# Showing papers in "Discrete Mathematics, Algorithms and Applications in 2016"

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TL;DR: Here, the behavior of this index under several graph operations is studied and the results are applied to find the F-index of different chemically interesting molecular graphs and nano-structures.

Abstract: The F-index of a graph is defined as the sum of cubes of the vertex degrees of the graph. This was introduced in 1972, in the same paper where the first and second Zagreb indices were introduced to study the structure-dependency of total π-electron energy. But this topological index was not further studied till then. Very recently, Furtula and Gutman [A forgotten topological index,J. Math. Chem. 53(4) (2015) 1184–1190.] reinvestigated the index and named it “forgotten topological index” or “F-index”. In that paper, they present some basic properties of this index and showed that this index can enhance the physico-chemical applicability of Zagreb index. Here, we study the behavior of this index under several graph operations and apply our results to find the F-index of different chemically interesting molecular graphs and nanostructures.

48 citations

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TL;DR: A new tool is used to obtain upper and lower bounds of $\prod(G)$ for all cactus graphs and characterize the corresponding extremal graphs.

Abstract: Let ∏ (G) be multiplicative Zagreb index of a graph G. A connected graph is a cactus graph if and only if any two of its cycles have at most one vertex in common, which is a generalization of trees and has been the interest of researchers in the field of material chemistry and graph theory. In this paper, we use a new tool to obtain the upper and lower bounds of ∏ (G) for all cactus graphs and characterize the corresponding extremal graphs.

15 citations

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TL;DR: The 2-distance chromatic number of G is the least integer k such that G has a k-2-distance coloring, denoted by χ2(G).

Abstract: A 2-distance coloring of G is a function φ: V (G) →{1, 2,…,k}, such that for every two distinct vertices u, v in G, φ(u)≠φ(v) if 0 < dG(u,v) ≤ 2. The 2-distance chromatic number of G is the least integer k such that G has a k-2-distance coloring, denoted by χ2(G). Similarly, the list 2-distance chromatic number of G is denoted by χ2l(G). In this paper, we proved that: (1) for every planar graph with g(G) ≥ 5 and Δ(G) ≥ 12, χ2l(G) ≤ Δ(G) + 6; (2) for every planar graph with g(G) ≥ 6 and Δ(G) ≥ 9, χ2l(G) ≤ Δ(G) + 3.

14 citations

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TL;DR: Let δ0(P,k) denote the degree k dilation of a point set P in the domain of plane geometric spanners and it is shown that all the authors' constructions are planar lattice tilings constrained to degree 3 or 4.

Abstract: Let δ0(P,k) denote the degree k dilation of a point set P in the domain of plane geometric spanners. If Λ is the infinite square lattice, it is shown that 1 + 2 ≤ δ0(Λ, 3) ≤ (3 + 22)5−1/2 = 2.6065… and δ0(Λ, 4) = 2. If Λ is the infinite hexagonal lattice, it is shown that δ0(Λ, 3) = 1 + 3 and δ0(Λ, 4) = 2. All our constructions are planar lattice tilings constrained to degree 3 or 4.

13 citations

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TL;DR: Tight upper bounds for γR (G) and bR(G) are obtained provided a graph G is in ℛUV R, and necessary and sufficient conditions for a tree to be in the class ℚUV R are presented.

Abstract: For a graph G = (V,E), a Roman dominating function (RDF) f : V →{0, 1, 2} has the property that every vertex v ∈ V with f(v) = 0 has a neighbor u with f(u) = 2. The weight of a RDF f is the sum f(V ) = Σv∈Vf(v), and the minimum weight of a RDF on G is the Roman domination number γR(G) of G. The Roman bondage number bR(G) of G is the minimum cardinality of all sets F ⊆ E for which γR(G − F) > γR(G). A graph G is in the class ℛUV R if the Roman domination number remains unchanged when a vertex is deleted. In this paper, we obtain tight upper bounds for γR(G) and bR(G) provided a graph G is in ℛUV R. We present necessary and sufficient conditions for a tree to be in the class ℛUV R. We give a constructive characterization of ℛUV R-trees using labelings.

13 citations

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TL;DR: This work introduces and study additive skew cyclic codes over the quaternary field GF(4), obtaining some structural properties of these codes and shows that many best known and optimal quantum codes can be obtained from this class.

Abstract: Additive codes received much attention due to their connections with quantum codes. On the other hand, skew cyclic codes proved to be a useful class of codes that contain many good codes. In this work, we introduce and study additive skew cyclic codes over the quaternary field GF(4), obtaining some structural properties of these codes. Moreover, we also show that many best known and optimal quantum codes can be obtained from this class.

11 citations

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TL;DR: The concepts of mean and variance are extended to a modified injective graph coloring and the values of these parameters for a number of standard graphs are determined.

Abstract: Coloring the vertices of a graph G according to certain conditions can be considered as a random experiment and a discrete random variable (r.v.) X can be defined as the number of vertices having a particular color in the proper coloring of G and a probability mass function for this random variable can be defined accordingly. In this paper, we extend the concepts of mean and variance to a modified injective graph coloring and determine the values of these parameters for a number of standard graphs.

10 citations

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CINVESTAV

^{1}TL;DR: An interesting and hard problem is to construct all the non-isomorphic CAs that exist for a particular combination of the parameters N, t, k and v, the results allow us to determine CAN(3,13,2) =16, CAN(2,10,3) =14, and the exact lower bound for these...

Abstract: A covering array CA(N; t,k,v) of strength t and order v is an N × k array over ℤv with the property that every N × t subarray covers all members of ℤvt at least once. When the value of N is the minimum possible it is named as the covering array number (CAN) i.e. N = CAN(t,k,v). Two CAs are isomorphic if one of them can be derived from the other by a combination of a row permutation, a column permutation, and a symbol permutation in a subset of columns. Isomorphic CAs have equivalent coverage properties, and can be considered as the same CA; the truly distinct CAs are those which are non-isomorphic among them. An interesting and hard problem is to construct all the non-isomorphic CAs that exist for a particular combination of the parameters N, t, k and v. We constructed the non-isomorphic CAs for 70 combinations of values of the parameters N, t, k and v, the results allow us to determine CAN(3,13,2) =16, CAN(3,14,2) =16, CAN(3,15,2) =17, CAN(3,16,2) =17, and CAN(2,10,3) =14. The exact lower bound for these...

10 citations

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TL;DR: Two new graph characteristics are introduced, the edge H-irregularity strength and the vertex H-IR irregularity strength of a graph, and the bounds of these parameters are estimated for several families of graphs.

Abstract: We introduce two new graph characteristics, the edge H-irregularity strength and the vertex H-irregularity strength of a graph. We estimate the bounds of these parameters and determine their exact values for several families of graphs namely, paths, ladders and fans.

9 citations

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TL;DR: It is proved that for every tree T, γd(T)/2 ≤ γsp(T), and the trees attaining this bound are characterized.

Abstract: A vertex of a graph G is said to dominate itself and all its neighbors. A double dominating set (DDS) of a graph G is a set D of vertices such that every vertex of G is dominated by at least two vertices of D. The double domination number of a graph G is the minimum cardinality of a DDS of G. For a graph G, a subset D of V (G) is a super dominating set SDS if for every vertex of V (G) \ D there exists an external private neighbor of v with respect to V (G) \ D. The super domination number of G is the minimum cardinality of a SDS of G. We prove that for every tree T, γd(T)/2 ≤ γsp(T), and we characterize the trees attaining this bound.

9 citations

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TL;DR: This paper studies certain chromatic sums for some standard cycle related graphs with respect to colouring sums such as $\chi$-chromatic sum, $\chi^+$-chrome sum, $b$- chrome sum, “b^+”, etc. are some of these types of Colouring sums that have been studied recently.

Abstract: Let 𝒞 = {c1,c2,c3,…,ck} be a certain type of proper k-coloring of a given graph G and θ(ci) denote the number of times a particular color ci is assigned to the vertices of G. Then, the coloring sum of a given graph G with respect to the coloring 𝒞, denoted by ω𝒞(G), is defined to be ω(𝒞) =∑i=1kiθ(c i). The coloring sums such as χ-chromatic sum, χ+-chromatic sum, b-chromatic sum, b+-chromatic sum, etc. are some of these types of coloring sums that have been studied recently. Motivated by these studies on certain chromatic sums of graphs, in this paper, we study certain chromatic sums for some standard cycle-related graphs.

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TL;DR: In this paper, the authors studied the monochromatic connection number on lexicographical, strong, Cartesian and direct products and presented several upper and lower bounds for these products of graphs.

Abstract: The concept of monochromatic connectivity was introduced by Caro and Yuster. A path in an edge-colored graph is called a monochromatic path if all the edges on the path are colored the same. An edge-coloring of G is a monochromatic connection coloring (MC-coloring, for short) if there is a monochromatic path joining any two vertices in G. The monochromatic connection number, denoted by mc(G), is defined to be the maximum number of colors used in an MC-coloring of a graph G. In this paper, we study the monochromatic connection number on the lexicographical, strong, Cartesian and direct products and present several upper and lower bounds for these products of graphs.

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TL;DR: A link between hyperstructures and TSPs is found and the definition of hypergroups is introduced by Marty.

Abstract: After introducing the definition of hypergroups by Marty, the study of hyperstructures and its applications has been of great importance. In this paper, we find a link between hyperstructures and t...

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TL;DR: This paper studies the problem of how to select the least number of TP, satisfying the requirement that the entire network is protected by the alerts that the TP send, and proposes an asymmetric trust (AT) information propagation model that is a constant-factor approximation algorithm.

Abstract: In a social network, rumor containment is vital, as the diffusion of a rumor will bring terrible results. Precautionary measure can be used to control rumor propagation: Anticipating the spread of a rumor, one can (1) select a set of trustworthy people (TP) in the network, (2) alert the TP about the rumor, and (3) ask the TP to protect their neighbors by sending out alerts. In this paper, we study the problem of how to select the least number of TP, satisfying the requirement that the entire network is protected by the alerts that the TP send. We propose an asymmetric trust (AT) information propagation model. Under this model, we study the Least Number TP Selection (LNTS) problem, establish its NP-hardness and reformulate it as a minimum submodular cover problem. As a result, the Greedy Algorithm is a constant-factor approximation algorithm. Using real-world data, we evaluate the performance of the Greedy Algorithm, and compare it with other algorithms. Experimental results indicate that the Greedy Algorithm performs the best among its competitors.

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TL;DR: The generators for these codes are obtained and a minimal spanning set is determined for even and odd t separately with arbitrary s.

Abstract: In this paper, we study some structural properties of ℤ2R-additive cyclic codes in ℤ2s×Rt as R[x]-submodules of ℤ2[x] 〈xs−1〉 × R[x] 〈xt−1〉, where R = ℤ2 + uℤ2,u2 = 0. The generators for these codes are obtained and a minimal spanning set is determined for even and odd t separately with arbitrary s. We also determine the generators of duals of the ℤ2R-additive cyclic codes for odd t. A necessary condition for a 1-generator ℤ2R-cyclic code to be a R-free module is obtained.

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TL;DR: The aim of this paper is to study the interplay between the ring theoretic properties of a ring R and the graph theoretic Properties of (Ω(R))c, where (Χ(R) c) is the complement of Ω( R), and it is shown that ( Ω (R)c is complemented if and only if R is reduced.

Abstract: The rings considered in this paper are commutative with identity which are not integral domains. Recall that an ideal I of a ring R is called an annihilating ideal if there exists r ∈ R\{0} such that Ir = (0). As in [M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10(4) (2011) 727–739], for any ring R, we denote by A(R) the set of all annihilating ideals of R and by A(R)∗ the set of all nonzero annihilating ideals of R. Let R be a ring. In [S. Visweswaran and H. D. Patel, A graph associated with the set of all nonzero annihilating ideals of a commutative ring, Discrete Math. Algorithm Appl. 6(4) (2014), Article ID: 1450047, 22pp], we introduced and studied the properties of a graph, denoted by Ω(R), which is an undirected simple graph whose vertex set is A(R)∗ and distinct elements I,J ∈ A(R)∗ are joined by an edge in this graph if and only if I + J ∈ A(R). The aim of this paper is to study the interplay between the ring theoretic properties of a ring R and the graph theoretic properties of (Ω(R))c, where (Ω(R))c is the complement of Ω(R). In this paper, we first determine when (Ω(R))c is connected and also determine its diameter when it is connected. We next discuss the girth of (Ω(R))c and study regarding the cliques of (Ω(R))c. Moreover, it is shown that (Ω(R))c is complemented if and only if R is reduced.

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TL;DR: A new coding and decoding method followed from Lucas p matrix, RpQpn is developed, which established the relations among the code matrix elements, error detection and correction for this coding theory.

Abstract: In [K. Kuhapatanakul, The Lucas p-matrix, Internat. J. Math. Ed. Sci. Tech. (2015), http://dx.doi.org/10.1080/0020739X.2015.1026612], Kuhapatanakul introduced Lucas p matrix, RpQpn whose elements are Lucas p numbers. In this paper, we developed a new coding and decoding method followed from Lucas p matrix, RpQpn. We established the relations among the code matrix elements, error detection and correction for this coding theory. Correction ability of this method is 93.33% for p = 1 and for p = 2, the correction ability is 99.80%. In general, correction ability of this method increases as p increases.

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TL;DR: In this article, the authors consider a simple graph associated with R denoted by ΩR∗, whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I and J are adjacent whenever JAnn(I) = (0) or IAnn(J) =(0).

Abstract: Let R be a commutative ring with identity. In this paper, we consider a simple graph associated with R denoted by ΩR∗, whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I and J are adjacent whenever JAnn(I) = (0) or IAnn(J) = (0). In this paper, we initiate the study of the graph ΩR∗ and we investigate its properties. In particular, we show that ΩR∗ is a connected graph with diam(ΩR∗) ≤ 3 unless R is isomorphic to a direct product of two fields. Moreover, we characterize all commutative rings R with at least two maximal ideals for which ΩR∗ are planar.

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TL;DR: This paper shows that a weakly triangulated graph without articulation points has at most 2nq different linear layouts, where nq is the number of quadrilaterals (4-cycles) in G, and exploits an edge order to identify the rigid and non-rigid components of G.

Abstract: A graph G = (V,E) is said to be triangulated if it has no chordless cycles of length 4 or more. Such a graph is said to be rigid if, for a valid assignment of edge lengths, it has a unique linear layout and non-rigid otherwise. Damaschke [Point placement on the line by distance data, Discrete Appl. Math. 127(1) (2003) 53–62] showed how to compute all linear layouts of a triangulated graph, for a valid assignment of lengths to the edges of G. In this paper, we extend this result to weakly triangulated graphs, resolving an open problem. A weakly triangulated graph can be constructively characterized by a peripheral ordering of its edges. The main contribution of this paper is to exploit such an edge order to identify the rigid and non-rigid components of G. We first show that a weakly triangulated graph without articulation points has at most 2nq different linear layouts, where nq is the number of quadrilaterals (4-cycles) in G. When G has articulation points, the number of linear layouts is at most 2nb−1+n...

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TL;DR: The number of edges, degrees of the vertices, independence number, dominating number, clique number, chromatic number, weakly perfectness, Eulerianness, Hamiltonicity of these graphs are investigated.

Abstract: In this paper, we study the structure of the permutability graphs of subgroups, and the permutability graphs of non-normal subgroups of the following groups: the dihedral groups Dn, the generalized quaternion groups Qn, the quasi-dihedral groups QD2n and the modular groups Mpn. Further, we investigate the number of edges, degrees of the vertices, independence number, dominating number, clique number, chromatic number, weakly perfectness, Eulerianness, Hamiltonicity of these graphs.

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TL;DR: It is proved that if n = n1 + n2 + ⋯ + nk with ni ≥ 2 and at most one ni odd, then Qn can be decomposed into k spanning subgraphs G1, G2,….,Gk such that Gi is ni-regular and ni-connected for i = 1, 2,…,k.

Abstract: It is known that the n-dimensional hypercube Qn, for n = n1 + n2 with ni ≥ 2, can be decomposed into two spanning bipancyclic subgraphs G1 and G2 such that Gi is ni-regular and ni-connected for i = 1, 2. In this paper, we prove that if n = n1 + n2 + ⋯ + nk with ni ≥ 2 and at most one ni odd, then Qn can be decomposed into k spanning subgraphs G1, G2,…,Gk such that Gi is ni-regular and ni-connected for i = 1, 2,…,k.

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TL;DR: Some results are generalized on linear codes over Z3[v]/〈v3 − v〉 in [15] to the ring Rp = Zp[v], where p is an odd prime number.

Abstract: Some results are generalized on linear codes over Z3[v]/〈v3 − v〉 in [15] to the ring Rp = Zp[v]/〈vp − v〉, where p is an odd prime number. The Gray images of cyclic and quasi-cyclic codes over Rp ar...

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TL;DR: In this paper, the exact value of the bondage number of the strong product of a complete graph and a path was derived for any two integers m ≥ 1 and n ≥ 2.

Abstract: The bondage number b(G) of a graph G is the cardinality of a minimum edge set whose removal from G results in a graph with the domination number greater than that of G. It is a parameter to measure the vulnerability of a communication network under link failure. In this paper, we obtain the exact value of the bondage number of the strong product of a complete graph and a path. That is, for any two integers m ≥ 1 and n ≥ 2, b(Km ⊠ Pn) = ⌈m 2 ⌉ if n ≡ 0(mod3); m if n ≡ 2 (mod 3); ⌈3m 2 ⌉ if n ≡ 1 (mod 3). Furthermore, we determine the exact value of the bondage number of the strong product of a complete graph and a special starlike tree.

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TL;DR: It is proved that the graphs such as subdivision of ladder, triangular ladder, shadow, total, flower, generalized prism, mΔn-snake, lotus inside a circle, square, gear, closed helm and antiprism are Zk-magic graphs.

Abstract: For any nontrivial abelian group A under addition a graph G is said to be A-magic if there exists a labeling f : E(G) → A −{0} such that the vertex labeling f+ defined as f+(v) =∑f(uv) taken over all edges uv incident at v is a constant. An A-magic graph G is said to be Zk-magic graph if the group A is Zk the group of integers modulo k. These Zk-magic graphs are referred to as k-magic graphs. In this paper, we prove that the graphs such as subdivision of ladder, triangular ladder, shadow, total, flower, generalized prism, mΔn-snake, lotus inside a circle, square, gear, closed helm and antiprism are Zk-magic graphs. Also we prove that if Gi(1 ≤ i ≤ t) be Zk-magic graphs with magic constant zero then Gt is also Zk-magic.

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TL;DR: This paper formally introduces the social-relation-based vaccine distribution planning problem (SVDP2) on the temporal graph and proposes an enumeration algorithm which will search the solution space using the evaluation technique and find the best possible solution within polynomial time.

Abstract: In this paper, we investigate the problem of how to distribute vaccines, which will be supplied over time, so that the number of the infected can be minimized during a given mission period. The concept of temporal graph is adopted to abstract the constantly changing social relations over time. Then, we formally introduce the social-relation-based vaccine distribution planning problem (SVDP2) on the temporal graph. To solve the problem, we first introduce a new graph induction technique to combine the subgraphs in the temporal graph into a single directed acyclic graph. Then, we design a new technique based on a maximum flow algorithm to evaluate the quality of any feasible solution of the problem. Finally, we propose an enumeration algorithm which will search the solution space using the evaluation technique and find the best possible solution within polynomial time. Our simulation result shows the proposed algorithm is more efficient than a simple strategy which randomly distributes vaccines.

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TL;DR: A subset U ⊂ V (D) is acyclic if it induces an acyClic subdigraph of a digraph D and the dichromatic number χd(D) of D is defined to be the minimum integer n such that V (C) can be partitioned into C.

Abstract: A subset U ⊂ V (D) is acyclic if it induces an acyclic subdigraph of a digraph D and the dichromatic number χd(D) of D is defined to be the minimum integer n such that V (D) can be partitioned into...

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TL;DR: A new public key scheme, which is a combination of RSA variant namely the DRSA and the generalization of generalized discrete logarithm problem (generalized GDLP), which is at least as secure as theDRSA and ElGamal schemes.

Abstract: In this paper, we propose a new public key scheme, which is a combination of RSA variant namely the DRSA and the generalization of generalized discrete logarithm problem (generalized GDLP). The security of this scheme depends equally on the integer factorization of n and the discrete logarithm problem (DLP) on ℤn∗, where n is the product of two large primes and ℤn∗ is the multiplicative group modulo n. The scheme is a randomized algorithm. It is at least as secure as the DRSA and ElGamal schemes. We also compare the encryption–decryption performance of the proposed scheme with the RSA and DRSA schemes.

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TL;DR: It is proved that (n,M,d,m)q codes based on m-dimensional totally isotropic subspaces in unitary space 𝔽q2(n) attain the Wang–Xing–Safavi-Naini bound if and only if they are certain Steiner structures in 𝓂(n).

Abstract: In this paper, the Sphere-packing bound, Singleton bound, Wang–Xing–Safavi-Naini bound, Johnson bound and Gilbert–Varshamov bound on the subspace codes (n,M,d,m)q based on m-dimensional totally isotropic subspaces in unitary space 𝔽q2(n) over finite fields 𝔽q2 are presented. Then, we prove that (n,M,d,m)q codes based on m-dimensional totally isotropic subspaces in unitary space 𝔽q2(n) attain the Wang–Xing–Safavi-Naini bound if and only if they are certain Steiner structures in 𝔽q2(n).

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TL;DR: This paper considers the model in which all guards move simultaneously when the configuration of guards induces total dominating sets (TDSs) and studies this model for some classes of graphs.

Abstract: Eternal domination of a graph requires the vertices of the graph to be protected, against infinitely long sequences of attacks, by guards located at vertices, with the requirement that the configuration of guards induces a dominating set at all times. Klostermeyer and Mynhardt studied some variations of this concept in which the configuration of guards induces total dominating sets (TDSs). They considered two models of the problem, one in which only one guard moves at a time and one in which more than one guard may move simultaneously. In this paper, we consider the model in which all guards move simultaneously when the configuration of guards induces TDSs and we study this model for some classes of graphs.

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Madura College

^{1}TL;DR: A two-valued function f : V →{−1, +1} defined on the vertices of a graph G = (V,E), is a majority dominating function if the sum of its function values over at least half the closed neighborhoods i...

Abstract: A two-valued function f : V →{−1, +1} defined on the vertices of a graph G = (V,E), is a majority dominating function if the sum of its function values over at least half the closed neighborhoods i...