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

Geodetic domination integrity in graphs

Reciprocal version of product degree distance of cactus graphs Let G be a simple graph. A subset S ⊆ V (G) is a said to be a geodetic set if every vertex u /∈ S lies on a shortest path between two vertices from S. The minimum cardinality of such a set S is the geodetic number g(G) of G. A subset D ⊆ V (G) is a dominating set of G if every vertex u /∈ D has at least one neighbor in D. The domination number γ(G) is the minimum cardinality of a dominating set of G. A subset is said to be a geodetic dominating set of G if it is both a geodetic and a dominating set. The geodetic domination number γg(G) is the minimum cardinality among all geodetic dominating sets in G. The geodetic domination integrity of a graph G is defined by DIg(G) = min{|S| + m(G − S) : S is a geodetic dominating set of G}, where m(G − S) denotes the order of the largest component in G−S. In this paper, we study the concepts of geodetic dominating integrity of some families of graphs and derive some bounds for the geodetic domination integrity. Also we obtain geodetic domination integrity of some cartesian product of graphs.Publisher's Versio

Monophonic Distance in Graphs

For any two vertices u and v in a connected graph G, a u − v path is a monophonic path if it contains no chords, and the monophonic distance dm(u, v) is the length of a longest u − v monophonic path in G. For any vertex v in G, the monophonic eccentricity of v is em(v) = max {dm(u, v) : u ∈ V}. The subgraph induced by the vertices of G having minimum monophonic eccentricity is the monophonic center of G, and it is proved that every graph is the monophonic center of some graph. Also it is proved that the monophonic center of every connected graph G lies in some block of G. With regard to convexity, this monophonic distance is the basis of some detour monophonic parameters such as detour monophonic number, upper detour monophonic number, forcing detour monophonic number, etc. The concept of detour monophonic sets and detour monophonic numbers by fixing a vertex of a graph would be introduced and discussed. Various interesting results based on these parameters are also discussed in this chapter

International Journal of Mathematical Combinatorics, Vol.2A

The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 460 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences

Geodetic Connected Domination Number of a graph

A pair x, y of vertices in a nontrivial connected graph is said to geodominate a vertex v of G if either v  {x,y} or v lies an x - y geodesic of G. A set S of vertices of G is a geodetic set if every vertex of G is geodominated by some pair of vertices of S. A subset S of vertices in a graph G is called a geodetic connected dominating set if S is both a geodetic set and a connected dominating set. We study geodetic connected domination on graphs

Algorithms to Find Linear Geodetic Numbers and Linear Edge Geodetic Numbers in Graphs

Given two vertices u and v of a connected graph G=(V, E), the closed interval I[u, v] is that set of all vertices lying in some u-v geodesic in G. A subset of V(G) S={v1,v2,v3,….,vk} is a linear geodetic set or sequential geodetic set if each vertex x of G lies on a vi – vi+1 geodesic where 1 ? i < k . A linear geodetic set of minimum cardinality in G is called as linear geodetic number lgn(G) or sequential geodetic number sgn(G). Similarly, an ordered set S={v1,v2,v3,….,vk} is a linear edge geodetic set if for each edge e = xy in G, there exists an index i, 1 ? i < k such that e lies on a vi – vi+1 geodesic in G. The cardinality of the minimum linear edge geodetic set is the linear edge geodetic number of G denoted by legn(G). The purpose of this paper is to introduce algorithms using dynamic programming concept to find minimum linear geodetic set and thereby linear geodetic number and linear edge geodetic set and number in connected graphs

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

The restrained geodetic number of a graph

Abstract A geodetic set S ⊆ V (G) of a graph G = (V, E) is a restrained geodetic set if the subgraph G[V \ S] has no isolated vertex. The minimum cardinality of a restrained geodetic set is the restrained geodetic number. In this paper, we initiate the study of the restrained geodetic number

Alliance free sets in Cartesian product graphs

Let $G=(V,E)$ be a graph. For a non-empty subset of vertices $S\subseteq V$, and vertex $v\in V$, let $\delta_S(v)=|\{u\in S:uv\in E\}|$ denote the cardinality of the set of neighbors of $v$ in $S$, and let $\bar{S}=V-S$. Consider the following condition: {equation}\label{alliancecondition} \delta_S(v)\ge \delta_{\bar{S}}(v)+k, \{equation} which states that a vertex $v$ has at least $k$ more neighbors in $S$ than it has in $\bar{S}$. A set $S\subseteq V$ that satisfies Condition (\ref{alliancecondition}) for every vertex $v \in S$ is called a \emph{defensive} $k$-\emph{alliance}; for every vertex $v$ in the neighborhood of $S$ is called an \emph{offensive} $k$-\emph{alliance}. A subset of vertices $S\subseteq V$, is a \emph{powerful} $k$-\emph{alliance} if it is both a defensive $k$-alliance and an offensive $(k +2)$-alliance. Moreover, a subset $X\subset V$ is a defensive (an offensive or a powerful) $k$-alliance free set if $X$ does not contain any defensive (offensive or powerful, respectively) $k$-alliance. In this article we study the relationships between defensive (offensive, powerful) $k$-alliance free sets in Cartesian product graphs and defensive (offensive, powerful) $k$-alliance free sets in the factor graphs