Outerplanar Graphs Research Articles

Further hardness results on rainbow and strong rainbow connectivity

A path in an edge-colored graph is rainbow if no two edges of it are colored the same. The graph is said to be rainbow connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph is strong rainbow connected. We consider the complexity of the problem of deciding if a given edge-colored graph is rainbow or strong rainbow connected. These problems are called Rainbow connectivity and Strong rainbow connectivity, respectively. We prove both problems remain NP-complete on interval outerplanar graphs and k-regular graphs for k≥3. Previously, no graph class was known where the complexity of the two problems would differ. We show that for block graphs, which form a subclass of chordal graphs, Rainbow connectivity is NP-complete while Strong rainbow connectivity is in P. We conclude by considering some tractable special cases, and show for instance that both problems are in XP when parameterized by tree-depth.

Bipartite minors

We introduce a notion of bipartite minors and prove a bipartite analog of Wagner's theorem: a bipartite graph is planar if and only if it does not contain K3,3 as a bipartite minor. Similarly, we provide a forbidden minor characterization for outerplanar graphs and forests. We then establish a recursive characterization of bipartite (2,2)-Laman graphs — a certain family of graphs that contains all maximal bipartite planar graphs.

Excluded-Minor Characterization of Apex-Outerplanar Graphs

The class of outerplanar graphs is minor-closed and can be characterized by two excluded minors: $$K_4$$K4 and $$K_{2,3}$$K2,3. 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.

On dominating sets of maximal outerplanar and planar graphs

A set D⊆V(G) of a graph G is a dominating set if every vertex v∈V(G) is either in D or adjacent to a vertex in D. The domination number γ(G) of a graph G is the minimum cardinality of a dominating set of G. Campos and Wakabayashi (2013) and Tokunaga (2013) proved independently that if G is an n-vertex maximal outerplanar graph having t vertices of degree 2, then γ(G)≤n+t4. We improve their upper bound by showing γ(G)≤n+k4, where k is the number of pairs of consecutive 2-degree vertices with distance at least 3 on the outer cycle. Moreover, we prove that γ(G)≤5n16 for a Hamiltonian maximal planar graph G with n≥7 vertices.

Outerplanar and Planar Oriented Cliques

AbstractThe 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 .

On [formula omitted]-edge-magic labelings of maximal outerplanar graphs

Let G be a graph with vertex set V and edge set E such that |V|=p and |E|=q. We denote this graph by (p,q)-graph. For integers k≥0, define a one-to-one map f from E to {k,k+1,…,k+q−1} and define the vertex sum for a vertex v as the sum of the labels of the edges incident to v. If such an edge labeling induces a vertex labeling in which every vertex has a constant vertex sum (modp), then G is said to be k-edge magic (k-EM). In this paper, we show that a maximal outerplanar graph of orders p = 4, 5, 7 are k-EM if and only if k≡2(modp) and obtain all maximal outerplanar graphs that are k-EM for k = 3, 4. Finally we characterize all (p,p−h)-graphs that are k-EM for h≥0. We conjecture that a maximal outerplanar graph of prime order p is k-EM if and only if k≡2(modp).

Contractible Edges in 2-Connected Locally Finite Graphs

In this paper, we prove that every contraction-critical 2-connected infinite graph has no vertex of finite degree and contains uncountably many ends. Then, by investigating the distribution of contractible edges in a 2-connected locally finite infinite graph $G$, we show that the closure of the subgraph induced by all the contractible edges in the Freudenthal compactification of $G$ is 2-arc-connected. Finally, we characterize all 2-connected locally finite outerplanar graphs nonisomorphic to $K_3$ as precisely those graphs such that every vertex is incident to exactly two contractible edges as well as those graphs such that every finite bond contains exactly two contractible edges.

Domination of maximal [formula omitted]-minor free graphs and maximal [formula omitted]-minor free graphs, and disproofs of two conjectures on planar graphs

It is well known that a graph is outerplanar if and only if it is K4-minor free and K2,3-minor free. Campos and Wakabayashi (2013) recently proved that γ(G)≤⌊n+k4⌋ for any maximal outerplanar graph G of order n≥3 with k vertices of degree 2, where γ(G) denotes the domination number of G. Tokunaga (2013) provided a short proof for the above theorem. Based on some structural properties of K2,3-minor free graphs and K4-minor free graphs, applying the idea of Tokunaga we extend the theorem of Campos and Wakabayashi to all maximal K4-minor free graphs and all maximal K2,3-minor free graphs. We also disprove two conjectures of Tokunaga on planar graphs.

Distance [formula omitted]-domination, distance [formula omitted]-guarding, and distance [formula omitted]-vertex cover of maximal outerplanar graphs

For positive integers k and n, let γk(n), gk(n), and βk(n) denote the maximum values of the distance k-domination number, the distance k-guarding number, and the distance k-vertex cover number of maximal outerplanar graphs of order n, respectively. Known results imply γ1(n)=⌊n/3⌋ for n≥3 (Matheson et al., 1996), g1(n)=⌊n/3⌋ for n≥3 (Chvátal, 1975), γ2(n)=g2(n)=⌊n/5⌋ for n≥5, and β2(n)=⌊n/4⌋ for n≥4 (Canales et al., 2015).We show γk(n)=gk(n)=⌊n/(2k+1)⌋ for n≥2k+1, and βk(n)=⌊n/2k⌋ for n≥2k≥4.

Broken circuit complexes of series–parallel networks

Let (h0,h1,…,hs) with hs≠0 be the h-vector of the broken circuit complex of a series–parallel network M. Let G be a graph whose cycle matroid is M. We give a formula for the difference hs−1−h1 in terms of an ear decomposition of G. A number of applications of this formula are provided, including several bounds for hs−1−h1, a characterization of outerplanar graphs, and a solution to a conjecture on A-graphs posed by Fenton. We also prove that hs−2≥h2 when s≥4.

SEFE without Mapping via Large Induced Outerplane Graphs in Plane Graphs

AbstractWe show that every n‐vertex planar graph admits a simultaneous embedding without mapping and with fixed edges with any ‐vertex planar graph. In order to achieve this result, we prove that every n‐vertex plane graph has an induced outerplane subgraph containing at least vertices. Also, we show that every n‐vertex planar graph and every n‐vertex planar partial 3‐tree admit a simultaneous embedding without mapping and with fixed edges.

Outer 1-Planar Graphs

A graph is outer 1-planar (o1p) if it can be drawn in the plane such that all vertices are in the outer face and each edge is crossed at most once. o1p graphs generalize outerplanar graphs, which can be recognized in linear time, and specialize 1-planar graphs, whose recognition is $${NP}$$NP-hard. We explore o1p graphs. Our first main result is a linear-time algorithm that takes a graph as input and returns a positive or a negative witness for o1p. If a graph $$G$$G is o1p, then the algorithm computes an embedding and can augment $$G$$G to a maximal o1p graph. Otherwise, $$G$$G includes one of six minors, which is detected by the recognition algorithm. Secondly, we establish structural properties of o1p graphs. o1p graphs are planar and are subgraphs of planar graphs with a Hamiltonian cycle. They are neither closed under edge contraction nor under subdivision. Several important graph parameters, such as treewidth, colorability, stack number, and queue number, increase by one from outerplanar to o1p graphs. Every o1p graph of size $$n$$n has at most $$\frac{5}{2} n - 4$$52n-4 edges and there are maximal o1p graphs with $$\frac{11}{5} n - \frac{18}{5}$$115n-185 edges, and these bounds are tight. Finally, every o1p graph has a straight-line grid drawing in $$\fancyscript{O}(n^2)$$O(n2) area with all vertices in the outer face, a planar visibility representation in $$\fancyscript{O}(n \log n)$$O(nlogn) area, and a 3D straight-line drawing in linear volume, and these drawings can be constructed in linear time.

On the planar, outer planar, cut vertices and end-regular comaximal graph of lattices

On the planar, outer planar, cut vertices and end-regular comaximal graph of lattices

Approximating the minimum hub cover problem on planar graphs

We study an approximation algorithm with a performance guarantee to solve a new $$\mathcal {NP}$$ -hard optimization problem on planar graphs. The problem, which is referred to as the minimum hub cover problem, has recently been introduced to the literature to improve query processing over large graph databases. Planar graphs also arise in various graph query processing applications, such as; biometric identification, image classification, object recognition, and so on. Our algorithm is based on a well-known graph decomposition technique that partitions the graph into a set of outerplanar graphs and provides an approximate solution with a proven performance ratio. We conduct a comprehensive computational experiment to investigate the empirical performance of the algorithm. Computational results demonstrate that the empirical performance of the algorithm surpasses its guaranteed performance. We also apply the same decomposition approach to develop a decomposition-based heuristic, which is much more efficient than the approximation algorithm in terms of computation time. Computational results also indicate that the efficacy of the decomposition-based heuristic in terms of solution quality is comparable to that of the approximation algorithm.

P-box: a new graph model

Graph Theory In this document, we study the scope of the following graph model: each vertex is assigned to a box in ℝd and to a representative element that belongs to that box. Two vertices are connected by an edge if and only if its respective boxes contain the opposite representative element. We focus our study on the case where boxes (and therefore representative elements) associated to vertices are spread in ℝ. We give both, a combinatorial and an intersection characterization of the model. Based on these characterizations, we determine graph families that contain the model (e. g., boxicity 2 graphs) and others that the new model contains (e. g., rooted directed path). We also study the particular case where each representative element is the center of its respective box. In this particular case, we provide constructive representations for interval, block and outerplanar graphs. Finally, we show that the general and the particular model are not equivalent by constructing a graph family that separates the two cases.

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

A treet-spanner of an unweighted graph G is a spanning tree T such that for every two vertices their distance in T is at most t times their distance in G. Given an unweighted graph G and a positive integer t as input, the Treet-spanner problem is to compute a tree t-spanner of G if one exists. This decision problem is known to be NP-complete even in the restricted class of unweighted planar graphs. We present a linear-time reduction from Treet-spanner in outerplanar graphs to the supply–demand tree partition problem. Based on this reduction, we obtain a linear-time algorithm to solve Treet-spanner in outerplanar graphs. Consequently, we show that the minimum value of t for which an input outerplanar graph on n vertices has a tree t-spanner can be found in O(nlogn) time.

Enumeration of labeled geodetic planar graphs

The labeled geodetic outerplanar and planar graphs with a prescribed number of vertices are enumerated and an asymptotic formula for the number of such graphs with a large number of vertices is obtained.

Star Edge Coloring of Some Classes of Graphs

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

An $$O(n\log n)$$ O ( n log n ) algorithm for finding edge span of cacti

Let $$G=(V,E)$$G=(V,E) be a nonempty graph and $$\xi :E\rightarrow \mathbb {N}$$?:E?N be a function. In the paper we study the computational complexity of the problem of finding vertex colorings $$c$$c of $$G$$G such that:(1)$$|c(u)-c(v)|\ge \xi (uv)$$|c(u)-c(v)|??(uv) for each edge $$uv\in E$$uv?E;(2)the edge span of $$c$$c, i.e. $$\max \{|c(u)-c(v)|:uv\in E\}$$max{|c(u)-c(v)|:uv?E}, is minimal. We show that the problem is NP-hard for subcubic outerplanar graphs of a very simple structure (similar to cycles) and polynomially solvable for cycles and bipartite graphs. Next, we use the last two results to construct an algorithm that solves the problem for a given cactus $$G$$G in $$O(n\log n)$$O(nlogn) time, where $$n$$n is the number of vertices of $$G$$G.

Chromatic sums for colorings avoiding monochromatic subgraphs

Abstract Given graphs G and H, a vertex coloring c : V ( G ) → N is an H-free coloring of G if no color class contains a subgraph isomorphic to H. The H-free chromatic number of G, χ ( H , G ) , is the minimum number of colors in an H-free coloring of G. The H-free chromatic sum of G , Σ ( H , G ) , is the minimum value achieved by summing the vertex colors of each H-free coloring of G. We provide a general bound for Σ ( H , G ) , discuss the computational complexity of finding this parameter for different choices of H, and prove an exact formulas for some graphs G. For every integer k and for every graph H, we construct families of graphs, G k with the property that k more colors than χ ( H , G ) are required to realize Σ ( H , G ) for H-free colorings. More complexity results and constructions of graphs requiring extra colors are given for planar and outerplanar graphs.