The Total Open Monophonic Number of a Graph
Journal Title: JOURNAL OF ADVANCES IN MATHEMATICS - Year 2014, Vol 9, Issue 3
Abstract
For a connected graph G of order n >- 2, a subset S of vertices of G is a monophonic set of G if each vertex v in G lies on a x-y monophonic path for some elements x and y in S. The minimum cardinality of a monophonic set of G is defined as the monophonic number of G, denoted by m(G). A monophonic set of cardinality m(G) is called a m-set of G. A set S of vertices of a connected graph G is an open monophonic set of G if for each vertex v in G, either v is an extreme vertex of G and v ˆˆ? S, or v is an internal vertex of a x-y monophonic path for some x, y ˆˆ? S. An open monophonic set of minimum cardinality is a minimum open monophonic set and this cardinality is the open monophonic number, om(G). A connected open monophonic set of G is an open monophonic set S such that the subgraph < S > induced by S is connected. The minimum cardinality of a connected open monophonic set of G is the connected open monophonic number of G and is denoted by omc(G). A total open monophonic set of a graph G is an open monophonic set S such that the subgraph < S > induced by S contains no isolated vertices. The minimum cardinality of a total open monophonic set of G is the total open monophonic number of G and is denoted by omt(G). A total open monophonic set of cardinality omt(G) is called a omt-set of G. The total open monophonic numbers of certain standard graphs are determined. Graphs with total open monphonic number 2 are characterized. It is proved that if G is a connected graph such that omt(G) = 3 (or omc(G) = 3), then G = K3 or G contains exactly two extreme vertices. It is proved that for any integer n 3, there exists a connected graph G of order n such that om(G) = 2, omt(G) = omc(G) = 3. It is proved that for positive integers r, d and k 4 with 2r, there exists a connected graph of radius r, diameter d and total open monophonic number k. It is proved that for positive integers a, b, n with 4 <_ a<_ b <_n, there exists a connected graph G of order n such that omt(G) = a and omc(G) = b.
Authors and Affiliations
A. P. Santhakumaran, M. Mahendran
Keywords
Related Articles
- EP ID EP651363
- DOI 10.24297/jam.v9i3.2424
- Views 169
- Downloads 0
SOME RESULTS OF GENERALIZED LEFT (θ,θ)-DERIVATIONS ON SEMIPRIME RINGS
Let R be an associative ring with center Z(R) . In this paper , we study the commutativity of semiprime rings under certain conditions , it comes through introduce the definition of generalized left(θ,θ)- derivation as...
A note on solvability of finite groups
Let G be a finite group. A subgroup H of G is said to be c-normal in G if there exists a normal subgroup K of G such that G = HK and H K -<HG, where HG is the largest normal subgroup of G contained in H. In this...
A characterization of the existence of generalized stable sets
The generalized stable sets solution introduced by van Deemen (1991) as a generalization of the von Neumann and Morgenstern stable sets solution for abstract systems. If such a solution concept exists, then it is e...
Weakly injective dimension and Almost perfect rings
In this paper, we study the weak-injective dimension and we characterize the global weak-injective dimension of rings. After we study the transfer of the global weak-injective dimension in some known ring construction. F...
Some Remarks on Restricted Panel Data Model
In this paper , we investigate some remarks on panel data model with linear constraints on the coefficients of the random panel data model. Furthermore, it investigates the inferences . The restricted maximum likelihood...