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

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 Detour Monophonic Convexity Number of a Graph

A set  is detour monophonic convexif  The detour monophonic convexity number is denoted by  is the cardinality of a maximum proper detour monophonic convex subset of  Some general properties satisfied by this concept are studied. The detour monophonic convexity number of certain classes of graphs are determined. It is shown that for every pair of integers   and  with  there exists a connected graph  such that   and , where  is the monophonic convexity number of

The Outer Connected Detour Monophonic Number of a Graph

For a connected graph ???? = (????, ????) of order  a set is called a monophonic set of ????if every vertex of ????is contained in a monophonic path joining some pair of vertices in ????. The monophonic number (????) of is the minimum cardinality of its monophonic sets. If  or the subgraph  is connected, then a detour monophonic set  of a connected graph is said to be an outer connected detour monophonic setof .The outer connecteddetourmonophonic number of , indicated by the symbol , is the minimum cardinality of an outer connected detour monophonic set of . The outer connected detour monophonic number of some standard graphs are determined. It is shown that for positive integers , and ???? ≥ 2 with ,there exists a connected graph ????with???????????????????? = , ????????????m???????? = and  = ????. Also, it is shown that for every pair of integers ????and b with 2 ≤ ???? ≤ ????, there exists a connected graph with and

Upper Vertex Triangle Free Detour Number of a Graph

For a graph G, the x-triangle free detour set, the x-triangle free detour number, the minimal x-triangle free detour set, the upper x-triangle free detour number, are defined and studied. Certain bounds are determined and the relation with the vertex triangle free detour number of a graph is found out. Some realization problems, properties related to the upper vertex detour number, the upper vertex detour monophonic number and the upper vertex geodetic number are also studied

Upper Vertex Triangle Free Detour Number of a Graph

For a graph G, the x-triangle free detour set, the x-triangle free detour number, the minimal x-triangle free detour set, the upper x-triangle free detour number, are defined and studied. Certain bounds are determined and the relation with the vertex triangle free detour number of a graph is found out. Some realization problems, properties related to the upper vertex detour number, the upper vertex detour monophonic number and the upper vertex geodetic number are also studied

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

Forcing vertex square free detour number of a graph

Let G be a connected graph and S a square free detour basis of G. A subset T\subseteq S is called a forcing subset for S if S is the unique square free detour basis of S containing T. A forcing subset for S of minimum order is a minimum forcing subset of G. The forcing square free detour number of G is fdn◻fu(G)=minfdn◻fuSu, where the minimum is taken over all square free detour bases S in G. In this paper, we introduce the forcing vertex square free detour sets. The general properties satisfied by these forcing subsets are discussed and the forcing square free detour number for a certain class of standard graphs are determined. We show that the two parameters dn◻fu(G) and fdn◻fu(G) satisfy the relationship 0\le fdn◻fu(G)≤dn◻fu(G). Also, we prove the existence of a graph G with fdn◻fu(G)=α and dn◻fu(G)=β, where 0\le\alpha\le\beta and \beta\geq2 for some vertex u in G

On The Study of Edge Monophonic Vertex Covering Number

For a connected graph G of order n ≥ 2, a set S of vertices of G is an edge monophonic vertex cover of G if S is both an edge monophonic set and a vertex covering set of G. The minimum cardinality of an edge monophonic vertex cover of G is called the edge monophonic vertex covering number of G and is denoted by . Any edge monophonic vertex cover of cardinality  is a -set of G. Some general properties satisfied by edge monophonic vertex cover are studied

Vertex Triangle Free Detour Number of a Graph

The \emph{$x$-triangle free detour number} $dn_{\triangle f_x}(G)$ of a connected graph $G$ is the minimum order of its $x$-triangle free detour sets and any $x$-triangle free detour set $S_{x} \subseteq V$ of order&nbsp; $dn_{\triangle f_x}(G)$ is a \emph{$x$-triangle free detour basis} of $G$. A connected graph of order $n$ with vertex triangle free detour number $n-1$ or $n-2$ for every vertex is characterized. Certain general properties satisfied by the vertex triangle free detour sets are studied