The Total Open Monophonic Number of a Graph

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

The connected detour monophonic number of a graph

For a connected graph G = (V, E) of order at least two, a chord of a path P is an edge joining two non-adjacent vertices of P. A path P is called a monophonic path if it is a chordless path. A longest x − y monophonic path is called an x − y detour monophonic path. A set S of vertices of G is a detour monophonic set of G if each vertex v of G lies on an x − y detour monophonic path, for some x and y in S. The minimum cardinality of a detour monophonic set of G is the detour monophonic number of G and is denoted by dm(G). A connected detour monophonic set of G is a detour monophonic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected detour monophonic set of G is the connected detour monophonic number of G and is denoted by dmc(G). We determine bounds for dmc(G) and characterize graphs which realize these bounds. It is shown that for positive integers r, d and k ≥ 6 with r < d, there exists a connected graph G with monophonic radius r, monophonic diameter d and dmc(G) = k. For each triple a, b, p of integers with 3 ≤ a ≤ b ≤ p − 2, there is a connected graph G of order p, dm(G) = a and dmc(G) = b. Also, for every pair a, b of positive integers with 3 ≤ a ≤ b, there is a connected graph G with mc(G) = a and dmc(G) = b, where mc(G) is the connected monophonic number of G.The first author is partially supported by DST Project No. SR/S4/MS:570/09.Publisher's Versio

A stochastic model for sero conversion times of HIV transmission

This paper focuses on the study of a Stochastic Model for predicting the seroconversion time of HIV transmission. As the immune capacities of an individual vary and also have its own resistance, the antigenic diversity threshold is different for different person. We propose a stochastic model to study the damage process acting on the immune system that is non- linear. The mean of seroconversion time of HIV and its variance are derived. A numerical example is given to illustrate the seroconversion times of HIV transmission

The restrained monophonic number of a graph

A set S of vertices of a connected graph G is a monophonic set of G if each vertex v of G lies on a x−y monophonic path for some x and y in S. The minimum cardinality of a monophonic set of G is the monophonic number of G and is denoted by m(G). A restrained monophonic set S of a graph G is a monophonic set such that either S = V or the subgraph induced by V − S has no isolated vertices. The minimum cardinality of a restrained monophonic set of G is the restrained monophonic number of G and is denoted by mr(G). We determine bounds for it and determine the same for some special classes of graphs. Further, several interesting results and realization theorems are proved.Publisher's Versio

Minimal restrained monophonic sets in graphs

For a connected graph G = (V, E) of order at least two, a restrained monophonic set S of a graph G is a monophonic set such that either S = V or the subgraph induced by V −S has no isolated vertices. The minimum cardinality of a restrained monophonic set of G is the restrained monophonic number of G and is denoted by mr(G). A restrained monophonic set S of G is called a minimal restrained monophonic set if no proper subset of S is a restrained monophonic set of G. The upper restrained monophonic number of G, denoted by m+r (G), is defined as the maximum cardinality of a minimal restrained monophonic set of G. We determine bounds for it and find the upper restrained monophonic number of certain classes of graphs. It is shown that for any two positive integers a, b with 2 ≤ a ≤ b, there is a connected graph G with mr(G) = a and m+r (G) = b. Also, for any three positive integers a, b and n with 2 ≤ a ≤ n ≤ b, there is a connected graph G with mr(G) = a, m+r (G) = b and a minimal restrained monophonic set of cardinality n. If p, d and k are positive integers such that 2 ≤ d ≤ p − 2, k ≥ 3, k 6= p − 1 and p − d − k ≥ 0, then there exists a connected graph G of order p, monophonic diameter d and m+r (G) = k.The third author’s research work has been supported by NBHM, India.Publisher's Versio

An Algorithm to Reconstruct the Missing Values for Diagnosing the Breast Cancer

The treatment of incomplete data is an important step in pre-processing data prior to later analysis. The main objective of this paper is to show how various methods can be used in such a way that they are able to process dataset with missing values. Computer–aided classification of Breast cancer using Back propagation neural network is discussed in this paper. The classification results have indicated that the network gave the good diagnostic performance of 99.06%

Restrained Double Monophonic Number of a Graph

For a connected graph G of order at least two, a double monophonic set S of a graph G is a restrained double monophonic set if either S=V or the subgraph induced by V−S has no isolated vertices. The minimum cardinality of a restrained double monophonic set of G is the restrained double monophonic number of G and is denoted by dmr(G). The restrained double monophonic number of certain classes graphs are determined. It is shown that for any integers a,b,c with 3≤a≤b≤c, there is a connected graph G with m(G)=a, mr(G)=b and dmr(G)=c, where m(G) is the monophonic number and mr(G) is the restrained monophonic number of a graph G.The second author research work was supported by National Board for Higher Mathematics, INDIA (Project No. NBHM/R.P.29/2015/Fresh/157).The authors are thankful to the reviewers for their useful comments for the improvement of this paper