LINEAR EDGE GEODETIC GRAPHS

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 $leg(G)=b$.

ON THE MONOPHONIC NUMBER OF A GRAPH

Santhakumaran, A.P.,Titus, P.,Ganesamoorthy, K. The Korean Society for Computational and Applied M 2014 Journal of applied mathematics & informatics Vol.32 No.1

For a connected graph G = (V,E) of order at least two, a set S of vertices of G is a monophonic set of G if each vertex v of G lies on an x - y monophonic path for some elements x and y in S. The minimum cardinality of a monophonic set of G is the monophonic number of G, denoted by m(G). Certain general properties satisfied by the monophonic sets are studied. Graphs G of order p with m(G) = 2 or p or p - 1 are characterized. For every pair a, b of positive integers with $2{\leq}a{\leq}b$, there is a connected graph G with m(G) = a and g(G) = b, where g(G) is the geodetic number of G. Also we study how the monophonic number of a graph is affected when pendant edges are added to the graph.

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.

On the edge -to -vertex detour number of a graph

A.P. Santhakumaran 장전수학회 2014 Advanced Studies in Contemporary Mathematics Vol.24 No.3

For two vertices u and v in a graph G=(V,E), The detour distance D(u,v) s the length of a longest, u-v path in G. A u-v path of length D(u,v) is called a u-v detour. For subsets A and B of V, the detour distance D(A,B) is defined as D(A,B)=(formula). A u-v path of length D(A,B) is called an A-B detour joining the sets A, B⊆V where u∈B. A vertex x is said to lie on an A-B detour is a vertex of A-B detour. A set S⊆E is called an detour joining a pair of edges of S. The edge-to-vertex detour number dn2(G) OF G is the minimum order of its edge-to-vertex detour sets and any edge-to-vertex detour set of order dn2(G) is an edge-to-vertex detour basis of G. Graphs G of size q for which dn2(G) = q or dn2(G) = q-1 or dn2(G) = q-2 are characterized.

Upper edge-to-vertex detour monophonic number of a graph

A. P. Santhakumaran,P. Titus,K. Ganesamoorthy 장전수학회 2016 Advanced Studies in Contemporary Mathematics Vol.26 No.2

For a connected graph G = (V,E) of order at least three, the monophonic distance dm(u, v) is the length of a longest u−v monophonic path in G. A u − v path of length dm(u, v) is called a u − v detour monophonic. For subsets A and B of V , the monophonic distance dm(A,B) is defined as dm(A,B) = min{dm(x, y) : x ∈ A, y ∈ B}. A u−v path of length dm(A, B) is called an A−B detour monophonic path joining the sets A,B ⊆ V, where u ∈ A and v ∈ B. A set S ⊆ E is called an edge-to-vertex detour monophonic set of G if every vertex of G is incident with an edge of S or lies on a detour monophonic path joining a pair of edges of S. The edge-to-vertex detour monophonic number dmev(G) of G is the minimum cardinality of its edge-to-vertex detour monophonic sets and any edge-to-vertex detour monophonic set of car- dinality dmev(G) is an edge-to-vertex detour monophonic basis of G. An edge-to-vertex detour monophonic set S in a connected graph G is called a minimal edge-to-vertex detour monophonic set of G if no proper subset of S is an edge-to-vertex detour monophonic set of G. The upper edge-to-vertex detour monophonic number dm+ ev(G) of G is the maxi- mum cardinality of a minimal edge-to-vertex detour monophonic set of G. We determine bounds for it and certain general properties of these concepts are studied. It is shown that for every pair a, b of integers with 2 ≤ a ≤ b, there exists a connected graph G with dmev(G) = a and dm+ ev(G) = b.

LINEAR EDGE GEODETIC GRAPHS

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.

Edge-to-vertex detour number of a graph

A. P. Santhakumaran,S. Athisayanathan 장전수학회 2011 Advanced Studies in Contemporary Mathematics Vol.21 No.4

For two vertices u and v in a graph G = (V,E), the detour distance D(u, v)is the length of a longest u–v path in G. A u–v path of length D(u, v) is called a u–v detour. For subsets A and B of V , the detour distance D(A,B)is defined as D(A,B) = min{D(x, y) : x ∈ A, y ∈ B}. A u–v path of length D(A,B) is called an A–B detour joining the sets A, B ⊆ V where u ∈ A and v ∈ B. A vertex x is said to lie on an A–B detour if x is a vertex of an A–B detour. A set S ⊆ E is called an edge-to-vertex detour set if every vertex of G is incident with an edge of S or lies on a detour joining a pair of edges of S. The edge-to-vertex detour number dn_2(G) of G is the minimum order of its edge-to-vertex detour sets and any edge-to-vertex detour set of order dn_2(G) is an edge-to-vertex detour basis of G. Certain general properties of these concepts are studied. The edge-to-vertex detour numbers of certain classes of graphs are determined. Its relationship with the detour diameter is discussed and it is proved that for each triple D, k, q of integers with 2 ≤ k ≤ q − D + 2 and D ≥ 4 there is a connected graph G of order p with detour diameter D and dn_2(G) = k. It is also proved that for any three positive integers a, b, k with k ≥ 2 and a < b ≤ 2a, there is a connected graph with detour radius a, detour diameter b and dn_2(G) = k.

ON THE MONOPHONIC NUMBER OF A GRAPH

A. P. Santhakumaran,P. Titus,K. Ganesamoorthy 한국전산응용수학회 2014 Journal of applied mathematics & informatics Vol.32 No.1

For a connected graph G = (V,E) of order at least two, a set S of vertices of G is a monophonic set of G if each vertex v of G lies on an χ−y monophonic path for some elements χ and y in S. The minimum cardinality of a monophonic set of G is the monophonic number of G, denoted by m(G). Certain general properties satisfied by the monophonic sets are studied. Graphs G of order p with m(G) = 2 or p or p −1 are characterized. For every pair a, b of positive integers with 2≤a≤b, there is a connected graph G with m(G) = a and g(G) = b, where g(G) is the geodetic number of G. Also we study how the monophonic number of a graph is affected when pendant edges are added to the graph.

The forcing open geodetic number of a graph

A. P. Santhakumaran,T. K. Latha 장전수학회 2012 Proceedings of the Jangjeon mathematical society Vol.15 No.2

For a connected graph G of order n ≥ 2, a set S of vertices of 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 ∈ S; 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). An open geodetic set of cardinality og(G) is called a og-set of G. A subset T of an og-set S is called a forcing subset for S if S is the unique minimum open geodetic set containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S and this cardinality is the forcing open geodetic number, fo(G). We determine bounds for fo(G) and determine the same for some standard graphs. Also, it is shown that for positive integers a, b with 0 ≤ a ≤ b−4and b ≥ 5, there exists a connected graph G such that fo(G) = a and og(G) = b.

Connected edge detour monophonic number of a graph

P. Titus,P. Balakrishnan,A. P. Santhakumaran,K. Ganesamoorthy 장전수학회 2013 Proceedings of the Jangjeon mathematical society Vol.16 No.4

Connected edge detour monophonic number of a graph