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    Tytuł:
    Total Roman {2}-Dominating Functions in Graphs
    Autorzy:
    Ahangar, H. Abdollahzadeh
    Chellali, M.
    Sheikholeslami, S.M.
    Valenzuela-Tripodoro, J.C.
    Powiązania:
    https://bibliotekanauki.pl/articles/32304142.pdf
    Data publikacji:
    2022-08-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    Roman domination
    Roman {2}-domination
    total Roman {2}-domination
    Opis:
    A Roman {2}-dominating function (R2F) is a function f : V → {0, 1, 2} with the property that for every vertex v ∈ V with f(v) = 0 there is a neighbor u of v with f(u) = 2, or there are two neighbors x, y of v with f(x) = f(y) = 1. A total Roman {2}-dominating function (TR2DF) is an R2F f such that the set of vertices with f(v) > 0 induce a subgraph with no isolated vertices. The weight of a TR2DF is the sum of its function values over all vertices, and the minimum weight of a TR2DF of G is the total Roman {2}-domination number γtR2(G). In this paper, we initiate the study of total Roman {2}-dominating functions, where properties are established. Moreover, we present various bounds on the total Roman {2}-domination number. We also show that the decision problem associated with γtR2(G) is possible to compute this parameter in linear time for bounded clique-width graphs (including trees).
    Źródło:
    Discussiones Mathematicae Graph Theory; 2022, 42, 3; 937-958
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Roman {2}-Domination Problem in Graphs
    Autorzy:
    Chen, Hangdi
    Lu, Changhong
    Powiązania:
    https://bibliotekanauki.pl/articles/32314051.pdf
    Data publikacji:
    2022-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    Roman {2}-domination
    domination
    algorithms
    Opis:
    For a graph G = (V, E), a Roman {2}-dominating function (R2DF) f : V → {0, 1, 2} has the property that for every vertex v ∈ V with f(v) = 0, either there exists a neighbor u ∈ N(v), with f(u) = 2, or at least two neighbors x, y ∈ N(v) having f(x) = f(y) = 1. The weight of an R2DF f is the sum f(V) = ∑v∈V f(v), and the minimum weight of an R2DF on G is the Roman {2}-domination number γ{R2}(G). An R2DF is independent if the set of vertices having positive function values is an independent set. The independent Roman {2}-domination number i{R2}(G) is the minimum weight of an independent Roman {2}-dominating function on G. In this paper, we show that the decision problem associated with γ{R2}(G) is NP-complete even when restricted to split graphs. We design a linear time algorithm for computing the value of i{R2}(T) in any tree T, which answers an open problem raised by Rahmouni and Chellali [Independent Roman {2}-domination in graphs, Discrete Appl. Math. 236 (2018) 408–414]. Moreover, we present a linear time algorithm for computing the value of γ{R2}(G) in any block graph G, which is a generalization of trees.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2022, 42, 2; 641-660
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Roman {2}-Bondage Number of a Graph
    Autorzy:
    Moradi, Ahmad
    Mojdeh, Doost Ali
    Sharifi, Omid
    Powiązania:
    https://bibliotekanauki.pl/articles/32083773.pdf
    Data publikacji:
    2020-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    domination
    Roman {2}-domination
    Roman {2}-bondage number
    Opis:
    For a given graph G=(V, E), a Roman {2}-dominating function f : V (G) → {0, 1, 2} has the property that for every vertex u with f(u) = 0, either u is adjacent to a vertex assigned 2 under f, or is adjacent to at least two vertices assigned 1 under f. The Roman {2}-domination number of G, γ{R2}(G), is the minimum of Σu∈V (G) f(u) over all such functions. In this paper, we initiate the study of the problem of finding Roman {2}-bondage number of G. The Roman {2}-bondage number of G, b{R2}, is defined as the cardinality of a smallest edge set E′ ⊆ E for which γ{R2}(G − E′) > γ{R2}(G). We first demonstrate complexity status of the problem by proving that the problem is NP-Hard. Then, we derive useful parametric as well as fixed upper bounds on the Roman {2}-bondage number of G. Specifically, it is known that the Roman bondage number of every planar graph does not exceed 15 (see [S. Akbari, M. Khatirinejad and S. Qajar, A note on the Roman bondage number of planar graphs, Graphs Combin. 29 (2013) 327–331]). We show that same bound will be preserved while computing the Roman {2}-bondage number of such graphs. The paper is then concluded by computing exact value of the parameter for some classes of graphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2020, 40, 1; 255-268
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-3 z 3

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