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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, 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 a connected graph G of order p >= 2, a set S subset of V(G) is a geodetic set of G if each vertex v is an element of V(G) lies on an x-y geodesic for some elements x and y in S. The minimum cardinality of a geodetic set of G i...
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For a connected graph G of order p >= 2, a set S subset of V(G) is a geodetic set of G if each vertex v is an element of V(G) lies on an x-y geodesic for some elements x and y in S. The minimum cardinality of a geodetic set of G is defined as the geodetic number of G, denoted by g(G). A geodetic set of cardinality g(G) is called a g-set of G. A connected geodetic set of G is a geodetic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected geodetic set of G is the connected geodetic number of G and is denoted by g(c)(G). A connected geodetic set of cardinality gc(G) is called a g(c)-set of G. A connected geodetic set S in a connected graph G is called a minimal connected geodetic set if no proper subset of S is a connected geodetic set of G. The upper connected geodetic number g(c)(+)(G) is the maximum cardinality of a minimal connected geodetic set of G. We determine bounds for gc+(G) and determine the same for some special classes of graphs. For positive integers r,d and n >= d+1 with r 收起
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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
摘要 :
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 is an element of S. A set of vert...
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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 is an element of S. A set of vertices S subset of V(G) is a geodetic set if I [S] = V(G), and the minimum cardinality of a geodetic set is the geodetic number g(G). A geodetic set S subset of V(G) is a total geodetic set if the subgraph G[S] induced by S has no isolated vertices. The minimum cardinality of a total geodetic set is the total geodetic number gt(G). A subset T of a minimum total geodetic set S is called a forcing subset for S if S is the unique minimum total geodetic set containing T. The forcing total geodetic number of S is the minimum cardinality among the forcing subsets of S, and the forcing total geodetic number (G, gt) of G is the minimum forcing total geodetic number among all minimum total geodetic sets of G. We determine sharp bounds for gt(G) and f (G, gt). The minimum cardinality of a geodetic set which induces a connected graph is the connected geodetic number gc(G). We present necessary and sufficient conditions which determine for every triple of integers a, b, c whether there exists a nontrivial connected graph G with: (i) a = g(G), b = g(t)(G), c = vertical bar V (G)vertical bar; (ii) a = rad G, b = diam G, c = g(t)(G); (iii) a = g(t)(G), b = g(e)(G), c = vertical bar V(G)vertical bar. We find all ordered pairs (a, b) of integers which are realizable as the forcing total geodetic number and total geodetic number for some nontrivial connected graph, respectively.
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For a connected graph G of order p ≥ 2, a set S is contained in V(G) is a geodetic set of G if each vertex v ∈ V(G) lies on an x-y geodesic for some elements x and y in S. The minimum cardinality of a geodetic set of G is defin...
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For a connected graph G of order p ≥ 2, a set S is contained in V(G) is a geodetic set of G if each vertex v ∈ V(G) lies on an x-y geodesic for some elements x and y in S. The minimum cardinality of a geodetic set of G is defined as the geodetic number of G, denoted by g(G). A geodetic set of cardinality g(G) is called a g-set of G. A connected geodetic set of G is a geodetic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected geodetic set of G is the connected geodetic number of G and is denoted by g_c(G). A connected geodetic set of cardinality g_c(G) is called a g_c-set of G. Connected graphs of order p with connected geodetic number 2 or p are characterized. It is shown that for positive integers r, d and n ≥ d + 1 with r ≤ d ≤ 2r, there exists a connected graph G of radius r, diameter d and g_c(G) = n. Also, for integers p, d and n with 2 ≤ d ≤ p - 1, d+1 ≤ n ≤ p, there exists a connected graph G of order p, diameter d and g_c(G) = n.
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For a connected graph G of order n ≥ 2, a set S of vertices of G is a geodetic set of G if each vertex v of G lies on a x-y geodesic for some elements x and y in 5. The geodetic number g(G) of G is the minimum cardinality of a ge...
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For a connected graph G of order n ≥ 2, a set S of vertices of G is a geodetic set of G if each vertex v of G lies on a x-y geodesic for some elements x and y in 5. The geodetic number g(G) of G is the minimum cardinality of a geodetic set of G. A geodetic set of cardinality g(G) is called a g-set of G. A set S of vertices of a connected graph G is an open geodetic set of G if for each vertex v in G, either v is an extreme vertex of G and v € 5; or v is an internal vertex of an x-y geodesic for some x, y ∈ S. An open geodetic set of minimum cardinality is a minimum open geodetic set and this cardinality is the open geodetic number, og(G). A connected open geodetic set of G is an open geodetic set 5 such that the subgraph < S > induced by S is connected. The minimum cardinality of a connected open geodetic set of G is the connected open geodetic number of G and is denoted by og_c(G). A total open geodetic set of a graph G is an open geodetic set S such that the subgraph < S > induced by S contains no isolated vertices. The minimum cardinality of a total open geodetic set of G is the total open geodetic number of G and is denoted by ogt(G). A total open geodetic set of cardinality og_t{G) is called og_t-set of G. Certain general properties satisfied by total open geodetic sets are discussed. Graphs with total open geodetic number 2 are characterized. The total open geodetic numbers of certain standard graphs are determined. It is proved that for positive integers r, d and k ≥ 4 with r ≤ d ≤ 2r, there exists a connected graph of radius r, diameter d and total open geodetic number k. It is also proved that for the positive integers a,b,n with 4 < a < b < n, there exists a connected graph G of order n such that og_t(G) = a and og_c(G) = b.
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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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