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

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

Double Geodetic Number of a line graph

Any line graph L(G), whose vertices correspond to the edges of G(V, E) and two vertices in L(G) are adjacent if and only if the corresponding edges in G are adjacent. If there are vertices u, v in S such that x, y ∈ I[u, v] for any pair of vertices x, y in G, then the set S of vertices of G is said to be a double geodetic set of G. The lowest cardinality of a double geodetic set is represented by the double geodetic number dg(G). In this study, we determine double geodetic number of several line graphs’ double geodetic numbers

ON DOUBLE SIGNAL NUMBER OF A GRAPH

A set $S$ of vertices in a connected graph $G=(V,E)$ is called a signal set if every vertex not in $S$ lies on a signal path between two vertices from $S$. A set $S$ is called a double signal set of $G$ if $S$ if for each pair of vertices $x,y \in G$ there exist $u,v \in S$ such that $x,y \in L[u,v]$. The double signal number $\mathrm{dsn}\,(G)$ of $G$ is the minimum cardinality of a double signal set. Any double signal set of cardinality $\mathrm{dsn}\,(G)$ is called $\mathrm{dsn}$-set of $G$. In this paper we introduce and initiate some properties on double signal number of a graph. We have also given relation between geodetic number, signal number and double signal number for some classes of graphs

Double geodetic number of a graph

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 of cardinality dg(G) is called dg-set of G. The double geodetic numbers of certain standard graphs are obtained. It is shown that for positive integers r,d such that r < d ≤ 2r and 3 ≤ a ≤ b there exists a connected graph G with rad G = r, diam G = d, g(G) = a and dg(G) = b. Also, it is proved that for integers n, d ≥ 2 and l such that 3 ≤ k ≤ l ≤ n and n-d-l+1 ≥ 0, there exists a graph G of order n diameter d, g(G) = k and dg(G) = l

The upper double geodetic number of a graph

For vertices x and y in a connected graph G of order n, the distance d(x,y) is the length of a shortest x -y path. An x-y path of length d( x,y) is called an x -y geodesic. The closed interval I[x,y] consists of all vertices lying on some x-y geodesi c of G, while for S?V,I[S] = ? ? ? x y S I[x,y]. A set S of vertices in G is called a double geodetic set of G if for each pair of vertices x,y 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 geodetic set of cardinality dg(G) is called dg -set of G. A double geodetic set in a connected graph G is called a minimal double geodetic set if no proper subset of S is a double geodetic set of G. The upper double geodetic number dg+(G) of G is the maximum cardinality of a minimal double geodetic set of G. The upper double geodetic numbers of certain standard graphs are obtained. It is proved that for a connect ed graph G of order n, dg(G) = n if and only i f dg+(G) = n. It is also proved that dg(G) = n ?1 if and only if dg+(G) = n ?1 for a non-complete graph G of order n having a vertex of degree n ?1. For every two positive integer s a and b, where 2 = a = b, there exists a connected graph G with dg(G) = a and dg+(G) = b