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    Wyświetlanie 1-7 z 7
    Tytuł:
    The edge geodetic number and Cartesian product of graphs
    Autorzy:
    Santhakumaran, A.
    Ullas Chandran, S.
    Powiązania:
    https://bibliotekanauki.pl/articles/744517.pdf
    Data publikacji:
    2010
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    geodetic number
    edge geodetic number
    linear edge geodetic set
    perfect edge geodetic set
    (edge, vertex)-geodetic set
    superior edge geodetic set
    Opis:
    For a nontrivial connected graph G = (V(G),E(G)), a set S⊆ V(G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g₁(G) of G is the minimum order of its edge geodetic sets. Bounds for the edge geodetic number of Cartesian product graphs are proved and improved upper bounds are determined for a special class of graphs. Exact values of the edge geodetic number of Cartesian product are obtained for several classes of graphs. Also we obtain a necessary condition of G for which g₁(G ☐ K₂) = g₁(G).
    Źródło:
    Discussiones Mathematicae Graph Theory; 2010, 30, 1; 55-73
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    An O(mn2) Algorithm for Computing the Strong Geodetic Number in Outerplanar Graphs
    Autorzy:
    Mezzini, Mauro
    Powiązania:
    https://bibliotekanauki.pl/articles/32317585.pdf
    Data publikacji:
    2022-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    outerplanar graph
    strong geodetic set
    strong geodetic number
    geodetic set
    geodetic number
    geodesic convexity
    Opis:
    Let $ G = (V (G), E(G)) $ be a graph and $S$ be a subset of vertices of $G$. Let us denote by $ \gamma [u, v] $ a geodesic between $u$ and $v$. Let $ \Gamma(S) = \{ γ [ v_i, v_j ] | v_i, v_j \in S} $ be a set of exactly $ |S|(|S|−1) // 2 $ geodesics, one for each pair of distinct vertices in $S$. Let \( V ( \Gamma (S)) = \bigcup_{ \gamma [x,y] \in \Gamma (S) } V ( \gamma [x, y]) \) be the set of all vertices contained in all the geodesics in $ \Gamma (S) $. If $ V ( \Gamma (S)) = V (G) $ for some $ \Gamma (S) $, then we say that $S$ is a strong geodetic set of $G$. The cardinality of a minimum strong geodetic set of a graph is the strong geodetic number of $G$. It is known that it is NP-hard to determine the strong geodetic number of a general graph. In this paper we show that the strong geodetic number of an outerplanar graph can be computed in polynomial time.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2022, 42, 2; 591-599
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Geodetic sets in graphs
    Autorzy:
    Chartrand, Gary
    Harary, Frank
    Zhang, Ping
    Powiązania:
    https://bibliotekanauki.pl/articles/743733.pdf
    Data publikacji:
    2000
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    geodetic set
    geodetic number
    upper geodetic number
    Opis:
    For two vertices u and v of a graph G, the closed interval I[u,v] consists of u, v, and all vertices lying in some u-v geodesic in G. If S is a set of vertices of G, then I[S] is the union of all sets I[u,v] for u, v ∈ S. If I[S] = V(G), then S is a geodetic set for G. The geodetic number g(G) is the minimum cardinality of a geodetic set. A set S of vertices in a graph G is uniform if the distance between every two distinct vertices of S is the same fixed number. A geodetic set is essential if for every two distinct vertices u,v ∈ S, there exists a third vertex w of G that lies in some u-v geodesic but in no x-y geodesic for x, y ∈ S and {x,y} ≠ {u,v}. It is shown that for every integer k ≥ 2, there exists a connected graph G with g(G) = k which contains a uniform, essential minimum geodetic set. A minimal geodetic set S has no proper subset which is a geodetic set. The maximum cardinality of a minimal geodetic set is the upper geodetic number g⁺(G). It is shown that every two integers a and b with 2 ≤ a ≤ b are realizable as the geodetic and upper geodetic numbers, respectively, of some graph and when a < b the minimum order of such a graph is b+2.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2000, 20, 1; 129-138
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The forcing geodetic number of a graph
    Autorzy:
    Chartrand, Gary
    Zhang, Ping
    Powiązania:
    https://bibliotekanauki.pl/articles/744241.pdf
    Data publikacji:
    1999
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    geodetic set
    geodetic number
    forcing geodetic number
    Opis:
    For two vertices u and v of a graph G, the set I(u, v) consists of all vertices lying on some u-v geodesic in G. If S is a set of vertices of G, then I(S) is the union of all sets I(u,v) for u, v ∈ S. A set S is a geodetic set if I(S) = V(G). A minimum geodetic set is a geodetic set of minimum cardinality and this cardinality is the geodetic number g(G). A subset T of a minimum geodetic set S is called a forcing subset for S if S is the unique minimum geodetic set containing T. The forcing geodetic number $f_G(S)$ of S is the minimum cardinality among the forcing subsets of S, and the forcing geodetic number f(G) of G is the minimum forcing geodetic number among all minimum geodetic sets of G. The forcing geodetic numbers of several classes of graphs are determined. For every graph G, f(G) ≤ g(G). It is shown that for all integers a, b with 0 ≤ a ≤ b, a connected graph G such that f(G) = a and g(G) = b exists if and only if (a,b) ∉ {(1,1),(2,2)}.
    Źródło:
    Discussiones Mathematicae Graph Theory; 1999, 19, 1; 45-58
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On graphs with a unique minimum hull set
    Autorzy:
    Chartrand, Gary
    Zhang, Ping
    Powiązania:
    https://bibliotekanauki.pl/articles/743417.pdf
    Data publikacji:
    2001
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    geodetic set
    geodetic number
    convex hull
    hull set
    hull number
    hull graph
    Opis:
    We show that for every integer k ≥ 2 and every k graphs G₁,G₂,...,Gₖ, there exists a hull graph with k hull vertices v₁,v₂,...,vₖ such that link $L(v_i) = G_i$ for 1 ≤ i ≤ k. Moreover, every pair a, b of integers with 2 ≤ a ≤ b is realizable as the hull number and geodetic number (or upper geodetic number) of a hull graph. We also show that every pair a,b of integers with a ≥ 2 and b ≥ 0 is realizable as the hull number and forcing geodetic number of a hull graph.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2001, 21, 1; 31-42
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Double geodetic number of a graph
    Autorzy:
    Santhakumaran, A.
    Jebaraj, T.
    Powiązania:
    https://bibliotekanauki.pl/articles/743673.pdf
    Data publikacji:
    2012
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    geodetic number
    weak-extreme vertex
    double geodetic set
    double geodetic number
    Opis:
    For a connected graph G of order n, a set S of vertices is called a double geodetic set of G if for each pair of vertices x,y in G there exist vertices u,v ∈ S such that x,y ∈ I[u,v]. The double geodetic number dg(G) is the minimum cardinality of a double geodetic set. Any double godetic of cardinality dg(G) is called dg-set of G. The double geodetic numbers of certain standard graphs are obtained. It is shown that for positive integers r,d such that r < d ≤ 2r and 3 ≤ a ≤ b there exists a connected graph G with rad G = r, diam G = d, g(G) = a and dg(G) = b. Also, it is proved that for integers n, d ≥ 2 and l such that 3 ≤ k ≤ l ≤ n and n-d-l+1 ≥ 0, there exists a graph G of order n diameter d, g(G) = k and dg(G) = l.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2012, 32, 1; 109-119
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On Minimal Geodetic Domination in Graphs
    Autorzy:
    Nuenay, Hearty M.
    Jamil, Ferdinand P.
    Powiązania:
    https://bibliotekanauki.pl/articles/31339437.pdf
    Data publikacji:
    2015-08-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    minimal geodetic dominating set
    upper geodetic domination number
    Opis:
    Let $G$ be a connected graph. For two vertices $u$ and $v$ in $G$, a $u$-$v$ geodesic is any shortest path joining $u$ and $v$. The closed geodetic interval $ I_G[u, v] $ consists of all vertices of $G$ lying on any $u$-$v$ geodesic. For $ S \subseteq V (G) $, $S$ is a geodetic set in $G$ if \( \bigcup_{u,v \in S} I_G [u, v] = V (G) \). Vertices $u$ and $v$ of $G$ are neighbors if $u$ and $v$ are adjacent. The closed neighborhood $ N_G[v]$ of vertex $v$ consists of $v$ and all neighbors of $v$. For $S \subseteq V (G)$, $S$ is a dominating set in $G$ if \( \bigcup_{u \in S} N_G[u] = V (G) \). A geodetic dominating set in $G$ is any geodetic set in $G$ which is at the same time a dominating set in $G$. A geodetic dominating set in $G$ is a minimal geodetic dominating set if it does not have a proper subset which is itself a geodetic dominating set in $G$. The maximum cardinality of a minimal geodetic dominating set in $G$ is the upper geodetic domination number of $G$. This paper initiates the study of minimal geodetic dominating sets and upper geodetic domination numbers of connected graphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2015, 35, 3; 403-418
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-7 z 7

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