be drawn in O(n ^3/2) area, and outerplanar graphs can be drawn in O(nlog n) area, but 2-outerplanar graphs and planar graphs of proper pathwidth 3 require Ω(n ²) area. Our drawings are visibility representations, which can be converted to polyline drawings of asymptotically the same area.     

The list chromatic numbers of some planar graphs

In this paper, the choosability of outerplanar graphs, 1-tree and strong 1-outerplanargraphs have been described completely. A precise upper bound of the list chromatic number of 1-outerplanar graphs is given, and that every 1-outerplanar graph with girth at least 4 is 3-choosable is proved.     

外平面图的对策色数

本文讨论了图的色对策 ,给出了外平面图的几个性质 ,并且利用性质证明了外平面图的对策色数至多是 6     

Outerplanar and Planar Oriented Cliques

The clique number of an undirected graph G is the maximum order of a complete subgraph of G and is a well‐known lower bound for the chromatic number of G. Every proper k‐coloring of G may be viewed as a homomorphism (an edge‐preserving vertex mapping) of G to the complete graph of order k. By considering homomorphisms of oriented graphs (digraphs without cycles of length at most 2), we get a natural notion of (oriented) colorings and oriented chromatic number of oriented graphs. An oriented clique is then an oriented graph whose number of vertices and oriented chromatic number coincide. However, the structure of oriented cliques is much less understood than in the undirected case. In this article, we study the structure of outerplanar and planar oriented cliques. We first provide a list of 11 graphs and prove that an outerplanar graph can be oriented as an oriented clique if and only if it contains one of these graphs as a spanning subgraph. Klostermeyer and MacGillivray conjectured that the order of a planar oriented clique is at most 15, which was later proved by Sen. We show that any planar oriented clique on 15 vertices must contain a particular oriented graph as a spanning subgraph, thus reproving the above conjecture. We also provide tight upper bounds for the order of planar oriented cliques of girth k for all .     

Every graph occurs as an induced subgraph of some hypohamiltonian graph

We prove the titular statement. This settles a problem of Chvátal from 1973 and encompasses earlier results of Thomassen, who showed it for K₃, and Collier and Schmeichel, who proved it for bipartite graphs. We also show that for every outerplanar graph there exists a planar hypohamiltonian graph containing it as an induced subgraph.     

一个求外平面图最小顶点赋权反馈点集的线性时间算法

若从一个图中去掉某些顶点后得到的导出子图是无圈图,则所去的那些顶点组成的集合就是原图的反馈点集.本文主要考虑外平面图中的反馈点集并给出了一个求外平面图最小顶点赋权反馈点集的线性时间算法.