Maximal Outerplanar Graphs Research Articles

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.

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.

Dominating sets inducing large components in maximal outerplanar graphs

AbstractFor a maximal outerplanar graph G of order n at least three, Matheson and Tarjan showed that G has domination number at most . Similarly, for a maximal outerplanar graph G of order n at least five, Dorfling, Hattingh, and Jonck showed, by a completely different approach, that G has total domination number at most unless G is isomorphic to one of two exceptional graphs of order 12.We present a unified proof of a common generalization of these two results. For every positive integer k, we specify a set of graphs of order at least and at most such that every maximal outerplanar graph G of order n at least that does not belong to has a dominating set D of order at most such that every component of the subgraph of G induced by D has order at least k.

Total Dominating Sets in Maximal Outerplanar Graphs

In this note we present an alternative proof of the result by Dorfling et al. (Discrete Math 339(3):1180–1188, 2016) establishing that any maximal outerplanar graph of order \(n \ge 5\) has a total dominating set of size at most \(\lfloor \frac{2n}{5}\rfloor \), apart from two exceptions. In addition, we briefly discuss a relation between total domination in maximal outerplanar graphs and the concept of watched guards in simple polygons.

An upper bound on the algebraic connectivity of outerplanar graphs

In this paper, we determine upper bounds on the algebraic connectivity, denoted as a(G), of maximal outerplanar graphs. We show that if G is a maximal outerplanar graph on n≥12 vertices not of the form K1∨Pn−1, then a(G)≤1 with equality holding for exactly two maximal outerplanar graphs on 12 vertices. We show this by assigning labels y1,…,yn to the vertices and showing the existence of vertex labellings such that ∑uv∈E(G)(yu−yv)2∕∑v∈V(G)yv2<1.

Remote Monitoring by Edges and Faces of Maximal Outerplanar Graphs

Remote Monitoring by Edges and Faces of Maximal Outerplanar Graphs

The general zeroth-order Randić index of maximal outerplanar graphs and trees with k maximum degree vertices

The general zeroth-order Randić index of maximal outerplanar graphs and trees with k maximum degree vertices

Total domination in maximal outerplanar graphs

We show that the total domination number of a maximal outerplanar graph G is bounded above by n+k3, where n is the order of G and k is the number of vertices of degree 2. For k>n3, a better bound is given by 2(n−k)3. For k>n3, we improve the upper bound of n+k4 on the usual domination number.

Combinatorial bounds on connectivity for dominating sets in maximal outerplanar graphs

In this article we study some variants of the domination concept attending to the connectivity of the subgraph generated by the dominant set. This study is restricted to maximal outerplanar graphs. We establish tight combinatorial bounds for connected domination, semitotal domination, independent domination and weakly connected domination for any n-vertex maximal outerplaner graph.

Algorithm on rainbow connection for maximal outerplanar graphs

In this paper, we consider rainbow connection number of maximal outerplanar graphs (MOPs) on algorithmic aspect. For the (MOP) G, we give sufficient conditions to guarantee that rc(G)=diam(G). Moreover, we produce the graph with given diameter D and give their rainbow coloring in linear time. X. Deng et al. [4] give a polynomial time algorithm to compute the rainbow connection number of MOPs by the Maximal fan partition method, but only obtain a compact upper bound. J. Lauri [11] proved that, for chordal outerplanar graphs given an edge-coloring, to verify whether it is rainbow connected is NP-complete under the coloring, it is so for MOPs. Therefore we construct Central-cut-spine of MOP G, by which we design an algorithm to give a rainbow edge coloring with at most 2rad(G)+2+c,0≤c≤rad(G)−2 colors in polynomial time.

Further characterizations and Helly-property in k-trees

The purpose of this paper is to obtain a characterization of $k$-trees in terms of $k$-connectivity and forbidden subgraphs. Also, we present the other characterizations of $k$-trees containing the full vertices by using the join operation. Further, we establish the property of $k$-trees dealing with the degrees and formulate the Helly-property for a family of nontrivial $k$-paths in a $k$-tree. We study the planarity of $k$-trees and express the maximal outerplanar graphs in terms of 2-trees and $K_2$-neighbourhoods. Finally, the similar type of results for the maximal planar graphs are obtained.

On the first-order edge tenacity of a graph

The first-order edge-tenacity T1(G) of a graph G is defined as T1(G)=min{|X|+τ(G−X)ω(G−X)−1} where the minimum is taken over every edge-cutset X that separates G into ω(G−X) components, and by τ(G−X) we denote the order (the number of edges) of a largest component of G−X.The objective of this paper is to study this concept of edge-tenacity and determining this quantity for some special classes of graphs. We calculate the first-order edge-tenacity of a complete n-partite graph. We shall obtain the first-order edge-tenacity of maximal planar graphs, maximal outerplanar graphs, and k-trees. Let G be a graph of order p and size q, we shall call the least integer r, 1≤r≤p−1, with Tr(G)=qp−r the balancity of G and denote it by b(G). Note that the balancity exists since Tr(G)=qp−r if r=p−1. In general, it is difficult to determine the balancity of a graph. In this paper, we shall first determine the balancity of a special class of graphs and use this to find an upper bound for the balancity of an arbitrary graph.

Super edge-magic sequence of maximal outer planer graph and its characteristics

AbstractIn this paper, we study specific striped maximal outer planar graph (MOP) in the view of super edge-magic sequences (SEMS). Also, we derive a formula for the Wiener index (WI) of above MOP graph which is having SEMS. We analyze graphical properties like independence number, chromatic number, dominance number, and matching number of specific striped MOP graph through SEMS.

Total domination in maximal outerplanar graphs II

The total domination number of a graph is the minimum size of a set S such that every vertex has a neighbor in S. We show that a maximal outerplanar graph of order n≥5 has total domination number at most 2n/5, apart from two exceptions, and this bound is best possible.

Gallai's conjecture for outerplanar graphs

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.

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.

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).

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.

A Self-stabilizing Algorithm for Locating the Center of Maximal Outerplanar Graphs

A Self-stabilizing Algorithm for Locating the Center of Maximal Outerplanar Graphs