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    Wyświetlanie 1-13 z 13
    Tytuł:
    The upper edge geodetic number and the forcing edge geodetic number of a graph
    Autorzy:
    Santhakumaran, A. P.
    John, J.
    Powiązania:
    https://bibliotekanauki.pl/articles/255845.pdf
    Data publikacji:
    2009
    Wydawca:
    Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
    Tematy:
    geodetic number
    edge geodetic basis
    edge geodetic number
    upper edge geodetic number
    forcing edge geodetic number
    Opis:
    An edge geodetic set of a connected graph G of order p ≥ 2 is a set S ⊆ V(G) such that 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 cardinality of its edge geodetic sets and any edge geodetic set of cardinality g1(G) is a minimum edge geodetic set of G or an edge geodetic basis of G. An edge geodetic set S in a connected graph G is a minimal edge geodetic set if no proper subset of S is an edge geodetic set of G. The upper edge geodetic number g1+(G) of G is the maximum cardinality of a minimal edge geodetic set of G. The upper edge geodetic number of certain classes of graphs are determined. It is shown that for every two integers a and b such that 2 ≤ a ≤ b, there exists a connected graph G with g1(G) = a and g1+(G) = b. For an edge geodetic basis S of G, a subset T ⊆ S is called a forcing subset for S if S is the unique edge geodetic basis containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing edge geodetic number of S denoted by ƒ1(S), is the cardinality of a minimum forcing subset of S. The forcing edge geodetic number of G, denoted by ƒ1(G), is ƒ1(G) = min{ ƒ1(S)}, where the minimum is taken over all edge geodetic bases S in G. Some general properties satisfied by this concept are studied. The forcing edge geodetic number of certain classes of graphs are determined. It is shown that for every pair a, b of integers with 0 ≤ a < b and b ≥ 2, there exists a connected graph G such thatƒ1(G) = a and g1(G) = b.
    Źródło:
    Opuscula Mathematica; 2009, 29, 4; 427-441
    1232-9274
    2300-6919
    Pojawia się w:
    Opuscula Mathematica
    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:
    The forcing steiner number of a graph
    Autorzy:
    Santhakumaran, A.
    John, J.
    Powiązania:
    https://bibliotekanauki.pl/articles/743843.pdf
    Data publikacji:
    2011
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    geodetic number
    Steiner number
    forcing geodetic number
    forcing Steiner number
    Opis:
    For a connected graph G = (V,E), a set W ⊆ V is called a Steiner set of G if every vertex of G is contained in a Steiner W-tree of G. The Steiner number s(G) of G is the minimum cardinality of its Steiner sets and any Steiner set of cardinality s(G) is a minimum Steiner set of G. For a minimum Steiner set W of G, a subset T ⊆ W is called a forcing subset for W if W is the unique minimum Steiner set containing T. A forcing subset for W of minimum cardinality is a minimum forcing subset of W. The forcing Steiner number of W, denoted by fₛ(W), is the cardinality of a minimum forcing subset of W. The forcing Steiner number of G, denoted by fₛ(G), is fₛ(G) = min{fₛ(W)}, where the minimum is taken over all minimum Steiner sets W in G. Some general properties satisfied by this concept are studied. The forcing Steiner numbers of certain classes of graphs are determined. It is shown for every pair a, b of integers with 0 ≤ a < b, b ≥ 2, there exists a connected graph G such that fₛ(G) = a and s(G) = b.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2011, 31, 1; 171-181
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On the forcing geodetic and forcing steiner numbers of a graph
    Autorzy:
    Santhakumaran, A.
    John, J.
    Powiązania:
    https://bibliotekanauki.pl/articles/743992.pdf
    Data publikacji:
    2011
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    geodetic number
    Steiner number
    forcing geodetic number
    forcing Steiner number
    Opis:
    For a connected graph G = (V,E), a set W ⊆ V is called a Steiner set of G if every vertex of G is contained in a Steiner W-tree of G. The Steiner number s(G) of G is the minimum cardinality of its Steiner sets and any Steiner set of cardinality s(G) is a minimum Steiner set of G. For a minimum Steiner set W of G, a subset T ⊆ W is called a forcing subset for W if W is the unique minimum Steiner set containing T. A forcing subset for W of minimum cardinality is a minimum forcing subset of W. The forcing Steiner number of W, denoted by fₛ(W), is the cardinality of a minimum forcing subset of W. The forcing Steiner number of G, denoted by fₛ(G), is fₛ(G) = min{fₛ(W)}, where the minimum is taken over all minimum Steiner sets W in G. The geodetic number g(G) and the forcing geodetic number f(G) of a graph G are defined in [2]. It is proved in [6] that there is no relationship between the geodetic number and the Steiner number of a graph so that there is no relationship between the forcing geodetic number and the forcing Steiner number of a graph. We give realization results for various possibilities of these four parameters.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2011, 31, 4; 611-624
    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:
    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:
    The geodetic number of strong product graphs
    Autorzy:
    Santhakumaran, A.
    Ullas Chandran, S.
    Powiązania:
    https://bibliotekanauki.pl/articles/744104.pdf
    Data publikacji:
    2010
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    geodetic number
    extreme vertex
    extreme geodesic graph
    open geodetic number
    double domination number
    Opis:
    For two vertices u and v of a connected graph G, the set $I_G[u,v]$ consists of all those vertices lying on u-v geodesics in G. Given a set S of vertices of G, the union of all sets $I_G[u,v]$ for u,v ∈ S is denoted by $I_G[S]$. A set S ⊆ V(G) is a geodetic set if $I_G[S] = V(G)$ and the minimum cardinality of a geodetic set is its geodetic number g(G) of G. Bounds for the geodetic number of strong product graphs are obtainted and for several classes improved bounds and exact values are obtained.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2010, 30, 4; 687-700
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Products of Geodesic Graphs and the Geodetic Number of Products
    Autorzy:
    Soloff, Jake A.
    Márquez, Rommy A.
    Friedler, Louis M.
    Powiązania:
    https://bibliotekanauki.pl/articles/31233148.pdf
    Data publikacji:
    2015-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    geodesic graph
    geodetic number
    Cartesian products
    Opis:
    Given a connected graph and a vertex $x ∈ V (G)$, the geodesic graph $P_x(G)$ has the same vertex set as $G$ with edges $uv$ iff either $v$ is on an $x − u$ geodesic path or $u$ is on an $x − v$ geodesic path. A characterization is given of those graphs all of whose geodesic graphs are complete bipartite. It is also shown that the geodetic number of the Cartesian product of $K_{m,n}$ with itself, where $m, n ≥ 4$, is equal to the minimum of $m, n$ and eight.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2015, 35, 1; 35-42
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    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:
    The hull number of strong product graphs
    Autorzy:
    Santhakumaran, A.
    Ullas Chandran, S.
    Powiązania:
    https://bibliotekanauki.pl/articles/743965.pdf
    Data publikacji:
    2011
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    strong product
    geodetic number
    hull number
    extreme hull graph
    Opis:
    For a connected graph G with at least two vertices and S a subset of vertices, the convex hull $[S]_G$ is the smallest convex set containing S. The hull number h(G) is the minimum cardinality among the subsets S of V(G) with $[S]_G = V(G)$. Upper bound for the hull number of strong product G ⊠ H of two graphs G and H is obtainted. Improved upper bounds are obtained for some class of strong product graphs. Exact values for the hull number of some special classes of strong product graphs are obtained. Graphs G and H for which h(G⊠ H) = h(G)h(H) are characterized.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2011, 31, 3; 493-507
    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:
    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 1-13 z 13

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