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For a connected graph G of order at least two, a connected outer connected geodetic set S of G is called a minimal connected outer connected geodetic set if no proper subset of S is a connected outer connected geodetic set of G. T...
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For a connected graph G of order at least two, a connected outer connected geodetic set S of G is called a minimal connected outer connected geodetic set if no proper subset of S is a connected outer connected geodetic set of G. The upper connected outer connected geodetic number cg(co)(+) (G) of G is the maximum cardinality of a minimal connected outer connected geodetic set of G. We determine bounds for it and certain general properties satisfied by this parameter are studied. It is shown that, for any two integers a, b with 3 <= a <= b, there exists a connected graph G with cg(co) (G) = a and cg(co)(+) (G) = b, where cg(co) (G) is the connected outer connected geodetic number of a graph. Also, another parameter forcing connected outer connected geodetic number f(cog)(G) of a graph G is introduced and several interesting results on this parameter are studied.
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For a connected graph G = (V, E) of order at least two, a total outer connected geodetic set S of a graph G is an outer connected geodetic set such that the subgraph induced by S has no isolated vertices. The minimum cardinality o...
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For a connected graph G = (V, E) of order at least two, a total outer connected geodetic set S of a graph G is an outer connected geodetic set such that the subgraph induced by S has no isolated vertices. The minimum cardinality of a total outer connected geodetic set of G is the total outer connected geodetic number of G and is denoted by cg(to) (G). We determine bounds for it and also find the total outer connected geodetic number for some special classes of graphs. It is shown that for positive integers r, d and k >= 4 with r < d = 4, d >= 2 and p - d - k + 2 >= 0, there exists a connected graph G of order p such that diam(G) = d and cg(to) (G) = k. It is also shown that for positive integers a, b such that 3 收起
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For a connected graph G - oV; ETHORN of order at least two, a set S of vertices in a graph G is said to be an outer connected geodetic set if S is a geodetic set of G and either S - V or the subgraph induced by V S is connected. T...
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For a connected graph G - oV; ETHORN of order at least two, a set S of vertices in a graph G is said to be an outer connected geodetic set if S is a geodetic set of G and either S - V or the subgraph induced by V S is connected. The minimum cardinality of an outer connected geodetic set of G is the outer connected geodetic number of G and is denoted by gocoGTHORN. We determine bounds for it and find the same for some special classes of graphs. Also some realization results for this parameter are studied. This concept can be mainly used in fault-tolerant network in order to ensure the communication between nodes.
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For a connected graph G of order at least two, a connected outer connected monophonic set S of G is called a minimal connected outer connected monophonic set if no proper subset of S is a connected outer connected monophonic set o...
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For a connected graph G of order at least two, a connected outer connected monophonic set S of G is called a minimal connected outer connected monophonic set if no proper subset of S is a connected outer connected monophonic set of G. The upper connected outer connected monophonic number cm_(co)~+(G) of G is the maximum cardinality of a minimal connected outer connected monophonic set of G. We determine bounds for it and find the upper connected outer connected monophonic number of certain classes of graphs. It is shown that for any two integers a, b with 4 ≤ a ≤ b ≤ p - 2, there is a connected graph G of order p with cm_(co)(G) = a and cm_(co)~+(G) = 6. Also, for any three integers a, b and n with 4 ≤a ≤ n≤ b, there is a connected graph G with cm_(co)(G) = a and cm_(co)~+(G) = b and a min-imal connected outer connected monophonic set of cardinality n, where cm_(co)(G) is the connected outer connected monophonic number of a graph.
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For a connected graph $G$ of order at least two, an outer connected geodetic set $S$ in a connected graph $G$ is called a \emph{minimal outer connected geodetic set} if no proper subset of $S$ is an outer connected geodetic set of...
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For a connected graph $G$ of order at least two, an outer connected geodetic set $S$ in a connected graph $G$ is called a \emph{minimal outer connected geodetic set} if no proper subset of $S$ is an outer connected geodetic set of $G$. The \emph{upper outer connected geodetic number} $g_{\operatorname{oc}}^{+}(G)$ of $G$ is the maximum cardinality of a minimal outer connected geodetic set of $G$. We determine bounds for it and find the upper outer connected geodetic number of some standard graphs. Some realization results on the upper outer connected geodetic number of a graph are studied. The proposed method can be extended to the identification of beacon vertices towards the network fault-tolerant in wireless local access network communication. Also, another parameter \emph{forcing outer connected geodetic number} $f_{\operatorname{og}}(G)$ of a graph $G$ is introduced and several interesting results and realization theorem are proved.
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For two vertices u and v of a nontrivial connected graph G, the set I[u, v] consists of all vertices lying on some u - v geodesic in G, including u and v. For S subset of V(G), the set I[S] is the union of all sets I[u, v] for u, ...
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For two vertices u and v of a nontrivial connected graph G, the set I[u, v] consists of all vertices lying on some u - v geodesic in G, including u and v. For S subset of V(G), the set I[S] is the union of all sets I[u, v] for u, v is an element of S. A set S subset of V(G) is a connected geodetic set of G if I[S] = V(G) and the subgraph in G induced by S is connected. The minimum cardinality of a connected geodetic set of G is the connected geodetic number g(c)(G) of G and a connected geodetic set of G whose cardinality equals g(c)(G) is a minimum connected geodetic set of G. A subset T of a minimum connected geodetic set S is a forcing subset for S if S is the unique minimum connected geodetic set of G containing T. The forcing connected geodetic number f(c)(S) of S is the minimum cardinality of a forcing subset of S and the forcing connected geodetic number f(c)(G) of G is the minimum forcing connected geodetic number among all minimum connected geodetic sets of G. Therefore, 0 收起
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A set S of vertices of a graph G is an outer-connected dominating set if every vertex not in S is adjacent to some vertex in S and the subgraph induced by Va?-S is connected. The outer-connected domination number γ?c(G) is the mi...
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A set S of vertices of a graph G is an outer-connected dominating set if every vertex not in S is adjacent to some vertex in S and the subgraph induced by Va?-S is connected. The outer-connected domination number γ?c(G) is the minimum size of such a set. We prove that if δ(G)≥2 and diam (G)≤2, then γ?c(G)≤ (n+1)/2, and we study the behavior of γ?c(G) under an edge addition.
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A set S ? V(G) is an independent set if no two vertices of S are adjacent. An independent set S such that 〈V - S〉 is connected is called an outer-connected independent set (ociset). An oc?-set is maximal if it is not a proper su...
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A set S ? V(G) is an independent set if no two vertices of S are adjacent. An independent set S such that 〈V - S〉 is connected is called an outer-connected independent set (ociset). An oc?-set is maximal if it is not a proper subset of any oci-set. The minimum and maximum cardinality of a maximal oci-set are called respectively the outer-connected independence number and the upper outer-connected independence number. This paper initiates a study of these parameters.
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A set S of vertices of a graph G is an edge geodetic set if every edge of G lies on x-y geodesic for some elements x and y in S. The minimum cardinality of an edge geodetic set of G is the edge geodetic number of G denoted by g_1 ...
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A set S of vertices of a graph G is an edge geodetic set if every edge of G lies on x-y geodesic for some elements x and y in S. The minimum cardinality of an edge geodetic set of G is the edge geodetic number of G denoted by g_1 (G). In this paper, we explore the concept of edge geodetic parameters in the context of various types of special graphs such as Cocktail party graph, Crown graph, Dutch windmill graph, Friendship graph, Shadow graph, Tadpole graph, Windmill graph, Jump graph.
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A set S of vertices of a graph G is a total outer-connected dominating set if every vertex in V (G) is adjacent to some vertex in S and the subgraph induced by V \ S is connected. The total outer-connected domination number γtoc(...
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A set S of vertices of a graph G is a total outer-connected dominating set if every vertex in V (G) is adjacent to some vertex in S and the subgraph induced by V \ S is connected. The total outer-connected domination number γtoc(G) is the minimum size of such a set. We give some properties and bounds for γtoc in general graphs and in trees. For graphs of order n, diameter 2 and minimum degree at least 3, we show that γtoc(G) ≤ 2n-2/3 and we determine the extremal graphs.
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