scholarly journals Complexity of domination in triangulated plane graphs

2019 ◽  
Vol 11 (2) ◽  
pp. 174-183
Author(s):  
Dömötör Pálvölgyi
Keyword(s):  
Domination Number ◽  
Plane Graph ◽  
Plane Graphs ◽  
Power Domination ◽  
Np Complete

Abstract We prove that for a triangulated plane graph it is NP-complete to determine its domination number and its power domination number.

scholarly journals FOLDING EQUILATERAL PLANE GRAPHS

2013 ◽  
Vol 23 (02) ◽  
pp. 75-92 ◽  
Author(s):  
ZACHARY ABEL ◽  
ERIK D. DEMAINE ◽  
MARTIN L. DEMAINE ◽  
SARAH EISENSTAT ◽  
JAYSON LYNCH ◽  
...  
Keyword(s):  
Plane Graph ◽  
Analogous Problem ◽  
Central Vertex ◽  
Polyhedral Complex ◽  
Plane Graphs ◽  
Single Vertex ◽  
Np Completeness ◽  
Np Complete ◽  
Polyhedral Manifold

We consider two types of folding applied to equilateral plane graph linkages. First, under continuous folding motions, we show how to reconfigure any linear equilateral tree (lying on a line) into a canonical configuration. By contrast, it is known that such reconfiguration is not always possible for linear (nonequilateral) trees and for (nonlinear) equilateral trees. Second, under instantaneous folding motions, we show that an equilateral plane graph has a noncrossing linear folded state if and only if it is bipartite. Furthermore, we show that the equilateral constraint is necessary for this result, by proving that it is strongly NP-complete to decide whether a (nonequilateral) plane graph has a linear folded state. Equivalently, we show strong NP-completeness of deciding whether an abstract metric polyhedral complex with one central vertex has a noncrossing flat folded state. By contrast, the analogous problem for a polyhedral manifold with one central vertex (single-vertex origami) is only weakly NP-complete.

scholarly journals Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 293
Author(s):  
Xinyue Liu ◽  
Huiqin Jiang ◽  
Pu Wu ◽  
Zehui Shao
Keyword(s):  
Linear Time ◽  
Minimum Weight ◽  
Domination Number ◽  
Time Algorithm ◽  
Simple Graph ◽  
Linear Time Algorithm ◽  
Domination Problem ◽  
Np Complete ◽  
Chordal Bipartite Graphs ◽  
Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

GENERA OF THE LINKS DERIVED FROM 2-CONNECTED PLANE GRAPHS

2012 ◽  
Vol 21 (14) ◽  
pp. 1250129 ◽  
Author(s):  
SHUYA LIU ◽  
HEPING ZHANG
Keyword(s):  
Explicit Formula ◽  
Large Family ◽  
Plane Graph ◽  
Plane Graphs ◽  
Degree Sum ◽  
Oriented Link

In this paper, we associate a plane graph G with an oriented link by replacing each vertex of G with a special oriented n-tangle diagram. It is shown that such an oriented link has the minimum genus over all orientations of its unoriented version if its associated plane graph G is 2-connected. As a result, the genera of a large family of unoriented links are determined by an explicit formula in terms of their component numbers and the degree sum of their associated plane graphs.

scholarly journals A New Algorithm for Embedding Plane Graphs at Fixed Vertex Locations

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Marcus Schaefer
Keyword(s):  
Plane Graph ◽  
Plane Graphs ◽  
Classic Result

We show that a plane graph can be embedded with its vertices at arbitrarily assigned locations in the plane and at most $6n-1$ bends per edge. This improves and simplifies a classic result by Pach and Wenger. The proof extends to orthogonal drawings.

scholarly journals Equitable power domination number of total graph of certain graphs

2020 ◽  
Vol 1531 ◽  
pp. 012073
Author(s):  
S Banu Priya ◽  
A Parthiban ◽  
P Abirami
Keyword(s):  
Domination Number ◽  
Total Graph ◽  
Power Domination

CONVEX GRID DRAWINGS OF FOUR-CONNECTED PLANE GRAPHS

2006 ◽  
Vol 17 (05) ◽  
pp. 1031-1060 ◽  
Author(s):  
KAZUYUKI MIURA ◽  
SHIN-ICHI NAKANO ◽  
TAKAO NISHIZEKI
Keyword(s):  
Convex Polygon ◽  
Linear Time ◽  
Time Algorithm ◽  
Plane Graph ◽  
Outer Face ◽  
Plane Graphs ◽  
Line Segments ◽  
Grid Drawing ◽  
Straight Line ◽  
Grid Points

A convex grid drawing of a plane graph G is a drawing of G on the plane such that all vertices of G are put on grid points, all edges are drawn as straight-line segments without any edge-intersection, and every face boundary is a convex polygon. In this paper we give a linear-time algorithm for finding a convex grid drawing of every 4-connected plane graph G with four or more vertices on the outer face. The size of the drawing satisfies W + H ≤ n - 1, where n is the number of vertices of G, W is the width and H is the height of the grid drawing. Thus the area W · H is at most ⌈(n - 1)/2⌉ · ⌊(n - 1)/2⌋. Our bounds on the sizes are optimal in a sense that there exist an infinite number of 4-connected plane graphs whose convex drawings need grids such that W + H = n - 1 and W · H = ⌈(n - 1)/2⌉ · ⌊(n - 1)/2⌋.

DETERMINING THE COMPONENT NUMBER OF LINKS CORRESPONDING TO LATTICES

2009 ◽  
Vol 18 (12) ◽  
pp. 1711-1726 ◽  
Author(s):  
XIAN'AN JIN ◽  
FENGMING DONG ◽  
ENG GUAN TAY
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Plane Graph ◽  
Link Diagram ◽  
Plane Graphs ◽  
Link Diagrams ◽  
One To One

It is well known that there is a one-to-one correspondence between signed plane graphs and link diagrams via the medial construction. The component number of the corresponding link diagram is however independent of the signs of the plane graph. Determining this number may be one of the first problems in studying links by using graphs. Some works in this aspect have been done. In this paper, we investigate the component number of links corresponding to lattices. Firstly we provide some general results on component number of links. Then, via these results, we proceed to determine the component number of links corresponding to lattices with free or periodic boundary conditions and periodic lattices with one cap (i.e. spiderweb graphs) or two caps.

scholarly journals Heredity for generalized power domination

2016 ◽  
Vol Vol. 18 no. 3 (Graph Theory) ◽  
Author(s):  
Paul Dorbec ◽  
Seethu Varghese ◽  
Ambat Vijayakumar
Keyword(s):  
Domination Number ◽  
Power Domination ◽  
Edge Contraction ◽  
Vertex Deletion ◽  
International Audience ◽  
Optimal Bounds

International audience In this paper, we study the behaviour of the generalized power domination number of a graph by small changes on the graph, namely edge and vertex deletion and edge contraction. We prove optimal bounds for $\gamma_{p,k}(G-e)$, $\gamma_{p,k}(G/e)$ and for $\gamma_{p,k}(G-v)$ in terms of $\gamma_{p,k}(G)$, and give examples for which these bounds are tight. We characterize all graphs for which $\gamma_{p,k}(G-e) = \gamma_{p,k}(G)+1$ for any edge $e$. We also consider the behaviour of the propagation radius of graphs by similar modifications.

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