Search results for: MINIMAL DOUBLE DOMINATING SET - Bridge of Knowledge

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Search results for: MINIMAL DOUBLE DOMINATING SET

Search results for: MINIMAL DOUBLE DOMINATING SET

  • Minimal double dominating sets in trees

    Publication

    - Year 2014

    We provide an algorithm for listing all minimal double dominating sets of a tree of order $n$ in time $\mathcal{O}(1.3248^n)$. This implies that every tree has at most $1.3248^n$ minimal double dominating sets. We also show that this bound is tight.

  • An algorithm for listing all minimal double dominating sets of a tree

    Publication

    We provide an algorithm for listing all minimal double dominating sets of a tree of order $n$ in time $\mathcal{O}(1.3248^n)$. This implies that every tree has at most $1.3248^n$ minimal double dominating sets. We also show that this bound is tight.

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  • Polynomial Algorithm for Minimal (1,2)-Dominating Set in Networks

    Publication

    - Electronics - Year 2022

    Dominating sets find application in a variety of networks. A subset of nodes D is a (1,2)-dominating set in a graph G=(V,E) if every node not in D is adjacent to a node in D and is also at most a distance of 2 to another node from D. In networks, (1,2)-dominating sets have a higher fault tolerance and provide a higher reliability of services in case of failure. However, finding such the smallest set is NP-hard. In this paper, we...

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  • Trees having many minimal dominating sets

    We provide an algorithm for listing all minimal dominating sets of a tree of order n in time O(1.4656^n). This leads to that every tree has at most 1.4656^n minimal dominating sets. We also give an infinite family of trees of odd and even order for which the number of minimal dominating sets exceeds 1.4167^n, thus exceeding 2^{n/2}. This establishes a lower bound on the running time of an algorithm for listing all minimal dominating...

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  • Minimal 2-dominating sets in Trees

    We provide an algorithm for listing all minimal 2-dominating sets of a tree of order n in time O(1.3247^n). This leads to that every tree has at most 1.3247^n minimal 2-dominating sets. We also show that thisbound is tight.

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  • An Algorithm for Listing All Minimal 2-Dominating Sets of a Tree

    We provide an algorithm for listing all minimal 2-dominating sets of a tree of order n in time O(1.3248n) . This implies that every tree has at most 1.3248 n minimal 2-dominating sets. We also show that this bound is tigh.

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  • On trees with double domination number equal to 2-outer-independent domination number plus one

    A vertex of a graph is said to dominate itself and all of its neighbors. A double dominating set of a graph G is a set D of vertices of G 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 double dominating set of G. For a graph G=(V,E), a subset D subseteq V(G) is a 2-dominating set if every vertex of V(G)D has at least two neighbors...

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  • On trees with double domination number equal to 2-domination number plus one

    A vertex of a graph is said to dominate itself and all of its neighbors. A subset D subseteq V(G) is a 2-dominating set of G if every vertex of V(G)D is dominated by at least two vertices of D, while it is a double dominating set of G if every vertex of G is dominated by at least two vertices of D. The 2-domination (double domination, respectively) number of a graph G is the minimum cardinality of a 2-dominating (double dominating,...

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  • On trees with double domination number equal to total domination number plus one

    Publication

    A total dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbor in D. A vertex of a graph is said to dominate itself and all of its neighbors. A double dominating set of a graph G is a set D of vertices of G such that every vertex of G is dominated by at least two vertices of D. The total (double, respectively) domination number of a graph G is the minimum cardinality of a total (double,...

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  • Graphs with equal domination and certified domination numbers

    Publication

    - Opuscula Mathematica - Year 2019

    A setDof vertices of a graphG= (VG,EG) is a dominating set ofGif every vertexinVG−Dis adjacent to at least one vertex inD. The domination number (upper dominationnumber, respectively) ofG, denoted byγ(G) (Γ(G), respectively), is the cardinality ofa smallest (largest minimal, respectively) dominating set ofG. A subsetD⊆VGis calleda certified dominating set ofGifDis a dominating set ofGand every vertex inDhas eitherzero...

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  • An upper bound for the double outer-independent domination number of a tree

    Publication

    A vertex of a graph is said to dominate itself and all of its neighbors. A double outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of G is dominated by at least two vertices of D, and the set V(G)\D is independent. The double outer-independent domination number of a graph G, denoted by γ_d^{oi}(G), is the minimum cardinality of a double outer-independent dominating set of G. We prove...

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  • Total Domination Versus Domination in Cubic Graphs

    Publication

    A dominating set in a graph G is a set S of vertices of G such that every vertex not in S has a neighbor in S. Further, if every vertex of G has a neighbor in S, then S is a total dominating set of G. The domination number,γ(G), and total domination number, γ_t(G), are the minimum cardinalities of a dominating set and total dominating set, respectively, in G. The upper domination number, \Gamma(G), and the upper total domination...

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  • A lower bound on the double outer-independent domination number of a tree

    Publication

    A vertex of a graph is said to dominate itself and all of its neighbors. A double outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of G is dominated by at least two vertices of D, and the set V(G)D is independent. The double outer-independent domination number of a graph G, denoted by gamma_d^{oi}(G), is the minimum cardinality of a double outer-independent dominating set of G. We...

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  • On the double bondage in graphs

    Publication

    A vertex of a graph is said to dominate itself and all of its neighbors. A double dominating set of a graph G=(V,E) is a set D of vertices of G such that every vertex of G is dominated by at least two vertices of D. The double domination number of a graph G, denoted by gamma_d(G), is the minimum cardinality of a double dominating set of G. The double bondage number of G, denoted by b_d(G), is the minimum cardinality among all sets...

  • Density functional theory calculations on entire proteins for free energies of binding: Application to a model polar binding site

    Publication

    - PROTEINS-STRUCTURE FUNCTION AND BIOINFORMATICS - Year 2014

    In drug optimization calculations, the molecular mechanics Poisson-Boltzmann surface area (MM-PBSA) method can be used to compute free energies of binding of ligands to proteins. The method involves the evaluation of the energy of configurations in an implicit solvent model. One source of errors is the force field used, which can potentially lead to large errors due to the restrictions in accuracy imposed by its empirical nature....

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  • Structural properties of mixed conductor Ba1−xGd1−yLax+yCo2O6−δ

    Publication

    - DALTON TRANSACTIONS - Year 2022

    Ba1−xGd1−yLax+yCo2O6−δ (BGLC) compositions with large compositional ranges of Ba, Gd, and La have been characterised with respect to phase compositions, structure, and thermal and chemical expansion. The results show a system with large compositional flexibility, enabling tuning of functional properties and thermal and chemical expansion. We show anisotropic chemical expansion and detailed refinements of emerging phases as La is...

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  • Control of the wind turbine generator

    Publication

    Wind power system consists of two main parts: wind turbine and electrical generator. Wind turbine converts the energy of the flowing air into mechanical energy, next generator converts this energy into electrical energy that is sent to the power system. These two processes should be realized with maximum efficiency and the following requirements for the control system can be formulated: opti-mal wind power conversion, compensation...

  • Design considerations for compact microstrip resonant cells dedicated to efficient branch-line miniaturization

    A conventional compact microstrip resonant cell (CMRC)has been thoroughly investigated to enhance its slow-wave properties and subsequently ensure an efficient miniaturization of a microstrip circuit. The geometry of a classic CMRC has been improved in terms of slowwave effect in two progressive steps: (i) a single-element topology has been replaced with a double-element one and (ii) a high-impedance section has been refined by...

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  • Determination of magnetisation conditions in a Double-Core Barkhausen Noise measurement set-up

    The magnetic Barkhausen effect is useful forassessing 1D and 2D stress states of ferromagnetic steelobjects. However, its extension to technically importantmaterials, such as duplex anisotropic steels, remains challenging. The determination of magnetisation inside the studied object and the electromagnet for various geometries, materials and magnetisation angles is a key issue.Three-dimensional, dynamic finite element analysis...

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  • On proper (1,2)‐dominating sets in graphs

    Publication

    In 2008, Hedetniemi et al. introduced the concept of (1,)-domination and obtained some interesting results for (1,2) -domination. Obviously every (1,1) -dominating set of a graph (known as 2-dominating set) is (1,2) -dominating; to distinguish these concepts, we define a proper (1,2) -dominating set of a graph as follows: a subset is a proper (1,2) -dominating set of a graph if is (1,2) -dominating and it is not a (1,1) -dominating...

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  • Unicyclic graphs with equal total and total outer-connected domination numbers

    Publication

    - ARS COMBINATORIA - Year 2015

    Let G = (V,E) be a graph without an isolated vertex. A set D ⊆ V (G) is a total dominating set if D is dominating and the in- duced subgraph G[D] does not contain an isolated vertex. The total domination number of G is the minimum cardinality of a total domi- nating set of G. A set D ⊆ V (G) is a total outer–connected dominating set if D is total dominating and the induced subgraph G[V (G)−D] is a connected graph. The total outer–connected...

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  • Weakly connected Roman domination in graphs

    Publication

    A Roman dominating function on a graph G=(V,E) is defined to be a function f :V → {0,1,2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v)=2. A dominating set D⊆V is a weakly connected dominating set of G if the graph (V,E∩(D×V)) is connected. We define a weakly connected Roman dominating function on a graph G to be a Roman dominating function such that the set...

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  • On the ratio between 2-domination and total outer-independent domination numbers of trees

    A 2-dominating set of a graph G is a set D of vertices of G such that every vertex of V(G)D has a at least two neighbors in D. A total outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbor in D, and the set V(G)D is independent. The 2-domination (total outer-independent domination, respectively) number of a graph G is the minimum cardinality of a 2-dominating (total...

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  • On trees with equal domination and total outer-independent domination numbers

    Publication

    For a graph G=(V,E), a subset D subseteq V(G) is a dominating set if every vertex of V(G)D has a neighbor in D, while it is a total outer-independent dominating set if every vertex of G has a neighbor in D, and the set V(G)D is independent. The domination (total outer-independent domination, respectively) number of G is the minimum cardinality of a dominating (total outer-independent dominating, respectively) set of G. We characterize...

  • On trees with equal 2-domination and 2-outer-independent domination numbers

    For a graph G = (V,E), a subset D \subseteq V(G) is a 2-dominating set if every vertex of V(G)\D$ has at least two neighbors in D, while it is a 2-outer-independent dominating set if additionally the set V(G)\D is independent. The 2-domination (2-outer-independent domination, respectively) number of G, is the minimum cardinality of a 2-dominating (2-outer-independent dominating, respectively) set of G. We characterize all trees...

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  • Total domination in versus paired-domination in regular graphs

    A subset S of vertices of a graph G is a dominating set of G if every vertex not in S has a neighbor in S, while S is a total dominating set of G if every vertex has a neighbor in S. If S is a dominating set with the additional property that the subgraph induced by S contains a perfect matching, then S is a paired-dominating set. The domination number, denoted γ(G), is the minimum cardinality of a dominating set of G, while the...

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  • The convex domination subdivision number of a graph

    Publication

    Let G = (V;E) be a simple graph. A set D\subset V is a dominating set of G if every vertex in V - D has at least one neighbor in D. The distance d_G(u, v) between two vertices u and v is the length of a shortest (u, v)-path in G. An (u, v)-path of length d_G(u; v) is called an (u, v)-geodesic. A set X\subset V is convex in G if vertices from all (a, b)-geodesics belong to X for any two vertices a, b \in X. A set X is a convex dominating...

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  • Weakly convex domination subdivision number of a graph

    Publication

    - FILOMAT - Year 2016

    A set X is weakly convex in G if for any two vertices a; b \in X there exists an ab–geodesic such that all of its vertices belong to X. A set X \subset V is a weakly convex dominating set if X is weakly convex and dominating. The weakly convex domination number \gamma_wcon(G) of a graph G equals the minimum cardinality of a weakly convex dominating set in G. The weakly convex domination subdivision number sd_wcon (G) is the minimum...

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  • Global defensive sets in graphs

    In the paper we study a new problem of finding a minimum global defensive set in a graph which is a generalization of the global alliance problem. For a given graph G and a subset S of a vertex set of G, we define for every subset X of S the predicate SEC ( X ) = true if and only if | N [ X ] ∩ S | ≥ | N [ X ] \ S | holds, where N [ X ] is a closed neighbourhood of X in graph G. A set S is a defensive alliance if and only if for...

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  • Similarities and Differences Between the Vertex Cover Number and the Weakly Connected Domination Number of a Graph

    Publication
    • M. Lemańska
    • J. A. RODRíGUEZ-VELáZQUEZ
    • R. Trujillo-Rasua

    - FUNDAMENTA INFORMATICAE - Year 2017

    A vertex cover of a graph G = (V, E) is a set X ⊂ V such that each edge of G is incident to at least one vertex of X. The ve cardinality of a vertex cover of G. A dominating set D ⊆ V is a weakly connected dominating set of G if the subgraph G[D]w = (N[D], Ew) weakly induced by D, is connected, where Ew is the set of all edges having at least one vertex in D. The weakly connected domination number γw(G) of G is the minimum cardinality...

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  • A lower bound on the total outer-independent domination number of a tree

    Publication

    A total outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbor in D, and the set V(G)D is independent. The total outer-independent domination number of a graph G, denoted by gamma_t^{oi}(G), is the minimum cardinality of a total outer-independent dominating set of G. We prove that for every nontrivial tree T of order n with l leaves we have gamma_t^{oi}(T) >= (2n-2l+2)/3,...

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  • An upper bound on the 2-outer-independent domination number of a tree

    Publication

    A 2-outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of V(G)D has a at least two neighbors in D, and the set V(G)D is independent. The 2-outer-independent domination number of a graph G, denoted by gamma_2^{oi}(G), is the minimum cardinality of a 2-outer-independent dominating set of G. We prove that for every nontrivial tree T of order n with l leaves we have gamma_2^{oi}(T) <= (n+l)/2,...

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  • An upper bound on the total outer-independent domination number of a tree

    Publication

    A total outer-independent dominating set of a graph G=(V(G),E(G)) is a set D of vertices of G such that every vertex of G has a neighbor in D, and the set V(G)D is independent. The total outer-independent domination number of a graph G, denoted by gamma_t^{oi}(G), is the minimum cardinality of a total outer-independent dominating set of G. We prove that for every tree T of order n >= 4, with l leaves and s support vertices we have...

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  • 2-outer-independent domination in graphs

    Publication

    We initiate the study of 2-outer-independent domination in graphs. A 2-outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of V(G)\D has at least two neighbors in D, and the set V(G)\D is independent. The 2-outer-independent domination number of a graph G is the minimum cardinality of a 2-outer-independent dominating set of G. We show that if a graph has minimum degree at least two,...

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  • Bounds on the vertex-edge domination number of a tree

    Publication

    - COMPTES RENDUS MATHEMATIQUE - Year 2014

    A vertex-edge dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every edge of $G$ is incident with a vertex of $D$ or a vertex adjacent to a vertex of $D$. The vertex-edge domination number of a graph $G$, denoted by $\gamma_{ve}(T)$, is the minimum cardinality of a vertex-edge dominating set of $G$. We prove that for every tree $T$ of order $n \ge 3$ with $l$ leaves and $s$ support vertices we have $(n-l-s+3)/4...

  • On trees attaining an upper bound on the total domination number

    A total dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbor in D. The total domination number of a graph G, denoted by γ_t(G), is the minimum cardinality of a total dominating set of G. Chellali and Haynes [Total and paired-domination numbers of a tree, AKCE International Journal of Graphs and Combinatorics 1 (2004), 69-75] established the following upper bound on the total domination...

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  • An Alternative Proof of a Lower Bound on the 2-Domination Number of a Tree

    A 2-dominating set of a graph G is a set D of vertices of G such that every vertex not in D has a at least two neighbors in D. The 2-domination number of a graph G, denoted by gamma_2(G), is the minimum cardinality of a 2-dominating set of G. Fink and Jacobson [n-domination in graphs, Graph theory with applications to algorithms and computer science, Wiley, New York, 1985, 283-300] established the following lower bound on the 2-domination...

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  • Database of the minimal sets of Lefschetz periods for Morse-Smale diffeomorphisms of a connected sum of g tori

    Open Research Data
    open access

    Morse–Smale diffeomorphisms, structurally stable and having relatively simple dynamics, constitute an important subclass of diffeomorphisms that have been carefully studied during past decades. For a given Morse–Smale diffeomorphism one can consider “Minimal set of Lefschetz periods”, which provides the information about the set of periodic points of...

  • Paired domination versus domination and packing number in graphs

    Publication

    Given a graph G = (V(G), E(G)), the size of a minimum dominating set, minimum paired dominating set, and a minimum total dominating set of a graph G are denoted by γ (G), γpr(G), and γt(G), respectively. For a positive integer k, a k-packing in G is a set S ⊆ V(G) such that for every pair of distinct vertices u and v in S, the distance between u and v is at least k + 1. The k-packing number is the order of a largest kpacking and...

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  • All graphs with paired-domination number two less than their order

    Publication

    Let G=(V,E) be a graph with no isolated vertices. A set S⊆V is a paired-dominating set of G if every vertex not in S is adjacent with some vertex in S and the subgraph induced by S contains a perfect matching. The paired-domination number γp(G) of G is defined to be the minimum cardinality of a paired-dominating set of G. Let G be a graph of order n. In [Paired-domination in graphs, Networks 32 (1998), 199-206] Haynes and Slater...

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  • Independent Domination Subdivision in Graphs

    Publication

    - GRAPHS AND COMBINATORICS - Year 2021

    A set $S$ of vertices in a graph $G$ is a dominating set if every vertex not in $S$ is adjacent to a vertex in~$S$. If, in addition, $S$ is an independent set, then $S$ is an independent dominating set. The independent domination number $i(G)$ of $G$ is the minimum cardinality of an independent dominating set in $G$. The independent domination subdivision number $\sdi(G)$ is the minimum number of edges that must be subdivided (each...

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  • Weakly convex and convex domination numbers of some products of graphs

    If $G=(V,E)$ is a simple connected graph and $a,b\in V$, then a shortest $(a-b)$ path is called a $(u-v)$-{\it geodesic}. A set $X\subseteq V$ is called {\it weakly convex} in $G$ if for every two vertices $a,b\in X$ exists $(a-b)$- geodesic whose all vertices belong to $X$. A set $X$ is {\it convex} in $G$ if for every $a,b\in X$ all vertices from every $(a-b)$-geodesic belong to $X$. The {\it weakly convex domination number}...

  • Periodic expansion in determining minimal sets of Lefschetz periods for Morse–Smale diffeomorphisms

    We apply the representation of Lefschetz numbers of iterates in the form of periodic expansion to determine the minimal sets of Lefschetz periods of Morse–Smale diffeomorphisms. Applying this approach we present an algorithmic method of finding the family of minimal sets of Lefschetz periods for Ng, a non-orientable compact surfaces without boundary of genus g. We also partially confirm the conjecture of Llibre and Sirvent (J Diff...

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  • Database of the minimal sets of Lefschetz periods for Morse-Smale diffeomorphisms of a connected sum of g real projective planes.

    Open Research Data
    open access

    Morse–Smale diffeomorphisms, structurally stable and having relatively simple dynamics, constitute an important subclass of diffeomorphisms that were carefully studied during past decades. For a given  Morse–Smale diffeomorphism one can consider “Minimal set of Lefschetz periods”, which provides the information about the set of periodic points of considered...

  • Isolation Number versus Domination Number of Trees

    Publication
    • M. Lemańska
    • M. J. Souto-Salorio
    • A. Dapena
    • F. Vazquez-Araujo

    - Mathematics - Year 2021

    If G=(VG,EG) is a graph of order n, we call S⊆VG an isolating set if the graph induced by VG−NG[S] contains no edges. The minimum cardinality of an isolating set of G is called the isolation number of G, and it is denoted by ι(G). It is known that ι(G)≤n3 and the bound is sharp. A subset S⊆VG is called dominating in G if NG[S]=VG. The minimum cardinality of a dominating set of G is the domination number, and it is denoted by γ(G)....

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  • Bipartite theory of graphs: outer-independent domination

    Publication

    - NATIONAL ACADEMY SCIENCE LETTERS-INDIA - Year 2015

    Let $G = (V,E)$ be a bipartite graph with partite sets $X$ and $Y$. Two vertices of $X$ are $X$-adjacent if they have a common neighbor in $Y$, and they are $X$-independent otherwise. A subset $D \subseteq X$ is an $X$-outer-independent dominating set of $G$ if every vertex of $X \setminus D$ has an $X$-neighbor in $D$, and all vertices of $X \setminus D$ are pairwise $X$-independent. The $X$-outer-independent domination number...

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  • Domination-Related Parameters in Rooted Product Graphs

    Abstract A set S of vertices of a graph G is a dominating set in G if every vertex outside of S is adjacent to at least one vertex belonging to S. A domination parameter of G is related to those sets of vertices of a graph satisfying some domination property together with other conditions on the vertices of G. Here, we investigate several domination-related parameters in rooted product graphs.

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  • On the super domination number of lexicographic product graphs

    Publication

    - DISCRETE APPLIED MATHEMATICS - Year 2019

    The neighbourhood of a vertexvof a graphGis the setN(v) of all verticesadjacent tovinG. ForD⊆V(G) we defineD=V(G)\D. A setD⊆V(G) is called a super dominating set if for every vertexu∈D, there existsv∈Dsuch thatN(v)∩D={u}. The super domination number ofGis theminimum cardinality among all super dominating sets inG. In this article weobtain closed formulas and tight bounds for the super dominating number oflexicographic product...

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  • 2-bondage in graphs

    A 2-dominating set of a graph G=(V,E) is a set D of vertices of G such that every vertex of V(G)D has at least two neighbors in D. The 2-domination number of a graph G, denoted by gamma_2(G), is the minimum cardinality of a 2-dominating set of G. The 2-bondage number of G, denoted by b_2(G), is the minimum cardinality among all sets of edges E' subseteq E such that gamma_2(G-E') > gamma_2(G). If for every E' subseteq E we have...

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  • Minimal Sets of Lefschetz Periods for Morse-Smale Diffeomorphisms of a Connected Sum of g Real Projective Planes

    Publication

    - Year 2022

    The dataset titled Database of the minimal sets of Lefschetz periods for Morse-Smale diffeomorphisms of a connected sum of g real projective planes contains all of the values of the topological invariant called the minimal set of Lefschetz periods, computed for Morse-Smale diffeomorphisms of a non-orientable compact surface without boundary of genus g (i.e. a connected sum of g real projective planes), where g varies from 1 to...

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