GUARDING RECTANGULAR PARTITIONS

2009 ◽  
Vol 19 (06) ◽  
pp. 579-594 ◽  
Author(s):  
YEFIM DINITZ ◽  
MATTHEW J. KATZ ◽  
ROI KRAKOVSKI
Keyword(s):  
Planar Graphs ◽  
Vertex Cover ◽  
Plane Graph ◽  
Common Point ◽  
Distinct Vertex ◽  
Np Hard ◽  
Worst Case ◽  
Minimum Cardinality ◽  
Graph Theoretic ◽  
Np Hardness

A rectangular partition is a partition of a rectangle into non-overlapping rectangles, such that no four rectangles meet at a common point. A vertex guard is a guard located at a vertex of the partition (i.e., at a corner of a rectangle); it guards the rectangles that meet at this vertex. An edge guard is a guard that patrols along an edge of the partition, and is thus equivalent to two adjacent vertex guards. We consider the problem of finding a minimum-cardinality guarding set for the rectangles of the partition. For vertex guards, we prove that guarding a given subset of the rectangles is NP-hard. For edge guards, we prove that guarding all rectangles, where guards are restricted to a given subset of the edges, is NP-hard. For both results we show a reduction from vertex cover in non-bipartite 3-connected cubic planar graphs of girth greater than three. For the second NP-hardness result, we obtain a graph-theoretic result which establishes a connection between the set of faces of a plane graph of vertex degree at most three and a vertex cover for this graph. More precisely, we prove that one can assign to each internal face a distinct vertex of the cover, which lies on the face's boundary. We show that the vertices of a rectangular partition can be colored red, green, or black, such that each rectangle has all three colors on its boundary. We conjecture that the above is also true for four colors. Finally, we obtain a worst-case upper bound on the number of edge guards that are sufficient for guarding rectangular partitions with some restrictions on their structure.

scholarly journals The Simultaneous Strong Resolving Graph and the Simultaneous Strong Metric Dimension of Graph Families

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 125
Author(s):  
Ismael González Yero
Keyword(s):  
Vertex Cover ◽  
Common Vertex ◽  
Metric Dimension ◽  
Minimum Cardinality ◽  
New Approach ◽  
Strong Metric Dimension ◽  
Vertex Set ◽  
Graph Families ◽  
Vertex Sets ◽  
Np Hardness

We consider in this work a new approach to study the simultaneous strong metric dimension of graphs families, while introducing the simultaneous version of the strong resolving graph. In concordance, we consider here connected graphs G whose vertex sets are represented as V ( G ) , and the following terminology. Two vertices u , v ∈ V ( G ) are strongly resolved by a vertex w ∈ V ( G ) , if there is a shortest w − v path containing u or a shortest w − u containing v. A set A of vertices of the graph G is said to be a strong metric generator for G if every two vertices of G are strongly resolved by some vertex of A. The smallest possible cardinality of any strong metric generator (SSMG) for the graph G is taken as the strong metric dimension of the graph G. Given a family F of graphs defined over a common vertex set V, a set S ⊂ V is an SSMG for F , if such set S is a strong metric generator for every graph G ∈ F . The simultaneous strong metric dimension of F is the minimum cardinality of any strong metric generator for F , and is denoted by Sd s ( F ) . The notion of simultaneous strong resolving graph of a graph family F is introduced in this work, and its usefulness in the study of Sd s ( F ) is described. That is, it is proved that computing Sd s ( F ) is equivalent to computing the vertex cover number of the simultaneous strong resolving graph of F . Several consequences (computational and combinatorial) of such relationship are then deduced. Among them, we remark for instance that we have proved the NP-hardness of computing the simultaneous strong metric dimension of families of paths, which is an improvement (with respect to the increasing difficulty of the problem) on the results known from the literature.

SIMULTANEOUS EMBEDDING OF EMBEDDED PLANAR GRAPHS

2013 ◽  
Vol 23 (02) ◽  
pp. 93-126 ◽  
Author(s):  
PATRIZIO ANGELINI ◽  
GIUSEPPE DI BATTISTA ◽  
FABRIZIO FRATI
Keyword(s):  
Planar Graphs ◽  
Np Hard ◽  
Line Segments ◽  
Planar Embedding ◽  
Straight Line ◽  
K 13 ◽  
Hard Problems ◽  
Simultaneous Embedding ◽  
Np Hardness ◽  
Np Hard Problems

A simultaneous embedding with fixed edges (SEFE) of a set of k planar graphs G1,…,Gk on the same set of vertices is a set of k planar drawings of G1,…,Gk, respectively, such that each vertex is placed on the same point in all the drawings and each edge is represented by the same Jordan curve in the drawings of all the graphs it belongs to. A simultaneous geometric embedding (SGE) is a SEFE in which the edges are represented by straight-line segments. Given k planar graphs G1,…,Gk, deciding whether they admit a SEFE and whether they admit an SGE are NP-hard problems, for k ≥ 3 and for k ≥ 2, respectively. In this paper we consider the complexity of SEFE and of SGE when the graphs G1,…,Gk have a fixed planar embedding. In sharp contrast with the NP-hardness of SEFE for three non-embedded planar graphs, we show that SEFE is quadratic-time solvable for three graphs with a fixed planar embedding. Furthermore, we show that, given k embedded planar graphs G1,…,Gk, deciding whether a SEFE of G1,…,Gk exists and deciding whether an SGE of G1,…,Gk exists are NP-hard problems, for k ≥ 14 and k ≥ 13, respectively.

scholarly journals Worst-case-optimal algorithms for guarding planar graphs and polyhedral surfaces

2003 ◽  
Vol 26 (3) ◽  
pp. 209-219 ◽  
Author(s):  
Prosenjit Bose ◽  
David Kirkpatrick ◽  
Zaiqing Li
Keyword(s):  
Planar Graphs ◽  
Optimal Algorithms ◽  
Worst Case ◽  
Polyhedral Surfaces

Evaluation of supply chain resilience index: a graph theory based approach

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Nishtha Agarwal ◽  
Nitin Seth ◽  
Ashish Agarwal
Keyword(s):  
Supply Chain ◽  
Adjacency Matrix ◽  
Theoretic Approach ◽  
Worst Case ◽  
Content Type ◽  
Supply Chain Resilience ◽  
Graph Theoretic ◽  
Graph Theoretic Approach ◽  
Resilience Index ◽  
Index Value

PurposeThe present study aims at developing a model to quantify supply chain resilience as a single numerical value. The numerical value is called resilience index that measures the resilience capability of the case company's supply chain. The model calculates the index value based on the interactions between the enablers of supply chain resilience and its dimensions.Design/methodology/approachGraph theoretic approach (GTA) is used to evaluate the resilience index for the case company's supply chain. In GTA, the dimensions of resilience enablers and their interdependencies are modelled through a digraph. The digraph depicting the influence of each dimension is converted into an adjacency matrix. The permanent function value of the adjacency matrix is called the resilience index (RI).FindingsThe proposed approach has been illustrated in context of an Indian automobile organization, and value of the RI is evaluated. The best case and the worst-case values are also obtained with the help of GTA. It is noted from the model that strategic level dimension of enablers is most important in contributing towards supply chain resilience. They are followed by tactical and operational level enablers. The GTA framework proposed will help supply chain practitioners to evaluate and benchmark the supply chain resilience of their respective organizations with the best in the industry.Originality/valueA firm can compare the RI of its own supply chain with other's supply chain or with the best in the industry for benchmarking purpose. Benchmarking of resilience will help organizations in developing strategies to compete in dynamic market scenario.

scholarly journals Solving NP-hard problems in ‘almost trees’: Vertex cover

1985 ◽  
Vol 10 (1) ◽  
pp. 27-45 ◽  
Author(s):  
Don Coppersmith ◽  
Uzi Vishkin
Keyword(s):  
Vertex Cover ◽  
Np Hard ◽  
Hard Problems ◽  
Np Hard Problems

Finding an Unknown Acyclic Orientation of a Given Graph

2009 ◽  
Vol 19 (1) ◽  
pp. 121-131 ◽  
Author(s):  
OLEG PIKHURKO
Keyword(s):  
Complete Graph ◽  
Discrete Mathematics ◽  
Np Hard ◽  
Worst Case ◽  
Acyclic Orientation ◽  
Multiplicative Factor ◽  
The Given

Let c(G) be the smallest number of edges we have to test in order to determine an unknown acyclic orientation of the given graph G in the worst case. For example, if G is the complete graph on n vertices, then c(G) is the smallest number of comparisons needed to sort n numbers.We prove that c(G) ≤ (1/4 + o(1))n2 for any graph G on n vertices, answering in the affirmative a question of Aigner, Triesch and Tuza [Discrete Mathematics144 (1995) 3–10]. Also, we show that, for every ϵ > 0, it is NP-hard to approximate the parameter c(G) within a multiplicative factor 74/73 − ϵ.

scholarly journals THE POINT-SET EMBEDDABILITY PROBLEM FOR PLANE GRAPHS

2013 ◽  
Vol 23 (04n05) ◽  
pp. 357-395 ◽  
Author(s):  
THERESE BIEDL ◽  
MARTIN VATSHELLE
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Line Drawing ◽  
Np Hard ◽  
Bounded Treewidth ◽  
Straight Line ◽  
Triangulated Graphs ◽  
Point Set ◽  
Fixed Embedding ◽  
Set Of Points

In this paper, we study the point-set embeddability problem, i.e., given a planar graph and a set of points, is there a mapping of the vertices to the points such that the resulting straight-line drawing is planar? It was known that this problem is NP-hard if the embedding can be chosen, but becomes polynomial for triangulated graphs of treewidth 3. We show here that in fact it can be answered for all planar graphs with a fixed combinatorial embedding that have constant treewidth and constant face-degree. We prove that as soon as one of the conditions is dropped (i.e., either the treewidth is unbounded or some faces have large degrees), point-set embeddability with a fixed embedding becomes NP-hard. The NP-hardness holds even for a 3-connected planar graph with constant treewidth, triangulated planar graphs, or 2-connected outer-planar graphs. These results also show that the convex point-set embeddability problem (where faces must be convex) is NP-hard, but we prove that it becomes polynomial if the graph has bounded treewidth and bounded maximum degree.

scholarly journals ON SPANNERS OF GEOMETRIC GRAPHS

2009 ◽  
Vol 20 (01) ◽  
pp. 135-149 ◽  
Author(s):  
JOACHIM GUDMUNDSSON ◽  
MICHIEL SMID
Keyword(s):  
Real Number ◽  
Upper Bound ◽  
Weighted Graph ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Np Hard ◽  
Minimum Number ◽  
Np Hardness

Given a connected geometric graph G, we consider the problem of constructing a t-spanner of G having the minimum number of edges. We prove that for every real number t with [Formula: see text], there exists a connected geometric graph G with n vertices, such that every t-spanner of G contains Ω(n1+1/t) edges. This bound almost matches the known upper bound, which states that every connected weighted graph with n vertices contains a t-spanner with O(n1+2/(t-1)) edges. We also prove that the problem of deciding whether a given geometric graph contains a t-spanner with at most K edges is NP-hard. Previously, this NP-hardness result was only known for non-geometric graphs.

Worst-case ration for planar graphs and the method of induction on faces

1981 ◽  
Author(s):  
Christos H. Papadimitriou ◽  
Mihalis Yannakakis
Keyword(s):  
Planar Graphs ◽  
Worst Case

scholarly journals ON THE ROMAN BONDAGE NUMBER OF A GRAPH

2013 ◽  
Vol 05 (01) ◽  
pp. 1350001 ◽  
Author(s):  
A. BAHREMANDPOUR ◽  
FU-TAO HU ◽  
S. M. SHEIKHOLESLAMI ◽  
JUN-MING XU
Keyword(s):  
Decision Problem ◽  
Maximum Degree ◽  
Minimum Weight ◽  
Domination Number ◽  
Np Hard ◽  
Roman Domination ◽  
Minimum Cardinality ◽  
Bondage Number ◽  
Roman Domination Number ◽  
Roman Dominating Function

A Roman dominating function (RDF) on a graph G = (V, E) is a function f : V → {0, 1, 2} such that every vertex v ∈ V with f(v) = 0 has at least one neighbor u ∈ V with f(u) = 2. The weight of a RDF is the value f(V(G)) = Σu∈V(G) f(u). The minimum weight of a RDF on a graph G is called the Roman domination number, denoted by γR(G). The Roman bondage number bR(G) of a graph G with maximum degree at least two is the minimum cardinality of all sets E′ ⊆ E(G) for which γR(G - E′) > γR(G). In this paper, we first show that the decision problem for determining bR(G) is NP-hard even for bipartite graphs and then we establish some sharp bounds for bR(G) and characterizes all graphs attaining some of these bounds.

Sign in / Sign up

Export Citation Format

Share Document