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A. P. Santhakumaran,T. Jebaraj,S. V. Ullas Chandran 한국전산응용수학회 2012 Journal of applied mathematics & informatics Vol.30 No.5
For a connected graph G of order n, an ordered set S ={u1, u2, . . . , uk} of vertices in G is a linear edge geodetic set of G if for each edge e = xy in G, there exists an index i, 1 ≤ i < k such that e lies on a ui − ui+1 geodesic in G, and a linear edge geodetic set of minimum cardinality is the linear edge geodetic number leg(G) of G. A graph G is called a linear edge geodetic graph if it has a linear edge geodetic set. The linear edge geodetic numbers of certain standard graphs are obtained. Let gl(G) and eg(G) denote the linear geodetic number and the edge geodetic number, respectively of a graph G. For positive integers r, d and k ≥ 2with r < d ≤ 2r, there exists a connected linear edge geodetic graph G with rad G = r, diam G = d, and gl(G) = leg(G) = k. It is shown that for each pair a, b of integers with 3 ≤ a ≤ b, there is a connected linear edge geodetic graph G with eg(G) = a and leg(G) = b.
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Santhakumaran, A.P.,Jebaraj, T.,Ullas Chandran, S.V. The Korean Society for Computational and Applied M 2012 Journal of applied mathematics & informatics Vol.30 No.5
For a connected graph G of order $n$, an ordered set $S=\{u_1,u_2,{\cdots},u_k\}$ of vertices in G is a linear edge geodetic set of G if for each edge $e=xy$ in G, there exists an index $i$, $1{\leq}i$ < $k$ such that e lie on a $u_i-u_{i+1}$ geodesic in G, and a linear edge geodetic set of minimum cardinality is the linear edge geodetic number $leg(G)$ of G. A graph G is called a linear edge geodetic graph if it has a linear edge geodetic set. The linear edge geodetic numbers of certain standard graphs are obtained. Let $g_l(G)$ and $eg(G)$ denote the linear geodetic number and the edge geodetic number, respectively of a graph G. For positive integers $r$, $d$ and $k{\geq}2$ with $r$ < $d{\leq}2r$, there exists a connected linear edge geodetic graph with rad $G=r$, diam $G=d$, and $g_l(G)=leg(G)=k$. It is shown that for each pair $a$, $b$ of integers with $3{\leq}a{\leq}b$, there is a connected linear edge geodetic graph G with $eg(G)=a$ and $leg(G)=b$.
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THE CONNECTED DOUBLE GEODETIC NUMBER OF A GRAPH
SANTHAKUMARAN, A.P.,JEBARAJ, T. The Korean Society for Computational and Applied M 2021 Journal of applied mathematics & informatics Vol.39 No.1
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 set of cardinality dg(G) is called a dg-set of G. A connected double geodetic set of G is a double geodetic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected double geodetic set of G is the connected double geodetic number of G and is denoted by dgc(G). A connected double geodetic set of cardinality dgc(G) is called a dgc-set of G. Connected graphs of order n with connected double geodetic number 2 or n are characterized. For integers n, a and b with 2 ≤ a < b ≤ n, there exists a connected graph G of order n such that dg(G) = a and dgc(G) = b. It is shown that for positive integers r, d and k ≥ 5 with r < d ≤ 2r and k - d - 3 ≥ 0, there exists a connected graph G of radius r, diameter d and connected double geodetic number k.
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