Outerplanar Graphs Research Articles

Polynomial time algorithm for computing a minimum geodetic set in outerplanar graphs

Given a graph G and a pair of vertices u,v the interval IG[u,v] is the set of all vertices that are in some shortest path between u and v. Given a subset X of vertices of G, the interval IG[X] of X, is the union of the intervals for all pairs of vertices in X and we say that X is geodetic if its interval do coincide with the set of vertices in the graph. A minimum geodetic set is a minimum cardinality geodetic set of G. The problem of computing a minimum geodetic set is known to be NP-Hard for general graphs but is known to be polynomially solvable for maximal outerplanar graphs. In this paper we show a polynomial time algorithm for finding a minimum geodetic set in general outerplanar graphs.

When the maximal graph is planar, outerplanar, and ring graph

Let [Formula: see text] be a commutative ring with nonzero identity. Let [Formula: see text] denote the maximal graph associated to [Formula: see text], that is, [Formula: see text] is a graph with vertices as non-units of [Formula: see text], where two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if there is a maximal ideal of [Formula: see text] containing both. In this paper, we characterize the finite commutative rings such that their maximal graph are planar graphs, and we also study the case where they are outerplanar and ring graphs. The equivalence of outerplanar graphs and ring graphs for [Formula: see text] is established.

Small Universal Point Sets for k-Outerplanar Graphs

A point set $$\mathcal{S} \subseteq \mathbb {R}^2$$ is universal for a class $$\mathcal G$$ of planar graphs if every graph of $$\mathcal{G}$$ has a planar straight-line embedding on $$\mathcal{S}$$ . It is well-known that the integer grid is a quadratic-size universal point set for planar graphs, while the existence of a subquadratic universal point set still remains one of the most fascinating open problems in Graph Drawing. In this paper we make a major step towards a solution for this problem. Motivated by the fact that each point set of size n in general position is universal for the class of n-vertex outerplanar graphs, we concentrate our attention on k-outerplanar graphs. We prove that they admit an $$O(n \log n)$$ -size universal point set in two distinct cases, namely when $$k=2$$ (2-outerplanar graphs) and when k is unbounded but each outerplanarity level is restricted to be a simple cycle (simply-nested graphs).

On interval number in cycle convexity

Recently, Araujo et al. [Manuscript in preparation, 2017] introduced the notion of Cycle Convexity of graphs. In their seminal work, they studied the graph convexity parameter called hull number for this new graph convexity they proposed, and they presented some of its applications in Knot theory. Roughly, the tunnel number of a knot embedded in a plane is upper bounded by the hull number of a corresponding planar 4-regular graph in cycle convexity. In this paper, we go further in the study of this new graph convexity and we study the interval number of a graph in cycle convexity. This parameter is, alongside the hull number, one of the most studied parameters in the literature about graph convexities. Precisely, given a graph G, its interval number in cycle convexity, denoted by $in_{cc} (G)$, is the minimum cardinality of a set S ⊆ V (G) such that every vertex w ∈ V (G) S has two distinct neighbors u, v ∈ S such that u and v lie in same connected component of G[S], i.e. the subgraph of G induced by the vertices in S. In this work, first we provide bounds on $in_{cc} (G)$ and its relations to other graph convexity parameters, and explore its behavior on grids. Then, we present some hardness results by showing that deciding whether $in_{cc} (G)$ ≤ k is NP-complete, even if G is a split graph or a bounded-degree planar graph, and that the problem is W[2]-hard in bipartite graphs when k is the parameter. As a consequence, we obtain that $in_{cc} (G)$ cannot be approximated up to a constant factor in the classes of split graphs and bipartite graphs (unless P = N P ). On the positive side, we present polynomial-time algorithms to compute $in_{cc} (G)$ for outerplanar graphs, cobipartite graphs and interval graphs. We also present fixed-parameter tractable (FPT) algorithms to compute it for (q, q − 4)-graphs when q is the parameter and for general graphs G when parameterized either by the treewidth or the neighborhood diversity of G. Some of our hardness results and positive results are not known to hold for related graph convexities and domination problems. We hope that the design of our new reductions and polynomial-time algorithms can be helpful in order to advance in the study of related graph problems.

The Number of Labeled Outerplanar k-Cyclic Graphs

A k-cyclic graph is a graph with cyclomatic number k. An explicit formula for the number of labeled connected outerplanar k-cyclic graphs with a given number of vertices is obtained. In addition, such graphs with fixed cyclomatic number k and a large number of vertices are asymptotically enumerated. As a consequence, it is found that, for fixed k, almost all labeled connected outerplanar k-cyclic graphs with a large number of vertices are cacti.

A study on oriented relative clique number

An oriented graph is a directed graph with no directed cycle of length one or two. The relative clique number of an oriented graph is the cardinality of a largest subset X of vertices such that each pair of vertices is either adjacent or connected by a directed 2-path. It is known that the oriented relative clique number of a planar graph is at most 80. Here we improve the upper bound to 32. We also prove an upper bound of 14 for oriented relative clique number of triangle-free planar graphs. Furthermore, we determine the exact values of oriented relative clique number for the families of outerplanar graphs with girth at least g and planar graphs with girth at least g+2 for all g≥3. Moreover, we study the relation of oriented relative clique number with oriented chromatic number, oriented absolute clique number and maximum degree of a graph. We also show that oriented relative clique number of a connected subcubic graph is at most seven which weakly supports a conjecture by Sopena (JGT 1997).

On pathos block line graph of a tree

Let T be a tree of order n ≥ 2. A pathos line graph of T, written PL(T), is a graph whose vertices are the edges and paths of a pathos of T, with two vertices of PL(T) adjacent whenever the corresponding edges of T are adjacent or the edge lies on the corresponding path of the pathos. An extension is called a pathos block line graph, denoted by PBL(T), and is similar to PL(T), except that the adjacency of pathos vertices. For this class of graphs we discuss the planarity, outer planarity, maximal outer planarity, minimally nonouter planarity, and crossing number one properties of these graphs.

Defective and Clustered Graph Colouring

Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has defect $d$ if each monochromatic component has maximum degree at most $d$. A colouring has clustering $c$ if each monochromatic component has at most $c$ vertices. This paper surveys research on these types of colourings, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. List colouring variants are also considered. The following graph classes are studied: outerplanar graphs, planar graphs, graphs embeddable in surfaces, graphs with given maximum degree, graphs with given maximum average degree, graphs excluding a given subgraph, graphs with linear crossing number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdière parameter, graphs with given circumference, graphs excluding a given immersion, graphs with given thickness, graphs with given stack- or queue-number, graphs excluding $K_t$ as a minor, graphs excluding $K_{s,t}$ as a minor, and graphs excluding an arbitrary graph $H$ as a minor. Several open problems are discussed.

Relative clique number of planar signed graphs

A signed relative clique number of signed graph (where edges are assigned positive or negative signs) is the size of a largest subset X of vertices such that every two vertices are either adjacent or are part of a 4-cycle with an odd number of negative edges. The signed relative clique number is sandwiched between two other parameters of signed graphs, namely, the signed absolute clique number and the signed chromatic number, all three notions defined in Naserasr et al. (2014). Thus, together with a result from Ochem et al. (2016) the lower bound of 8 and upper bound of 40 has already been proved for the signed relative clique number of the family of planar graphs. Here we improve the upper bound to 15. Furthermore, we determine the exact values of signed relative clique number of the families of outerplanar graphs and triangle-free planar graphs.

Thickness and outerthickness for embedded graphs

We prove new upper bounds for the thickness and outerthickness of a graph in terms of its orientable and nonorientable genus by applying the method of deleting spanning disks of embeddings to approximate the thickness and outerthickness. We also show that every non-planar toroidal graph can be edge partitioned into a planar graph and an outerplanar graph. This implies that the outerthickness of the torus (the maximum outerthickness of all toroidal graphs) is 3. Finally, we show that all graphs embeddable in the double torus have thickness at most 3 and outerthickness at most 5.

A simple linear time algorithm for the locally connected spanning tree problem on maximal planar chordal graphs

A locally connected spanning tree (LCST) T of a graph G is a spanning tree of G such that, for each node, its neighborhood in T induces a connected subgraph in G. The problem of determining whether a graph contains an LCST or not has been proved to be NP-complete, even if the graph is planar or chordal. The main result of this paper is a simple linear time algorithm that, given a maximal planar chordal graph, determines in linear time whether it contains an LCST or not, and produces one if it exists. We give an analogous result for the case when the input graph is a maximal outerplanar graph.

Realizability of graphs as triangle cover contact graphs

Let S={p1,p2,…,pn} be a set of pairwise disjoint geometric objects of some type and let C={c1,c2,…,cn} be a set of closed objects of some type with the property that each element in C covers exactly one element in S and any two elements in C can intersect only on their boundaries. We call an element in S a seed and an element in C a cover. A cover contact graph (CCG) consists of a set of vertices and a set of edges where each of the vertex corresponds to each of the covers and each edge corresponds to a connection between two covers if and only if they touch at their boundaries. A triangle cover contact graph (TCCG) is a cover contact graph whose cover elements are triangles. In this paper, we show that every Halin graph has a realization as a TCCG on a given set of collinear seeds. We introduce a new class of graphs which we call super-Halin graphs. We also show that the classes super-Halin graphs, cubic planar Hamiltonian graphs and a×b grid graphs have realizations as TCCGs on collinear seeds. We also show that some non-planar graphs have planar realizations as TCCGs. Every complete graph and clique-3 graph have realizations as TCCGs on any given set of seeds. Note that only trees and cycles are known to be realizable as CCGs and outerplanar graphs are known to be realizable as TCCGs.

Rainbow numbers for small cycles

The rainbow number $rb(G,H)$ is the minimum number $k$ such that any $k$-edge-coloring of $G$ contains a rainbow copy of $H$. In this paper, we determine the rainbow numbers of small cycles in the complete split graph and maximal outerplanar graph.

Acquisition of characteristic sets of block preserving outerplanar graph patterns by a two-stage evolutionary learning method for graph pattern sets

Knowledge acquisition from graph structured data is an important task in machine learning and data mining. Block preserving outerplanar graph patterns are graph structured patterns having structured variables and are suited to represent characteristic graph structures of graph data modelled as outerplanar graphs. We propose a learning method for acquiring characteristic sets of block preserving outerplanar graph patterns by a two-stage evolutionary learning method for graph pattern sets as individuals, from positive and negative outerplanar graph data, in order to represent characteristic graph structures more concretely.

Bounding the Order of a Graph Using Its Diameter and Metric Dimension: A Study Through Tree Decompositions and VC Dimension

The metric dimension of a graph is the minimum size of a set of vertices such that each vertex is uniquely determined by the distances to the vertices of that set. Our aim is to upper-bound the order $n$ of a graph in terms of its diameter $d$ and metric dimension $k$. In general, the bound $n\leq d^k+k$ is known to hold. We prove a bound of the form $n=\mathcal{O}(kd^2)$ for trees and outerplanar graphs (for trees we determine the best possible bound and the corresponding extremal examples). More generally, for graphs having a tree decomposition of width $w$ and length $\ell$, we obtain a bound of the form $n=\mathcal{O}(kd^2(2\ell+1)^{3w+1})$. This implies in particular that $n=\mathcal{O}(kd^{\mathcal{O}(1)})$ for graphs of constant treewidth and $n=\mathcal{O}(f(k)d^2)$ for chordal graphs, where $f$ is a doubly-exponential function. Using the notion of distance-VC dimension (introduced in 2014 by Bousquet and Thomass\'e) as a tool, we prove the bounds $n\leq (dk+1)^{t-1}+1$ for $K_t$-minor-free graphs, and $n\leq (dk+1)^{d(3\cdot 2^{r}+2)}+1$ for graphs of rankwidth at most $r$.

Orthogonal Tree Decompositions of Graphs

This paper studies graphs that have two tree decompositions with the property that every bag from the first decomposition has a bounded-size intersection with every bag from the second decomposition. We show that every graph in each of the following classes has a tree decomposition and a linear-sized path decomposition with bounded intersections: (1) every proper minor-closed class, (2) string graphs with a linear number of crossings in a fixed surface, (3) graphs with linear crossing number in a fixed surface. Here `linear size' means that the total size of the bags in the path decomposition is $O(n)$ for $n$-vertex graphs. We then show that every $n$-vertex graph that has a tree decomposition and a linear-sized path decomposition with bounded intersections has $O(\sqrt{n})$ treewidth. As a corollary, we conclude a new lower bound on the crossing number of a graph in terms of its treewidth. Finally, we consider graph classes that have two path decompositions with bounded intersections. Trees and outerplanar graphs have this property. But for the next most simple class, series parallel graphs, we show that no such result holds.

Drawing Planar Graphs with Few Geometric Primitives

We define the visual complexity of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw a path with an arbitrary number of edges). Let $n$ denote the number of vertices of a graph. We show that trees can be drawn with $3n/4$ straight-line segments on a polynomial grid, and with $n/2$ straight-line segments on a quasi-polynomial grid. Further, we present an algorithm for drawing planar 3-trees with $(8n-17)/3$ segments on an $O(n)\times O(n^2)$ grid. This algorithm can also be used with a small modification to draw maximal outerplanar graphs with $3n/2$ edges on an $O(n)\times O(n^2)$ grid. We also study the problem of drawing maximal planar graphs with circular arcs and provide an algorithm to draw such graphs using only $(5n - 11)/3$ arcs. This is significantly smaller than the lower bound of $2n$ for line segments for a nontrivial graph class.

Rainbow numbers for small cycles

The rainbow number $rb(G,H)$ is the minimum number $k$ such that any $k$-edge-coloring of $G$ contains a rainbow copy of $H$. In this paper, we determine the rainbow numbers of small cycles in the complete split graph and maximal outerplanar graph.

Acquisition of characteristic sets of block preserving outerplanar graph patterns by a two-stage evolutionary learning method for graph pattern sets

Acquisition of characteristic sets of block preserving outerplanar graph patterns by a two-stage evolutionary learning method for graph pattern sets

On facial unique-maximum (edge-)coloring

A facial unique-maximum coloring of a plane graph is a vertex coloring where on each face α the maximal color appears exactly once on the vertices of α. If the coloring is required to be proper, then the upper bound for the minimal number of colors required for such a coloring is set to 5. Fabrici and Göring (2016) even conjectured that 4 colors always suffice. Confirming the conjecture would hence give a considerable strengthening of the Four Color Theorem. In this paper, we prove that the conjecture holds for subcubic plane graphs, outerplane graphs and plane quadrangulations. Additionally, we consider the facial edge-coloring analogue of the aforementioned coloring and prove that every 2-connected plane graph admits such a coloring with at most 4 colors.