- Kuwait City, Adeliah, Kuwait
Paul Manuel
Kuwait University, Information Science, Faculty Member
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A set S of vertices in a graph G is said to be a perfect k-dominating set if every vertex in V −S is adjacent to exactly k vertices of S. The perfect k-domination number, γkp(G) is the minimum cardinality of a perfect k-dominating set of... more
A set S of vertices in a graph G is said to be a perfect k-dominating set if every vertex in V −S is adjacent to exactly k vertices of S. The perfect k-domination number, γkp(G) is the minimum cardinality of a perfect k-dominating set of G. In this paper, we construct a minimum perfect k-dominating set where k = 1, 2, 3, 4 for infinite diamond lattice. AMS Subject Classification: 05C69
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Graph is a mathematical model represented by points and lines joining certain pairs of points. These points are addressed as vertices or nodes and the lines are addressed as edges or links. Graph embedding is a mapping of guest graph G... more
Graph is a mathematical model represented by points and lines joining certain pairs of points. These points are addressed as vertices or nodes and the lines are addressed as edges or links. Graph embedding is a mapping of guest graph G into host graph H satisfying certain conditions. Embedding has been studied for many networks in the literature. The Recursive Circulant has several attractive topological properties. Though the embedding of parallel architectures such as Hypercubes and Mesh into Recursive Circulant has been studied, the embedding of Recursive Circulant into other architectures has not been taken up so far. In this paper, we compute the wirelength of embedding even into paths (MinLA), 1-rooted complete binary trees, regular caterpillars and ladders.
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The aim of this paper was to generate nanotopological structure on the power set of vertices of simple digraphs using new definition neighbourhood of vertices on out linked of digraphs. Based on the neighbourhood we define the... more
The aim of this paper was to generate nanotopological structure on the power set of vertices of simple digraphs using new definition neighbourhood of vertices on out linked of digraphs. Based on the neighbourhood we define the approximations of the subgraphs of a graph. A new nanotopological graph reduction to symbolic circuit analysis is developed in this paper. By means of structural equivalence on nanotopology induced by graph we have framed an algorithm for detecting patent infringement suit.
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Graph embedding is an important technique that maps a logical graph into a host graph, usually an interconnection network. In this paper, we compute the exact wirelength of embedding Christmas trees into trees. Moreover, we present an... more
Graph embedding is an important technique that maps a logical graph into a host graph, usually an interconnection network. In this paper, we compute the exact wirelength of embedding Christmas trees into trees. Moreover, we present an algorithm for embedding Christmas trees into caterpillars with dilation 3 proving that the lower bound obtained in [30] is sharp. Further, we solve the maximum subgraph problem for Christmas trees and provide a linear time algorithm to compute the exact wirelength of embedding Christmas trees into trees.
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Graph embedding problems have gained importance in the field of interconnection networks for parallel computer architectures. Interconnection networks provide an effective mechanism for exchanging data between processors in a parallel... more
Graph embedding problems have gained importance in the field of interconnection networks for parallel computer architectures. Interconnection networks provide an effective mechanism for exchanging data between processors in a parallel computing system. In this paper, we introduce a technique to obtain a lower bound for dilation of an embedding. Moreover, we give algorithms to compute exact dilation of embedding circulant network into a triangular grid, Tower of Hanoi graph and Sierpinski gasket graph, proving that the lower bound obtained is sharp.
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The paper introduces a graph theory variation of the general position problem: given a graph $G$, determine a largest set $S$ of vertices of $G$ such that no three vertices of $S$ lie on a common geodesic. Such a set is a max-gp-set of... more
The paper introduces a graph theory variation of the general position problem: given a graph $G$, determine a largest set $S$ of vertices of $G$ such that no three vertices of $S$ lie on a common geodesic. Such a set is a max-gp-set of $G$ and its size is the gp-number $\text{gp}(G)$ of $G$. Upper bounds on $\text{gp}(G)$ in terms of different isometric covers are given and used to determine the gp-number of several classes of graphs. Connections between general position sets and packings are investigated and used to give lower bounds on the gp-number. It is also proved that the general position problem is NP-complete.
Research Interests: Mathematics, Computational Complexity, Combinatorics, Pure Mathematics, Graph, and 3 morePacking, C, and Q
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Geodesic covering problems form a widely researched topic in graph theory. One such problem is geodetic problem introduced by Harary et al. [Math. Comput. Modelling, 1993, 17, 89-95]. Here we introduce a variation of the geodetic problem... more
Geodesic covering problems form a widely researched topic in graph theory. One such problem is geodetic problem introduced by Harary et al. [Math. Comput. Modelling, 1993, 17, 89-95]. Here we introduce a variation of the geodetic problem and call it strong edge geodetic problem. We illustrate how this problem is evolved from social transport networks. It is shown that the strong edge geodetic problem is NP-complete. We derive lower and upper bounds for the strong edge geodetic number and demonstrate that these bounds are sharp. We produce exact solutions for trees, block graphs, silicate networks and glued binary trees without randomization.
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ABSTRACT
An incomplete recursive circulant possesses virtually every advantage of a complete recursive circulant, including simple deadlock-free routing, a small diameter, a good support of parallel algorithms, and so on. It is natural to... more
An incomplete recursive circulant possesses virtually every advantage of a complete recursive circulant, including simple deadlock-free routing, a small diameter, a good support of parallel algorithms, and so on. It is natural to reconfigure a faulty recursive circulant into a maximum incomplete recursive circulant so as to lower potential performance degradation. For [Formula: see text], the maximum incomplete subgraph problem is to identify a subgraph [Formula: see text] of a graph [Formula: see text] on [Formula: see text] vertices having the maximum number of edges among all subgraphs on [Formula: see text] vertices and is NP-complete. In this paper we identify maximum incomplete recursive circulants and use them as a tool to compute the exact wirelength of embedding recursive circulants into special classes of trees, such as [Formula: see text]-rooted complete binary trees, [Formula: see text]-rooted sibling trees, binomial trees, certain caterpillars and path.
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Given four distinct vertices s 1 ,s 2 , t 1 and t 2 of a directed graph G, the 2-disjoint path problem (DPP) is to determine two disjoints paths, P 1 from s 1 to t 1 and P 2 from s 2 to t 2 if such paths exist. Disjoint can mean vertex-... more
Given four distinct vertices s 1 ,s 2 , t 1 and t 2 of a directed graph G, the 2-disjoint path problem (DPP) is to determine two disjoints paths, P 1 from s 1 to t 1 and P 2 from s 2 to t 2 if such paths exist. Disjoint can mean vertex- or edge-disjoint. Both the vertex- or edge-path disjoint problems are NP-hard for directed graphs. In this paper we solve the 2-DPP for Circulant digraphs.
A kernel in a directed graph D(V,E) is a set S of vertices of D such that no two vertices in S are adjacent and for every vertex u in V �\ S there is a vertex v in S, such that (u,v) is an arc of D. The definition of kernel implies that... more
A kernel in a directed graph D(V,E) is a set S of vertices of D such that no two vertices in S are adjacent and for every vertex u in V �\ S there is a vertex v in S, such that (u,v) is an arc of D. The definition of kernel implies that the vertices in the kernel form an independent set. If the vertices of the kernel induce an independent set of edges we obtain a variation of the definition of the kernel, namely a total-kernel. The problem of existence of a kernel is itself a NP−complete problem for a general digraph. But in this paper, we solve the strong total-kernel problem of an oriented Circular Ladder and Mobius Ladder.in polynomial time.
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We consider the problem of finding k disjoint dominating sets with a minimum size of their union, in a given network. We show that this problem can be solved in polynomial time for interval graphs and odd-sun-free graphs. We also relate... more
We consider the problem of finding k disjoint dominating sets with a minimum size of their union, in a given network. We show that this problem can be solved in polynomial time for interval graphs and odd-sun-free graphs. We also relate this question to the so called k-fold domination in graphs. 1 Problem definition and motivation We consider finite undirected graphs without loops or multiple edges. A set of vertices of a graph is called dominating if every vertex not in the set is adjacent (i.e., dominated) by at least one vertex from the set. Finding a dominating set of minimum cardinality is one of the basic optimization problems in computational graph theory. Many variants of this problem are studied in what is now commonly called domination theory in graphs [11, 12]. We propose to study the following problem *Partially supported by Czech research grants GAUK 194 and GACR 194/1996. Parts of the research were carried on while this author was visiting University of Oregon as a Ful...
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The number of matching's in a graph is known as the Hosoya index of the graph. The problem of computing Hosoya index is #P-complete. If the adjacent edges are sequentially ordered, then we show that a polynomial algorithm can be... more
The number of matching's in a graph is known as the Hosoya index of the graph. The problem of computing Hosoya index is #P-complete. If the adjacent edges are sequentially ordered, then we show that a polynomial algorithm can be designed. The significance of this algorithm is demonstrated by computing Hosoya index for certain chemical compounds such as Pyroxene. This algorithm can be applied to grid like chemical compounds such as sodium chloride, carbon nanotubes, naphtalenic nanotube etc.
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A compact power driven ski bob is attained by locating the driving tracks of a rear power ski beneath the seat and pivotally mounting the front end of the power ski longitudinally forward of the seat. The rear end of the power ski is... more
A compact power driven ski bob is attained by locating the driving tracks of a rear power ski beneath the seat and pivotally mounting the front end of the power ski longitudinally forward of the seat. The rear end of the power ski is biased into engagement with the snow and is located lower than the front end of the ski to enhance driving contact with the snow. The seated operator is cushioned from impact forces and up and down movements of the power ski by a rear attachment means comprising a lost motion connection and a shock absorber. To aid in transportation of the snow bob, the front ski and the fuel tanks are detachably connected to the frame.
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A recent variation of the classical geodetic problem, the strong geodetic problem, is defined as follows. If G is a graph, then sg(G) is the cardinality of a smallest vertex subset S, such that one can assign a fixed geodesic to each pair... more
A recent variation of the classical geodetic problem, the strong geodetic problem, is defined as follows. If G is a graph, then sg(G) is the cardinality of a smallest vertex subset S, such that one can assign a fixed geodesic to each pair {x,y}⊆ S so that these |S|2 geodesics cover all the vertices of G. In this paper, the strong geodesic problem is studied on Cartesian product graphs. A general upper bound is proved on the Cartesian product of a path with an arbitrary graph and showed that the bound is tight on flat grids and flat cylinders.
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Given a graph G, the (graph theory) general position problem is to find the maximum number of vertices such that no three vertices lie on a common geodesic. This graph invariant is called the general position number (gp-number for short)... more
Given a graph G, the (graph theory) general position problem is to find the maximum number of vertices such that no three vertices lie on a common geodesic. This graph invariant is called the general position number (gp-number for short) of G and denoted by gp(G). In this paper, the gp-number is determined for a large class of subgraphs of the infinite grid graph and for the infinite diagonal grid. To derive these results, we introduce monotone-geodesic labeling and prove a Monotone Geodesic Lemma that is in turn developed using the Erdös-Szekeres theorem on monotone sequences. The gp-number of the 3-dim infinite grid is bounded. Using isometric path covers, the gp-number is also determined for Beneš networks.
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Covering problems belong to the foundation of graph theory. There are several types of covering problems in graph theory such as covering the vertex set by stars (domination problem), covering the vertex set by cliques (clique covering... more
Covering problems belong to the foundation of graph theory. There are several types of covering problems in graph theory such as covering the vertex set by stars (domination problem), covering the vertex set by cliques (clique covering problem), covering the vertex set by independent sets (coloring problem), and covering the vertex set by paths or cycles. A similar concept which is partitioning problem is also equally important. Lately research in graph theory has produced unprecedented growth because of its various application in engineering and science. The covering and partitioning problem by paths itself have produced a sizable volume of literatures. The research on these problems is expanding in multiple directions and the volume of research papers is exploding. It is the time to simplify and unify the literature on different types of the covering and partitioning problems. The problems considered in this article are path cover problem, induced path cover problem, isometric pat...
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The isometric path cover (partition) problem of a graph is to find a minimum set of isometric paths which cover (partition) the vertex set of the graph. The isometric path cover (partition) number of a graph is the cardinality a minimum... more
The isometric path cover (partition) problem of a graph is to find a minimum set of isometric paths which cover (partition) the vertex set of the graph. The isometric path cover (partition) number of a graph is the cardinality a minimum isometric path cover (partition). We prove that the isometric path partition problem and the isometric k-path partition problem for k≥ 3 are NP-complete on general graphs. Fisher and Fitzpatrick FiFi01 have shown that the isometric path cover number of (r× r)-dimensional grid is 2r/3. We show that the isometric path cover (partition) number of (r× s)-dimensional grid is s when r ≥ s(s-1). We establish that the isometric path cover (partition) number of (r× r)-dimensional torus is r when r is even and is either r or r+1 when r is odd. Then, we demonstrate that the isometric path cover (partition) number of an r-dimensional Benes network is 2^r. In addition, we provide partial solutions for the isometric path cover (partition) problems for cylinder and...
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– The induced matching partition number of a graph G, denoted by imp(G), is the minimum integer k such that V(G) has a k-partition (V1, V2 … Vk) such that, for each i, 1 ≤ i ≤ k, G[Vi], the subgraph of G induced by Vi, is a 1-regular... more
– The induced matching partition number of a graph G, denoted by imp(G), is the minimum integer k such that V(G) has a k-partition (V1, V2 … Vk) such that, for each i, 1 ≤ i ≤ k, G[Vi], the subgraph of G induced by Vi, is a 1-regular graph. The induced matching k-partition problem is NP-complete even for k = 2. In this paper we investigate the induced matching partition problem for butterfly networks. We identify hypercubes, cube-connected cycles, grids of order m x n, where at least one of m and n is even, as graphs for which imp(G) = 2. In the sequel we prove that imp(G) does not exist for grids of order m x n where m and n are both odd and Mesh of trees MT(n), n ≥ 2.
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Covering problems belong to the foundation of graph theory. There are several types of covering problems in graph theory such as covering the vertex set by stars (domination problem), covering the vertex set by cliques (clique covering... more
Covering problems belong to the foundation of graph theory. There are several types of covering problems in graph theory such as covering the vertex set by stars (domination problem), covering the vertex set by cliques (clique covering problem), covering the vertex set by independent sets (coloring problem), and covering the vertex set by paths or cycles. A similar concept which is partitioning problem is also equally important. Lately research in graph theory has produced unprecedented growth because of its various application in engineering and science. The covering and partitioning problem by paths itself have produced a sizable volume of literatures. The research on these problems is expanding in multiple directions and the volume of research papers is exploding. It is the time to simplify and unify the literature on different types of the covering and partitioning problems. The problems considered in this article are path cover problem, induced path cover problem, isometric pat...
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The classical no-three-in-line problem is to find the maximum number of points that can be placed in the $n \times n$ grid so that no three points lie on a line. Given a set $S$ of points in an Euclidean plane, the General Position Subset... more
The classical no-three-in-line problem is to find the maximum number of points that can be placed in the $n \times n$ grid so that no three points lie on a line. Given a set $S$ of points in an Euclidean plane, the General Position Subset Selection Problem is to find a maximum subset $S'$ of $S$ such that no three points of $S'$ are collinear. Motivated by these problems, the following graph theory variation is introduced: Given a graph $G$, determine a largest set $S$ of vertices of $G$ such that no three vertices of $S$ lie on a common geodesic. Such a set is a gp-set of $G$ and its size is the gp-number ${\rm gp}(G)$ of $G$. Upper bounds on ${\rm gp}(G)$ in terms of different isometric covers are given and used to determine the gp-number of several classes of graphs. Connections between general position sets and packings are investigated and used to give lower bounds on the gp-number. It is also proved that the general position problem is NP-complete.
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In the modern world, social media private data evolves as a great asset to business and governments. While social media private data is a boon to the business, it is also causing concern to privacy regulators. We classify the social media... more
In the modern world, social media private data evolves as a great asset to business and governments. While social media private data is a boon to the business, it is also causing concern to privacy regulators. We classify the social media data as the business and governments require. The social media private data is classified into two layers: macro level and micro level. The macro level classification is Static Private Data and Dynamic Private Data. The micro level classification includes four types: Personal Identity Data (Static), Relational Identity Data (Static), Personal Identity Data (Dynamic), and Relational Identity Data (Dynamic). Two software metrics “complexity” and “relevancy” are considered. Based on the macro and micro level classification, we measure the complexity and relevancy of social media private data from the perspectives of business and police communities. By conducting extensive experimental research, we study the relationship between different types of social media private data and different communities by the means of the two-metrics relevancy and complexity and justify the necessity of macro and micro level classification. The outcome of the experimental survey is interesting. Police officers are more interested in static private data than dynamic private data. Business managers are more interested in dynamic private data than static private data. While the police are interested in static private data, the business communities are interested in dynamic private data.
AbstractThis paper considers a continuous review perishable inventory system with demands ar-rive according to a Markovian arrival process (MAP). We model, in this paper, the situationin which not all the ordered items are usable and the... more
AbstractThis paper considers a continuous review perishable inventory system with demands ar-rive according to a Markovian arrival process (MAP). We model, in this paper, the situationin which not all the ordered items are usable and the supply may contain a fraction of defec-tive items. The number of usable items is a random quantity. We consider a modified (s,S)policy which allows a finite number of pending order to be placed. We assume full back-logging of demands that occurred during stock out periods and that the recent backloggeddemand may renege the system after an exponentially distributed amount of time. Thelimiting distribution of the inventory level is derived and shown to have matrix geometricform. The measures of system performance in the steady state are derived. Keywords: Stochastic Inventory System, Perishable Items, Positive Lead Time, Modified(s,S) Policy, Matrix Geometric Solution.1. IntroductionInventory systems with stochastic input and output processes have been a...
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Given a graph G, the general position problem is to find a largest set S of vertices of G such that no three vertices of S lie on a common geodesic. Such a set is called a gp-set of G and its cardinality is the gp-number, gp(G), of G. In... more
Given a graph G, the general position problem is to find a largest set S of vertices of G such that no three vertices of S lie on a common geodesic. Such a set is called a gp-set of G and its cardinality is the gp-number, gp(G), of G. In this paper, the edge general position problem is introduced as the edge analogue of the general position problem. The edge general position number, gpe(G), is the size of a largest edge general position set of G. It is proved that gpe(Qr) = 2 r and that if T is a tree, then gpe(T ) is the number of its leaves. The value of gpe(Pr Ps) is determined for every r, s ≥ 2. To derive these results, the theory of partial cubes is used. Mulder’s meta-conjecture on median graphs is also discussed along the way.
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The geodetic problem was introduced by Harary et al. In order to model some social network problems, a similar problem is introduced in this paper and named the strong geodetic problem. The problem is solved for complete Apollonian... more
The geodetic problem was introduced by Harary et al. In order to model some social network problems, a similar problem is introduced in this paper and named the strong geodetic problem. The problem is solved for complete Apollonian networks. It is also proved that in general the strong geodetic problem is NP-complete.
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The Wiener index, or the Wiener number, also known as the "sum of distances" of a connected graph, is one of the quantities associated with a molecular graph that correlates nicely to physical and chemical properties, and has... more
The Wiener index, or the Wiener number, also known as the "sum of distances" of a connected graph, is one of the quantities associated with a molecular graph that correlates nicely to physical and chemical properties, and has been studied in depth in the literature. In this paper we develop a method to compute the Wiener index of molecular graphs and thereby obtain the Wiener index of all classes of regular plane tessellations composed of the same kind of regular polygons namely triangular, square, and hexagonal. Further we study the Szeged index of regular tessellations.
A set S of vertices of a graph G is a geodesic transversal of G if every maximal geodesic of G contains at least one vertex of S. We determine a smallest geodesic transversal in certain interconnection networks such as mesh of trees, and... more
A set S of vertices of a graph G is a geodesic transversal of G if every maximal geodesic of G contains at least one vertex of S. We determine a smallest geodesic transversal in certain interconnection networks such as mesh of trees, and some wellknown chemical structures such as silicate networks and carbon nanosheets. Some useful general bounds for the corresponding graph invariant are obtained along the way.
24 In order to model certain social network problems, the strong geodetic 25 problem and its related invariant, the strong geodetic number, are intro26 duced. The problem is conceptually similar to the classical geodetic prob27 lem but... more
24 In order to model certain social network problems, the strong geodetic 25 problem and its related invariant, the strong geodetic number, are intro26 duced. The problem is conceptually similar to the classical geodetic prob27 lem but seems intrinsically more difficult. The strong geodetic number is 28 compared with the geodetic number and with the isometric path number. It 29 is determined for several families of graphs including Apollonian networks. 30 2 P. Manuel, S. Klavžar, A. Xavier, A. Arokiaraj and E. Thomas Applying Sierpiński graphs, an algorithm is developed that returns a mini31 mum path cover of Apollonian networks corresponding to the strong geodetic 32 number. It is also proved that the strong geodetic problem is NP-complete. 33
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Covering problems belong to the foundation of graph theory. There are several types of covering problems in graph theory such as covering the vertex set by stars (domination problem), covering the vertex set by cliques (clique covering... more
Covering problems belong to the foundation of graph theory. There are several types of covering problems in graph theory such as covering the vertex set by stars (domination problem), covering the vertex set by cliques (clique covering problem), covering the vertex set by independent sets (coloring problem), and covering the vertex set by paths or cycles. A similar concept which is partitioning problem is also equally important. Lately research in graph theory has produced unprecedented growth because of its various application in engineering and science. The covering and partitioning problem by paths itself have produced a sizable volume of literatures. The research on these problems is expanding in multiple directions and the volume of research papers is exploding. It is the time to simplify and unify the literature on different types of the covering and partitioning problems. The problems considered in this article are path cover problem, induced path cover problem, isometric path cover problem, path partition problem, induced path partition problem and isometric path partition problem. The objective of this article is to summarize the recent developments on these problems, classify their literatures and correlate the inter-relationship among the related concepts.
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The classical no-three-in-line problem is to find the maximum number of points that can be placed in the n×n grid so that no three points lie on a line. Given a set S of points in an Euclidean plane, the General Position Subset Selection... more
The classical no-three-in-line problem is to find the maximum number of points that can be placed in the n×n grid so that no three points lie on a line. Given a set S of points in an Euclidean plane, the General Position Subset Selection Problem is to find a maximum subset S of S such that no three points of S are collinear. Motivated by these problems, the following graph theory variation is introduced: Given a graph G, determine a largest set S of vertices of G such that no three vertices of S lie on a common geodesic. Such a set is a gp-set of G and its size is the gp-number gp(G) of G. Upper bounds on gp(G) in terms of different isometric covers are given and used to determine the gp-number of several classes of graphs. Connections between general position sets and packings are investigated and used to give lower bounds on the gp-number. It is also proved that the general position problem is NP-complete.
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Geodesic covering problems form a widely researched topic in graph theory. One such problem is geodetic problem introduced by Harary et al. [18]. Here we introduce a variation of the geodetic problem and call it strong edge geode-tic... more
Geodesic covering problems form a widely researched topic in graph theory. One such problem is geodetic problem introduced by Harary et al. [18]. Here we introduce a variation of the geodetic problem and call it strong edge geode-tic problem. We illustrate how this problem is evolved from social transport networks. It is shown that the strong edge geodetic problem is NP-complete. We derive lower bounds for the strong edge geodetic number and demonstrate that these bounds are sharp and non-trivial. We produce exact solutions for trees, block graphs, silicate networks and glued binary trees without random-ization.