Spanning cactus and spanning cactus extension of outerplanar graphs

An outerplanar graph is a planar graph that can be embedded in the plane in such a way that all vertices lie on the outer boundary. Outerplanar graphs express many chemical compounds. An externally extensible outerplanar graph pattern (eeo-graph pattern for short) represents a graph pattern common to a finite set of outerplanar graphs, like a dataset of chemical compounds. The eeo-graph pattern can express a substructure common to blocks that appear in outerplanar graph structured data. In this paper, we propose a polynomial time algorithm for deciding whether or not a given eeo-graph pattern matches a given connected outerplanar graph. Moreover, we show the expressiveness of the pattern class by experiments on a chemical compound database.

A proximity drawing of a graph is one in which pairs of adjacent vertices are drawn relatively close together according to some proximity measure while pairs of non-adjacent vertices are drawn relatively far apart. The fundamental question concerning proximity drawability is: Given a graph G and a definition of proximity, is it possible to construct a proximity drawing of G? We consider this question for outerplanar graphs with respect to an infinite family of proximity drawings called β-drawings. These drawings include as special cases the well-known Gabriel drawings (when β=1), and relative neighborhood drawings (when β=2). We first show that all biconnected outerplanar graphs are β-drawable for all values of β such that 1 ≤ β ≤ 2. As a side effect, this result settles in the affirmative a conjecture by Lubiw and Sleumer [15, 17], that any biconnected outerplanar graph admits a Gabriel drawing. We then show that there exist biconnected outerplanar graphs that do not admit any convex β-drawing for 1 ≤ β ≤ 2. We also provide upper bounds on the maximum number of biconnected components sharing the same cut-vertex in a β-drawable connected outerplanar graph. This last result is generalized to arbitrary connected planar graphs and is the first non-trivial characterization of connected β-drawable graphs. Finally, a weaker definition of proximity drawings is applied and we show that all connected outerplanar graphs are drawable under this definition.

Tree [formula omitted]-spanners in outerplanar graphs via supply demand partition

This paper addresses the classic online Steiner tree problem on edge-weighted graphs. It is known that a greedy (nearest neighbor) online algorithm has a tight competitive ratio for wide classes of graphs, such as trees, rings, any class including series-parallel graphs, and unweighted graphs with bounded diameter. However, we do not know any greedy or non-greedy tight deterministic algorithm for other classes of graphs. In this paper, we observe that a greedy algorithm is $$\Omega (\log n)$$ -competitive on outerplanar graphs, where n is the number of vertices, and propose a 5.828-competitive deterministic algorithm on outerplanar graphs. Our algorithm connects a requested vertex and the tree constructed thus far using a path that is constant times longer than the distance between them. We also present a lower bound of 4 for arbitrary deterministic online Steiner tree algorithms on outerplanar graphs.

The concept of a crisp graph is essential in the study of outerplanar graphs because outerplanar graphs are a unique type of planar graphs containing special characteristics. One of the core concepts of crisp graphs, the notion of a subgraph, is utilized primarily to solve difficult data problems. In this paper, researching outerplanar graphs in fuzzy graphs is the main aim of the work. By allowing edges connecting vertices to have varying levels of membership or fuzziness, a fuzzy graph is a mathematical framework that develops on the concept of crisp graphs. Because of their capacity to describe unclear or imprecise relationships between items, fuzzy outerplanar graphs might have potential applications. The subgraphs of fuzzy outerplanar graphs are revealed by eliminating specific vertices or edges from the fuzzy graphs. Furthermore, maximum, and maximal fuzzy outerplanar subgraphs defined for both vertex and edge deletion are examined using examples. Some of the connections between the concepts discussed are represented as theorems and conclusions. An application for the layout of a bypass road is discussed using the proposed concepts.

We consider straight-line outerplanar drawings of outerplanar graphs in which the segments representing edges are parallel to a small number of directions. We prove that Δ − 1 directions suffice for every outerplanar graph with maximum degree Δ ≥ 4. This improves the previous bound of O(Δ5), which was shown for planar partial 3-trees, a superclass of outerplanar graphs. The bound is tight: for every Δ ≥ 4 there is an outerplanar graph of maximum degree Δ which requires at least Δ − 1 distinct edge slopes for an outerplanar straight-line drawing.

Let G be a connected simple graph on n vertices. The Gallai's conjecture asserts that the edges of G can be decomposed into paths. In this paper, we show that the conjecture is true if G is a maximal outerplanar graph or G ia a 2-connected outerplanar graph.

Embedding between metric spaces is a very powerful algorithmic tool and has been used for finding good approximation algorithms for several problems. In particular, embedding to an [cursive l]₁ norm has been used as the key step in an approximation algorithm for the sparsest cut problem. The sparsest cut problem, in turn, is the main ingredient of many algorithms that have a divide and conquer nature and are used in various fields. While every metric is embeddable into [cursive l]₁ with distortion O (log n) [13], and the bound is tight [39], for special classes of metrics better bounds exist. Shortest path metrics for trees and outerplanar graphs are isometrically embeddable into [cursive l]₁ [41]. Series-parallel graphs [28] and k-outerplanar graphs [19] (for constant k) are embeddable into[cursive l]₁ with constant distortion planar graphs and bounded tree-width graphs are conjectured to have constant distortion in embedding to [cursive l]₁ . Bounded tree-width graphs are one of most general graph classes on which several hard problems are tractable. We study the embedding of series-parallel graphs (or, more generally, graphs with tree-width two) into [cursive l]₁ and also the embedding between two line metrics. We then move our attention to the generalization of tree-width to digraphs and hypergraphs and study several relevant problems.

We improve the best previously-known lower bound on the size of largest homometric sets in sparse diameter-two graphs. Along with the proof, we provide a method for constructing the desired homometric sets. We also study outerplanar graphs and show how to construct homometric sets in a given outerplanar graph of the size at least clogn, where n is the number of vertices in the graph, and c a positive fixed constant.

Let G be a graph. A circular distance two labeling with span k is a function $f: V(G) \to \{0, 1, 2, \ldots, k-1\}$ such that (1) $2 \leq |f(u)-f(v)| \leq k-2$ if u and v are adjacent and (2) $f(u) \neq f(v)$ if u and v are of distance two apart. We denote by $\lambda_c(G)$ the smallest span of a circular distance two labeling for G. Let $\Delta(G)$ be the maximum degree of G. We prove, for any outerplanar graph G with $\Delta(G) \geq 15$, $\lambda_c(G)=\Delta(G) +3$. It is also shown that there exist outerplanar graphs G with $\Delta(G) = 2, 3, 4, 5$ for which $\lambda_c(G) = \Delta(G) +4$. Moreover, we prove that $\lambda_c(G) \leq \Delta(G) +5$ for any triangulated outerplanar graph, $\lambda_c(G) \leq \Delta(G) +7$ for any outerplanar graph, and $\lambda_c(G) \leq \Delta(G) +4$ for any outerplanar graph with $\Delta(G) \geq 11$. Immediate consequences of our results include that $\lambda(G) \leq \Delta(G) + 2$ for any outerplanar graphs with $\Delta(G) \geq 15$, where $\lambda(G)$ is the minimum k of a k-L(2, 1)-labeling (or distance two labeling) for G.

Minimum rank of outerplanar graphs

We give the first known optimal algorithm that computes a minimum cycle basis for any weighted outerplanar graph. Specifically, for any n-node edge-weighted outerplanar graph G, we give an O(n)-time algorithm to obtain an O(n)-space compact representation Z(?) for a minimum cycle basis ? of G. Each cycle in ? can be computed from Z(?) in O(1) time per edge. Our result works for directed and undirected outerplanar graphs G.

We show three linear-time algorithms for constructing planar straight-line grid drawings of outerplanar graphs. The first and the second algorithm are for balanced outerplanar graphs. Both require linear area. The drawings produced by the first algorithm are not outerplanar while those produced by the second algorithm are. On the other hand, the first algorithm constructs drawings with better angular resolution. The third algorithm constructs outerplanar drawings of general outerplanar graphs with O(n 1.48) area. Further, we study the interplay between the area requirements of the drawings of an outerplanar graph and the area requirements of a special class of drawings of its dual tree.

Zigzag pocket machining (or 2D-milling) plays an important role in the manufacturing industry. The objective is to minimize the number of tool retractions in the zigzag machining path for a given pocket (i.e., a planar domain). We give an optimal linear time dynamic programming algorithm for simply connected pockets, and a linear plus O(1)O(h) time optimal algorithm for pockets with h holes. If the dual graph of the zigzag line segment partition of the given pocket is a partial k-tree of bounded degree or a k- outerplanar graph, for a fixed k, we solve the problem optimally in time O(n logn). Finally, we propose a polynomial time algorithm for finding a machining path for a general pocket with h holes using at most OPT+eh retractions, where OPT is the smallest possible number of retractions and e>0 is any constant.

Recently, due to the rapid growth of electronic data having graph structures such as HTML and XML texts and chemical compounds, many researchers have been interested in data mining and machine learning techniques for finding useful patterns from graph-structured data (graph data). Since graph data contain a huge number of substructures and it tends to be computationally expensive to decide whether or not such data have given structural features, graph mining problems face computational difficulties. Let ${\mathcal{C}}$ be a graph class which satisfies a connected hereditary property and contains infinitely many different biconnected graphs, and for which a special kind of the graph isomorphism problem can be computed in polynomial time. In this paper, we consider learning and mining problems for ${\mathcal{C}}$ . Firstly, we define a new graph pattern, which is called a block preserving graph pattern (bp-graph pattern) for ${\mathcal{C}}$ . Secondly, we present a polynomial time algorithm for deciding whether or not a given bp-graph pattern matches a given graph in ${\mathcal{C}}$ . Thirdly, by giving refinement operators over bp-graph patterns, we present a polynomial time algorithm for finding a minimally generalized bp-graph pattern for ${\mathcal{C}}$ . Outerplanar graphs are planar graphs which can be embedded in the plane in such a way that all of vertices lie on the outer boundary. Many pharmacologic chemical compounds are known to be represented by outerplanar graphs. The class of connected outerplanar graphs ${\mathcal{O}}$ satisfies the above conditions for ${\mathcal{C}}$ . Next, we propose two incremental polynomial time algorithms for enumerating all frequent bp-graph patterns with respect to a given finite set of graphs in ${\mathcal{O}}$ . Finally, by reporting experimental results obtained by applying the two graph mining algorithms to a subset of the NCI dataset, we evaluate the performance of the two graph mining algorithms.